{
  "schema": 1,
  "manifest_sha256": "4f3e205bc468457a1d1e6337c4efaf3416331a6f50d0e6b93aca77cc53ae4a6e",
  "status": "PASS",
  "corpora": {
    "alphaproof-nexus": {
      "repo": "google-deepmind/alphaproof-nexus-results",
      "commit": "0647711a71183c1ea492ad60860776617ce1ea88",
      "overlay_sha256": "78300f8b46fe344f30cf5ee1e3c501bde9ca980fe6dbefa0387b82e171b463ae",
      "annotations": 721,
      "generated_source_matched": 373,
      "generated": 373,
      "curated_source_linked_unreviewed": 348,
      "shared_reference_links": 3
    },
    "carleson": {
      "repo": "fpvandoorn/carleson",
      "commit": "8972acdfd953c63bb2501ace20d61ced7c0697ec",
      "overlay_sha256": "3fbae432a77547b976b9d7ff2d103711e7abdf1bf8bc4e7e79eea24eed6daf65",
      "annotations": 991,
      "generated_source_matched": 981,
      "generated": 981,
      "curated_source_linked_unreviewed": 10,
      "shared_reference_links": 2
    },
    "compfiles": {
      "repo": "dwrensha/compfiles",
      "commit": "47b19bbfe5f38dc49379a6b53008d2909c4d71c5",
      "overlay_sha256": "ff0f1531fc8ae88346ac7c8097ad9dbc5c70c245bb634e3730153f64b6bc01e1",
      "annotations": 3313,
      "generated_source_matched": 3305,
      "generated": 3305,
      "curated_source_linked_unreviewed": 8,
      "shared_reference_links": 2
    },
    "flt": {
      "repo": "ImperialCollegeLondon/FLT",
      "commit": "f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff",
      "overlay_sha256": "f97fb372d0d5774602b844d6683127051539cab2b5967f2fbedba299b323e828",
      "annotations": 1212,
      "generated_source_matched": 1204,
      "generated": 1204,
      "curated_source_linked_unreviewed": 8,
      "shared_reference_links": 2
    },
    "formal-conjectures": {
      "repo": "google-deepmind/formal-conjectures",
      "commit": "1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89",
      "overlay_sha256": "93691f95d92b6ae9f8b94442dd49fcab7650bd6479291906daf8a9f1cd737bc9",
      "annotations": 7200,
      "generated_source_matched": 7190,
      "generated": 7190,
      "curated_source_linked_unreviewed": 10,
      "shared_reference_links": 2
    },
    "mil": {
      "repo": "leanprover-community/mathematics_in_lean",
      "commit": "dd6d752fedb14082f557913c2dccb2d4851e5173",
      "overlay_sha256": "62acbee84173d187e55569637477c285d58d9b801c00ccf3863767de8954fca2",
      "annotations": 16,
      "generated_source_matched": 7,
      "generated": 7,
      "curated_source_linked_unreviewed": 9,
      "shared_reference_links": 2
    },
    "oai": {
      "repo": "openai/math",
      "commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
      "overlay_sha256": "267c867ced4e465c243af36988649c08d23024da3060b32cd49a2da267d3f02a",
      "annotations": 1393,
      "generated_source_matched": 1273,
      "generated": 1273,
      "curated_source_linked_unreviewed": 120,
      "shared_reference_links": 2
    },
    "pfr": {
      "repo": "teorth/pfr",
      "commit": "ce7ac95e620afe0ae448df63477de0b29e4dcf4d",
      "overlay_sha256": "6665572f89a2e4a9c24325b8cd337a9fd1580eedf2fcf14b6f5a3f476daba1bf",
      "annotations": 494,
      "generated_source_matched": 485,
      "generated": 485,
      "curated_source_linked_unreviewed": 9,
      "shared_reference_links": 2
    },
    "tpil": {
      "repo": "leanprover/theorem_proving_in_lean4",
      "commit": "4e28129fdd58037f8f5857548d5e99fe4fb0cc57",
      "overlay_sha256": "7f65bec764e79335ed95622f398d7fb5ff50fb1cec824269f87303b85706fb86",
      "annotations": 34,
      "generated_source_matched": 26,
      "generated": 26,
      "curated_source_linked_unreviewed": 8,
      "shared_reference_links": 2
    }
  },
  "curated_entries": [
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "Conjecture2",
      "line": 1069,
      "title": "Reconstructing a typed bipartite graph from its incidence-deletion deck",
      "intuition": "Let `G` and `H` use the same finite bipartition. If `G` remains connected after deleting any vertex, its vertex types are pairwise distinct, and the two incidence-deletion decks agree, then a bijection preserving the two parts also preserves every edge. Thus the deck determines `G` up to bipartite isomorphism under these strong distinguishability hypotheses.",
      "source_statement": "theorem Conjecture2\n    -- G, H are finite simple bipartite graphs on at least 3 vertices.\n  (V : Type*) [Fintype V] [DecidableEq V]\n  (G H : SimpleGraph V) (leftU rightV : Finset V)\n  (hGBipartite : G.IsBipartiteWith leftU rightV)\n  (hHBipartite : H.IsBipartiteWith leftU rightV)\n  (hV : Fintype.card V ≥ 3)\n  (hV_parts : (univ : Finset V) = leftU ∪ rightV)\n    -- G is 2-connected means G is connected, and the subgraph after removal of any vertex is still connected.\n  (G_2_connected : G.Connected ∧ ∀ (v : V), (G.induce {x | x ≠ v}).Connected)\n    -- G is not a regular graph\n  (G_not_regular : ∀ d : ℕ, ¬ G.IsRegularOfDegree d)\n…",
      "source_statement_sha256": "4bdac91325e6e0f5c966f1f81413f116728d845b02f32987493f899adf4058cd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L1069",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "BipartiteIso",
      "line": 52,
      "title": "A graph isomorphism that preserves both sides of the bipartition",
      "intuition": "`BipartiteIso G H leftU rightV` strengthens an ordinary graph isomorphism by requiring its vertex equivalence to preserve membership in `leftU` and `rightV` separately. Swapping the two sides is not allowed.",
      "source_statement": "structure BipartiteIso (G H : SimpleGraph V) (leftU rightV : Finset V) extends G ≃g H where\n…",
      "source_statement_sha256": "59e1853875c33d8345cd4cb5115abc6826793e5dfecb8597ca12360353401261",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L52",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "bipartiteDeck",
      "line": 69,
      "title": "The multiset of incidence-deleted cards up to bipartite isomorphism",
      "intuition": "`bipartiteDeck G leftU rightV` records one card for every vertex `v`: all edges incident to `v` are deleted, `v` remains as an isolated vertex, and the result is stored as its entire bipartite-isomorphism class. Repeated isomorphism classes retain multiplicity.",
      "source_statement": "def bipartiteDeck : Multiset (Set (SimpleGraph V)) :=\n…",
      "source_statement_sha256": "d5e032928394f0e014b48f256a82e95d9f1fbbd672be0b44c6849f82da758bf7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L69",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "degreeProfile",
      "line": 79,
      "title": "The multiset of degrees seen around a vertex",
      "intuition": "`degreeProfile G x` lists `G.degree y` once for every neighbor `y` of `x`. It remembers the degrees surrounding `x` but deliberately forgets which neighboring vertex supplied each degree.",
      "source_statement": "def degreeProfile (x : V) : Multiset ℕ :=\n…",
      "source_statement_sha256": "55c956464bd32c5d30d81ef806da2e5321308249fb2d2b3579e7f1bbdee3362a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L79",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "τ",
      "line": 85,
      "title": "A vertex type combines its degree with its neighbors' degrees",
      "intuition": "`τ G x` is the ordered pair consisting of the degree of `x` and its degree profile. Two vertices have the same type exactly when they have the same degree and the same multiset of neighboring degrees.",
      "source_statement": "def τ (x : V) : ℕ × Multiset ℕ :=\n…",
      "source_statement_sha256": "c9336860b2c06c9e072dbeea1aad081b9552d7e474eeedd6d913bc734db2f741",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L85",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "typeDrop",
      "line": 494,
      "title": "How a neighbor's type changes when one incident edge is removed",
      "intuition": "`typeDrop t d` lowers the type's degree coordinate by one using natural-number subtraction and erases one occurrence of `d` from its degree-profile multiset. It models deleting the edge to a vertex of degree `d`.",
      "source_statement": "def typeDrop (t : ℕ × Multiset ℕ) (d : ℕ) : ℕ × Multiset ℕ :=\n…",
      "source_statement_sha256": "4f5639eab9bf436bd5a6bab00ad536d6fa5b3e37753acdaf6f6db46d895a9e8e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L494",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "multiset_map_eq_implies_bij",
      "line": 110,
      "title": "Equal unlabeled decks can be matched card by card",
      "intuition": "If two functions on the same finite type produce equal multisets of values, this lemma constructs a permutation pairing every input of the first function with an input of the second having the same value. Applied to decks, it recovers a bijection between the deleted vertices of matched cards.",
      "source_statement": "lemma multiset_map_eq_implies_bij {α β : Type*} [Fintype α] (f g : α → β)\n    (h : Multiset.map f (Finset.univ : Finset α).val = Multiset.map g (Finset.univ : Finset α).val) :\n    ∃ (e : α ≃ α), ∀ a, f a = g (e a) := by\n…",
      "source_statement_sha256": "3329daba49067d55d1d02b75bf20c5842a5833a8575f8094f2d260f61285e656",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "edgeCount_deleteIncidenceSet",
      "line": 144,
      "title": "An incidence-deleted card loses exactly the degree of its isolated vertex",
      "intuition": "Deleting every edge incident to `v` changes the edge count from `e(G)` to `e(G) − degree(v)`. Each incident undirected edge is removed once, even though degree is defined through the neighbor set.",
      "source_statement": "lemma edgeCount_deleteIncidenceSet (v : V) [DecidableRel G.Adj] :\n    edgeCount (G.deleteIncidenceSet v) = edgeCount G - G.degree v := by\n…",
      "source_statement_sha256": "20642b2a3def67be7564066068ffc2b4f729012780911ce9eff1c092dff9ea9d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L144",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "bipartiteDeck_determines_edgeCount",
      "line": 203,
      "title": "Recover the original number of edges from the deck",
      "intuition": "Equal bipartite decks of graphs on at least three vertices force equal total edge counts. Summing the edge counts of all cards counts each original edge once for every deleted vertex not incident to it, hence exactly `n−2` times.",
      "source_statement": "lemma bipartiteDeck_determines_edgeCount {G H : SimpleGraph V} (hV : Fintype.card V ≥ 3)\n    (h : bipartiteDeck G leftU rightV = bipartiteDeck H leftU rightV) :\n    edgeCount G = edgeCount H := by\n…",
      "source_statement_sha256": "0300b629d05f103934c3f5aa25369281cc0714179b9b56951e9d59f49ec7c72d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L203",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "degreeMultiset",
      "line": 219,
      "title": "The unlabeled multiset of all vertex degrees",
      "intuition": "`degreeMultiset G` lists the degree of every vertex of `G` while forgetting which vertex has which degree. This is the intermediate invariant reconstructed from the edge counts of the deck's cards.",
      "source_statement": "def degreeMultiset (G : SimpleGraph V) : Multiset ℕ :=\n…",
      "source_statement_sha256": "aedaa3a7d549645e91155f7ce40709e167d92dc40b07059b4b3cb2cdb99d3b34",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L219",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "bipartiteDeck_determines_degreeMultiset",
      "line": 250,
      "title": "Equal decks determine the complete degree distribution",
      "intuition": "Once the original edge count and the multiset of card edge counts are known, the identity `degree(v)=e(G)−e(Δv(G))` recovers the multiset of vertex degrees. Equal decks therefore give equal degree multisets.",
      "source_statement": "lemma bipartiteDeck_determines_degreeMultiset {G H : SimpleGraph V} (hV : Fintype.card V ≥ 3)\n    [DecidableRel G.Adj] [DecidableRel H.Adj]\n    (h : bipartiteDeck G leftU rightV = bipartiteDeck H leftU rightV) :\n    degreeMultiset G = degreeMultiset H := by\n…",
      "source_statement_sha256": "00e56f8a396fda83e4759d9fc8f500da7771acab61a23cdca2e4d0bf5e1d617e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L250",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "unique_isolated_of_2_connected",
      "line": 371,
      "title": "The deleted vertex is the unique isolated vertex of its card",
      "intuition": "Under the development's 2-connectivity hypothesis and `|V| ≥ 3`, the incidence-deleted card at `v` has degree zero exactly at `v`. Every other vertex retains an edge to someone other than `v`.",
      "source_statement": "lemma unique_isolated_of_2_connected {V : Type*} [Fintype V] [DecidableEq V]\n  (G : SimpleGraph V) (h2 : G.Connected ∧ ∀ v, (G.induce {x | x ≠ v}).Connected) (hV : Fintype.card V ≥ 3)\n  (v : V) : ∀ x : V, (G.deleteIncidenceSet v).degree x = 0 ↔ x = v := by\n…",
      "source_statement_sha256": "c0c21f52b8c7d31170ae141bbd72b3b569af2f244c8ac0bc4452d35b6aacdedf",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L371",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "H_is_2_connected",
      "line": 462,
      "title": "Matched incidence-deleted cards transfer 2-connectivity to H",
      "intuition": "If every incidence-deleted card of `G` is graph-isomorphic to a matched card of `H` and `G` is 2-connected, then every vertex-deleted induced subgraph of `H` is connected and hence `H` itself is connected.",
      "source_statement": "lemma H_is_2_connected {V : Type*} [Fintype V] [DecidableEq V]\n  (G H : SimpleGraph V)\n  (hG_2_conn : G.Connected ∧ ∀ v, (G.induce {x | x ≠ v}).Connected)\n  (hV : Fintype.card V ≥ 3)\n  (f : V ≃ V)\n  (hf_iso : ∀ v, Nonempty (G.deleteIncidenceSet v ≃g H.deleteIncidenceSet (f v))) :\n  H.Connected ∧ ∀ v, (H.induce {x | x ≠ v}).Connected := by\n…",
      "source_statement_sha256": "23bd6485232281c4c6f95c0feb83f19e0412188b2a42dcb9810a84a377299239",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L462",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "f_preserves_partitions",
      "line": 397,
      "title": "Card matching preserves the named left and right parts",
      "intuition": "The permutation matching deleted cards sends each vertex to a vertex in the same bipartition part. The key is that the deleted vertex is recognizable inside its card as the unique isolated vertex.",
      "source_statement": "lemma f_preserves_partitions {V : Type*} [Fintype V] [DecidableEq V]\n  (G H : SimpleGraph V) (leftU rightV : Finset V)\n  (hG_2_conn : G.Connected ∧ ∀ v, (G.induce {x | x ≠ v}).Connected)\n  (hH_2_conn : H.Connected ∧ ∀ v, (H.induce {x | x ≠ v}).Connected)\n  (hV : Fintype.card V ≥ 3)\n  (f : V ≃ V)\n  (hf_iso : ∀ v, Nonempty (BipartiteIso (G.deleteIncidenceSet v) (H.deleteIncidenceSet (f v)) leftU rightV)) :\n  (∀ v, v ∈ leftU ↔ f v ∈ leftU) ∧ (∀ v, v ∈ rightV ↔ f v ∈ rightV) := by\n…",
      "source_statement_sha256": "86388db4126671acf5c6118aab1dcf7d1685abcc8448e1f56699b4df33ee20c3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L397",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "count_degreeMultiset_deleteIncidenceSet_add",
      "line": 317,
      "title": "The degree-count recurrence before and after deleting one incidence set",
      "intuition": "For each degree `d`, this identity balances four effects of isolating `v`: the new card counts, the isolated deleted vertex, neighbors dropping from degree `d+1` to `d`, and the original global degree counts.",
      "source_statement": "lemma count_degreeMultiset_deleteIncidenceSet_add {V : Type*} [Fintype V] [DecidableEq V]\n  (G : SimpleGraph V) [DecidableRel G.Adj] (v : V) (d : ℕ) :\n  Multiset.count d (degreeMultiset (G.deleteIncidenceSet v)) + (if G.degree v = d then 1 else 0) + Multiset.count d (degreeProfile G v) =\n  (if d = 0 then 1 else 0) + Multiset.count (d + 1) (degreeProfile G v) + Multiset.count d (degreeMultiset G) := by\n…",
      "source_statement_sha256": "e54f8bd6402c36045841fcd114116e65e9e16f22b1373b6e3475bba43c49a330",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L317",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "profile_eq_of_iso",
      "line": 340,
      "title": "A matched card and global degrees determine the deleted vertex's degree profile",
      "intuition": "When matched deleted vertices have the same degree, their cards are bipartite-isomorphic, and the original graphs have equal degree multisets, the multisets of their neighbors' original degrees are equal.",
      "source_statement": "lemma profile_eq_of_iso {G H : SimpleGraph V} [DecidableRel G.Adj] [DecidableRel H.Adj]\n    (leftU rightV : Finset V)\n    (h_degG_eq_degH : degreeMultiset G = degreeMultiset H)\n    (v : V) (u : V)\n    (h_deg_vu : G.degree v = H.degree u)\n    (iso : BipartiteIso (G.deleteIncidenceSet v) (H.deleteIncidenceSet u) leftU rightV) :\n    degreeProfile G v = degreeProfile H u := by\n…",
      "source_statement_sha256": "b3eb251a2a776fc6c7756841cafada9e7d0809ec4918f0e7c671478d47217b19",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L340",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "rightV_types_del_eq",
      "line": 660,
      "title": "Split the opposite side into changed neighbor types and unchanged non-neighbor types",
      "intuition": "After deleting all edges incident to a left-side vertex `u`, every right-side neighbor has its type transformed by `typeDrop`, while every right-side non-neighbor keeps its original type. The equality records those two multisets separately.",
      "source_statement": "lemma rightV_types_del_eq {V : Type*} [Fintype V] [DecidableEq V]\n  (G : SimpleGraph V) (leftU rightV : Finset V)\n  (hBipartite : G.IsBipartiteWith leftU rightV)\n  (u : V) (hu : u ∈ leftU) :\n  rightV_types (G.deleteIncidenceSet u) rightV =\n  Multiset.map (fun t => typeDrop t (G.degree u)) (S_multiset G rightV u) + T_multiset G rightV u := by\n…",
      "source_statement_sha256": "4137a0ae4be4be90473ead1fb591edd5f411903c40f691dff1a637018d05e525",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L660",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "S_add_map_T_eq",
      "line": 867,
      "title": "Matched cards give a cancellable equation for neighboring-type multisets",
      "intuition": "The global types on the opposite side and the local types in the matched deleted cards are equal. Decomposing both by adjacency yields the manuscript's balance equation between the two neighboring-type multisets and their type-dropped images.",
      "source_statement": "lemma S_add_map_T_eq {V : Type*} [Fintype V] [DecidableEq V]\n  (G H : SimpleGraph V) (leftU rightV : Finset V)\n  (hBipartiteG : G.IsBipartiteWith leftU rightV)\n  (hBipartiteH : H.IsBipartiteWith leftU rightV)\n  (f : V ≃ V)\n  (hf_iso : ∀ v, Nonempty (BipartiteIso (G.deleteIncidenceSet v) (H.deleteIncidenceSet (f v)) leftU rightV))\n  (hf_type : ∀ v, τ G v = τ H (f v))\n  (hf_left : ∀ v, v ∈ leftU ↔ f v ∈ leftU)\n  (hf_right : ∀ v, v ∈ rightV ↔ f v ∈ rightV)\n  (u : V) (hu : u ∈ leftU) :\n  S_multiset G rightV u + Multiset.map (fun t => typeDrop t (G.degree u)) (S_multiset H rightV (f u)) =\n  S_multiset H rightV (f u) + Multiset.map (fun t => typeDrop t (G.degree u)) (S_multiset G rightV u) := by\n…",
      "source_statement_sha256": "93f6883830d7694d0ddf6cad263ead530654cfa18ffb7a3dcb79e77a7285b275",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L867",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "S_multiset_eq",
      "line": 910,
      "title": "Descending degree induction recovers each neighboring-type multiset",
      "intuition": "The balance equation involving `S_G`, `S_H`, and `typeDrop` forces `S_G = S_H`. Type dropping strictly lowers the first coordinate, so the equality can be solved from the highest degrees downward.",
      "source_statement": "lemma S_multiset_eq {V : Type*} [Fintype V] [DecidableEq V]\n  (G H : SimpleGraph V) (leftU rightV : Finset V)\n  (hBipartiteG : G.IsBipartiteWith leftU rightV)\n  (hBipartiteH : H.IsBipartiteWith leftU rightV)\n  (f : V ≃ V)\n  (hf_iso : ∀ v, Nonempty (BipartiteIso (G.deleteIncidenceSet v) (H.deleteIncidenceSet (f v)) leftU rightV))\n  (hf_type : ∀ v, τ G v = τ H (f v))\n  (hf_left : ∀ v, v ∈ leftU ↔ f v ∈ leftU)\n  (hf_right : ∀ v, v ∈ rightV ↔ f v ∈ rightV)\n  (u : V) (hu : u ∈ leftU) :\n  S_multiset G rightV u = S_multiset H rightV (f u) := by\n…",
      "source_statement_sha256": "09b125551db61cf5673db08e61ae537bf2384431f282eac67787f06aa4297f93",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L910",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "typeProfile",
      "line": 947,
      "title": "The multiset of full vertex types around one vertex",
      "intuition": "`typeProfile G x` maps the vertex-type invariant `τ G` over every neighbor of `x`. Compared with `degreeProfile`, it remembers both each neighbor's degree and that neighbor's own surrounding degree multiset.",
      "source_statement": "def typeProfile (G : SimpleGraph V) (x : V) : Multiset (ℕ × Multiset ℕ) :=\n…",
      "source_statement_sha256": "8e02f86ccfb15e96cf70ee1a1d58e0d00cd5dbf50df1b04fe06dbb46dc16cb8a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L947",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "count_typeProfile_eq_one_iff_adj",
      "line": 968,
      "title": "Distinct vertex types turn profile membership into adjacency",
      "intuition": "If every vertex type in `G` is distinct, the type of `y` appears exactly once in the type profile of `x` precisely when `x` is adjacent to `y`. No other neighbor can impersonate `y` by carrying the same type.",
      "source_statement": "lemma count_typeProfile_eq_one_iff_adj {V : Type*} [Fintype V] [DecidableEq V]\n  (G : SimpleGraph V)\n  (h_distinct : ∀ x y : V, x ≠ y → τ G x ≠ τ G y)\n  (x y : V) :\n  Multiset.count (τ G y) (typeProfile G x) = 1 ↔ G.Adj x y := by\n…",
      "source_statement_sha256": "b2e3f14ff4d05e5912f682b48dd0cb1438a5a298cd93ad77f72adf104886535b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L968",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "adjacency_preserved_left",
      "line": 978,
      "title": "Type and type-profile preservation force adjacency preservation from the left side",
      "intuition": "For a left-part vertex `x`, equality of type profiles and pointwise preservation of `τ` show that `y` occurs as a neighbor of `x` exactly when `f y` occurs as a neighbor of `f x`.",
      "source_statement": "lemma adjacency_preserved_left {V : Type*} [Fintype V] [DecidableEq V]\n  (G H : SimpleGraph V) (leftU rightV : Finset V)\n  (f : V ≃ V)\n  (h_distinct : ∀ x y : V, x ≠ y → τ G x ≠ τ G y)\n  (h_tau : ∀ v, τ G v = τ H (f v))\n  (h_typeProfile : ∀ u, u ∈ leftU → typeProfile G u = typeProfile H (f u))\n  (x y : V) (hx : x ∈ leftU) :\n  G.Adj x y ↔ H.Adj (f x) (f y) := by\n…",
      "source_statement_sha256": "16477a4aff094545562c16c43325682e21e7df107bd1558721571eef190e9856",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L978",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean",
      "declaration": "type_preserving_bijection_is_iso",
      "line": 994,
      "title": "Upgrade the card-matching bijection to a bipartite graph isomorphism",
      "intuition": "A vertex bijection that preserves both partition sides, all vertex types, and the matched incidence-deleted cards must preserve every edge. The proof packages that adjacency equivalence with the side-preservation witnesses as a `BipartiteIso`.",
      "source_statement": "lemma type_preserving_bijection_is_iso {V : Type*} [Fintype V] [DecidableEq V]\n  (G H : SimpleGraph V) (leftU rightV : Finset V)\n  (hG_bipartite : G.IsBipartiteWith leftU rightV)\n  (hH_bipartite : H.IsBipartiteWith leftU rightV)\n  (hG_distinct : ∀ x y : V, x ≠ y → τ G x ≠ τ G y)\n  (hG_2_conn : G.Connected ∧ ∀ v, (G.induce {x | x ≠ v}).Connected)\n  (hV : Fintype.card V ≥ 3)\n  (hV_parts : (univ : Finset V) = leftU ∪ rightV)\n  (f : V ≃ V)\n  (hf_iso : ∀ v, Nonempty (BipartiteIso (G.deleteIncidenceSet v) (H.deleteIncidenceSet (f v)) leftU rightV))\n  (hf_type : ∀ v, τ G v = τ H (f v))\n  (hf_left : ∀ v, v ∈ leftU ↔ f v ∈ leftU)\n…",
      "source_statement_sha256": "a30e7be4bed0030e66c4b366301b2f225b0cb66561d41781bf1701292a7864f4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/bipartite_graph_reconstruction_conjecture_2.lean#L994",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "P5",
      "line": 2132,
      "title": "Pure O-sequences of codimension 3 and type 2 are log-concave",
      "intuition": "A pure order ideal in three variables with exactly two maximal monomials has a degree-counting sequence `h` satisfying `h(i-1)·h(i+1) ≤ h(i)²` at every internal degree. This closes the codimension-3, type-2 pure O-sequence case identified as open in the cited literature.",
      "source_statement": "theorem P5 : ProblemP5 := by\n…",
      "source_statement_sha256": "32e613b9a2f43b8e3913aa1a623a6d1c4b2ccb465f65863942e6cc39a650735b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L2132",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "Monomial",
      "line": 39,
      "title": "A monomial as its vector of natural-number exponents",
      "intuition": "The Lean type `Monomial(r)` is the finitely supported exponent vector for a monomial in `r` variables. For the manuscript's three-variable theorem, a monomial such as `x₀^a x₁^b x₂^c` is therefore represented by the vector `(a,b,c)` in `ℕ³`.",
      "source_statement": "abbrev Monomial (r : ℕ) := Fin r →₀ ℕ",
      "source_statement_sha256": "ba028d617438bb9b13bbda4dd1bd6d6c61198f94d77f033f7b78200cb836ed0a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "totalDeg",
      "line": 42,
      "title": "Total degree is the sum of a monomial's exponents",
      "intuition": "`totalDeg(m)`—written `totalDeg m` in Lean—adds every coordinate of the exponent vector. In three variables this is `m(0)+m(1)+m(2)`, the usual total degree of the represented monomial.",
      "source_statement": "def totalDeg {r : ℕ} (m : Monomial r) : ℕ :=\n…",
      "source_statement_sha256": "07320fed24bc7ee391c851c6f4759f7f0bc86a339e7944bcb140e420c3a81ac8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "OrderIdeal",
      "line": 47,
      "title": "A finite set of monomials closed downward under divisibility",
      "intuition": "An `OrderIdeal(r)` (Lean: `OrderIdeal r`) stores a finite collection of monomials together with the guarantee that every divisor of a stored monomial is stored too. In exponent-vector form, divisibility is coordinatewise `≤`.",
      "source_statement": "structure OrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "ba122a64b08ac5cf04b4b37d3c8522deb453ddb1d4672639ef155e023ad5efa4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "IsMaximalIn",
      "line": 82,
      "title": "Maximality inside an order ideal under monomial divisibility",
      "intuition": "`IsMaximalIn(I,m)` (Lean: `IsMaximalIn I m`) says first that `m` belongs to `I`, and then that no strictly larger monomial of `I` is divisible by `m`. Any member `n` above `m` must also lie below `m`, hence equals it in the partial order.",
      "source_statement": "def IsMaximalIn {r : ℕ} (I : OrderIdeal r) (m : Monomial r) : Prop :=\n…",
      "source_statement_sha256": "a60c8e1aeee59741870c750d748a65267eff8911c395e5aa6d957019b83c6ef7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "maximalMonomials",
      "line": 87,
      "title": "The finite set of maximal monomials of an order ideal",
      "intuition": "`maximalMonomials(I)` filters the carrier of `I`, keeping exactly the members satisfying `IsMaximalIn(I,·)`. In Lean these applications are written `maximalMonomials I` and `IsMaximalIn I`. The result turns the mathematical notion of maximal generator into a finite set that Lean can count and select from.",
      "source_statement": "noncomputable def maximalMonomials {r : ℕ} (I : OrderIdeal r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ad916102ecc030ec78ad780af9ec9e9646b79eddf1c9799982c3114c9250e7ac",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L87",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "IsPure",
      "line": 91,
      "title": "Purity means every maximal monomial has the same total degree",
      "intuition": "`IsPure(I)` (Lean: `IsPure I`) requires any two maximal monomials of `I` to have equal total degree. It does not itself assert that a maximal monomial exists; that nonemptiness is added by `PureOrderIdeal`.",
      "source_statement": "def IsPure {r : ℕ} (I : OrderIdeal r) : Prop :=\n…",
      "source_statement_sha256": "03a865f2ea87c5e18c0d95ebd9a4a40c761c6c84812507389b001eff8a0222c5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "PureOrderIdeal",
      "line": 95,
      "title": "An order ideal packaged with purity and a maximal monomial",
      "intuition": "`PureOrderIdeal(r)` (Lean: `PureOrderIdeal r`) is the structure used when later statements need a pure finite order ideal as data. It contains the ideal, a proof that its maximal monomials share one degree, and a proof that the maximal set is nonempty.",
      "source_statement": "structure PureOrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "6d350f4719325c70dec54ff5b88b3153fe545231a269e9e89eb4e1dcf9320ed4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "typePO",
      "line": 101,
      "title": "The type is the number of maximal monomials",
      "intuition": "`typePO(P)` is the cardinality of `maximalMonomials(P.ideal)`. In Lean these are written `typePO P` and `maximalMonomials P.ideal`. Thus “type 2” in the theorem means that the pure order ideal has exactly two maximal monomial generators.",
      "source_statement": "noncomputable def typePO {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "0d966ca8aba605e0d57dd15479643315e3556c2ece0a816b26838e844e8f8973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "socleDegree",
      "line": 106,
      "title": "The shared total degree of a pure ideal's maximal monomials",
      "intuition": "`socleDegree(P)` (Lean: `socleDegree P`) chooses one maximal monomial and returns its total degree. Purity proves that every other maximal monomial has the same degree, so the result is independent of which witness is chosen.",
      "source_statement": "noncomputable def socleDegree {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "5650301724fb40e83c63529839c28aa33de260f51f059acab3ea978409b21e54",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "pureOSequence",
      "line": 110,
      "title": "Degree-by-degree counts of monomials in an order ideal",
      "intuition": "`pureOSequence(I,d)` (Lean: `pureOSequence I d`) counts the monomials in `I` whose total degree is exactly `d`. Varying `d` produces the Hilbert-function sequence studied by the theorem.",
      "source_statement": "def pureOSequence {r : ℕ} (I : OrderIdeal r) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f9067ca146a946dfce29d85869385ca16043308dd3b50b841447653bbb8429b8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "IsLogConcaveUpTo",
      "line": 122,
      "title": "Log-concavity at every internal degree through a chosen endpoint",
      "intuition": "`IsLogConcaveUpTo(e,h)` (Lean: `IsLogConcaveUpTo e h`) requires `h(i-1)·h(i+1) ≤ h(i)²` for every integer index strictly between `0` and `e`. Each middle term must therefore dominate the geometric mean of its two neighbours.",
      "source_statement": "def IsLogConcaveUpTo (e : ℕ) (h : ℕ → ℕ) : Prop :=\n…",
      "source_statement_sha256": "eaac3ce65f7cb9899a0f38b012844ec5a0a69d7f7766524de03a160408b669b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "divisorsFinset",
      "line": 56,
      "title": "All monomial divisors of one generator as a finite set",
      "intuition": "`divisorsFinset(g)` (Lean: `divisorsFinset g`) is the principal lower set of `g`: it contains exactly the monomials `u` whose exponent vector is coordinatewise at most that of `g`. Equivalently, these are the monomial divisors of `g`.",
      "source_statement": "noncomputable def divisorsFinset {r : ℕ} (g : Monomial r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ed048c02d80a7110d87b10e92d84789d3bc0a4c363fdab8ab54ae3df1710eb6e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L56",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "type_two_implies_two_maximals",
      "line": 221,
      "title": "Type two produces exactly two distinct maximal monomials",
      "intuition": "If a pure order ideal has type `2`, its finite set of maximal monomials can be written as the pair `{m₁,m₂}` with `m₁ ≠ m₂`. This is the bridge from the numerical type hypothesis to the two concrete generators used throughout the proof.",
      "source_statement": "lemma type_two_implies_two_maximals {r : ℕ} (P : PureOrderIdeal r) (h_type : typePO P = 2) :\n    ∃ m1 m2, m1 ≠ m2 ∧ maximalMonomials P.ideal = {m1, m2} := by\n…",
      "source_statement_sha256": "a1b6f9243323f0823d473a7344cc3ce4ed63532822df7eccdd37c525db6a11b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L221",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "ideal_carrier_eq_union",
      "line": 226,
      "title": "A type-two pure ideal is the union of two principal divisor sets",
      "intuition": "Once `m₁` and `m₂` are known to be all the maximal monomials, every monomial in the order ideal divides one of them. The carrier is therefore exactly `divisorsFinset(m₁) ∪ divisorsFinset(m₂)`; Lean writes the two applications with spaces.",
      "source_statement": "lemma ideal_carrier_eq_union {r : ℕ} (P : PureOrderIdeal r) (m1 m2 : Monomial r)\n    (h_max_eq : maximalMonomials P.ideal = {m1, m2}) :\n    P.ideal.carrier = divisorsFinset m1 ∪ divisorsFinset m2 := by\n…",
      "source_statement_sha256": "0dec6595474817d03b522aa296138b22d2057b73188e4b9980d8f8c870bf7007",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L226",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "exists_single_greater_var",
      "line": 258,
      "title": "Equal-degree distinct triples can be oriented around one exceptional coordinate",
      "intuition": "For two distinct three-variable monomials of equal total degree, the proof may order the pair so that the second exponent is larger at one coordinate `x` and no larger at either other coordinate. The Lean disjunction records the two possible orientations instead of silently swapping names.",
      "source_statement": "lemma exists_single_greater_var {m1 m2 : Monomial 3} (h_deg : totalDeg m1 = totalDeg m2) (h_neq : m1 ≠ m2) :\n    (∃ x : Fin 3, m1 x > m2 x ∧ ∀ y, y ≠ x → m1 y ≤ m2 y) ∨\n    (∃ x : Fin 3, m2 x > m1 x ∧ ∀ y, y ≠ x → m2 y ≤ m1 y) := by\n…",
      "source_statement_sha256": "d922e57f4182b7b57f2d50b71409e152b24ed3c05876feb09d8e257498b00202",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L258",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "reduced_monomial",
      "line": 276,
      "title": "The residual monomial after removing the exceptional-coordinate shift",
      "intuition": "Given oriented maxima `m₁,m₂` and the exceptional coordinate `x`, `reduced_monomial` lowers the `x` exponent of `m₂` by `m₁(x)+1` and leaves its other exponents unchanged. Its degree is correspondingly reduced by the same shift.",
      "source_statement": "noncomputable def reduced_monomial (m1 m2 : Monomial 3) (x : Fin 3) : Monomial 3 :=\n…",
      "source_statement_sha256": "016db1aefa6ff81c339f40c70c67af9e00b4b38fecb159a6d9a7c5a839210bbb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L276",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "pureOSeq_decomposition",
      "line": 291,
      "title": "Split the union's degree count into a principal sequence and a shifted residual sequence",
      "intuition": "The degree-`d` monomials below `m₁` contribute the first summand. Monomials below `m₂` but not `m₁` exist only when `d ≥ m₁(x)+1`; subtracting that amount in coordinate `x` bijects them with degree-`d-(m₁(x)+1)` divisors of the reduced monomial.",
      "source_statement": "lemma pureOSeq_decomposition {m1 m2 : Monomial 3} (x : Fin 3)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (d : ℕ) :\n    (Finset.filter (fun m => totalDeg m = d) (divisorsFinset m1 ∪ divisorsFinset m2)).card =\n    (Finset.filter (fun m => totalDeg m = d) (divisorsFinset m1)).card +\n    if d ≥ m1 x + 1 then\n      (Finset.filter (fun m => totalDeg m = d - (m1 x + 1)) (divisorsFinset (reduced_monomial m1 m2 x))).card\n    else 0 := by\n…",
      "source_statement_sha256": "5d57510260aac0333c142c25c0890cf1681398e8b2c5f9d115678292ebb0c030",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L291",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq",
      "line": 321,
      "title": "Degree counts for the divisors of one monomial",
      "intuition": "`h_seq(m,d)` (Lean: `h_seq m d`) counts the divisors of `m` having total degree `d`. Geometrically this is a slice of the exponent box below `m`; it is the principal sequence denoted `h_m(d)` in the companion proof.",
      "source_statement": "noncomputable def h_seq (m : Monomial 3) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "8b5d29e44dcdafaf4e96bca3bc2bb8e4980d0f625f958f5274df81354cacb679",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L321",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff",
      "line": 324,
      "title": "The first backward difference of an integer-valued sequence",
      "intuition": "At a positive index, `diff(h,d)` (Lean: `diff h d`) is `h(d)-h(d-1)`, the change from the preceding value. At index zero the definition uses `h(0)`, giving the difference operator a total boundary value on natural-number indices.",
      "source_statement": "def diff (h : ℕ → ℤ) (d : ℕ) : ℤ :=\n…",
      "source_statement_sha256": "910e1324bd30367287283778127debba878bcb1dc5eb05989bce5a89ace39112",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L324",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2",
      "line": 327,
      "title": "The second backward difference of an integer-valued sequence",
      "intuition": "At a positive index, `diff2(h,d)` (Lean: `diff2 h d`) measures how the first difference changes: `Δh(d)-Δh(d-1)`. This discrete curvature is the quantity bounded by `1` for each principal sequence and by `2` for their sum.",
      "source_statement": "def diff2 (h : ℕ → ℤ) (d : ℕ) : ℤ :=\n…",
      "source_statement_sha256": "5835364203905519376617291e4d488a14ee0d301e0a221dd3cd9b3162f561e2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L327",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_h_seq_le_one",
      "line": 476,
      "title": "A single monomial's degree-count sequence has second difference at most one",
      "intuition": "For every three-variable monomial and degree, the discrete second difference of its divisor-count sequence is at most `1`. This is the uniform curvature bound obtained from the companion's explicit seven-indicator box-count formula.",
      "source_statement": "lemma diff2_h_seq_le_one (m : Monomial 3) (d : ℕ) :\n  diff2 (fun k => (h_seq m k : ℤ)) d ≤ 1 := by\n…",
      "source_statement_sha256": "435d7b0673a06ba06b0100874790c27bed45557a7ab62b288b09e95eb4ca221b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L476",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_seq",
      "line": 552,
      "title": "The decomposed degree sequence for the two-generator order ideal",
      "intuition": "`H_seq` adds the principal degree counts below `m₁` to a shifted residual degree count derived from `m₂`. The second term is zero below the threshold `m₁(x)+1`, exactly encoding the piecewise decomposition of the manuscript's equation (2).",
      "source_statement": "noncomputable def H_seq {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) : ℤ :=\n…",
      "source_statement_sha256": "30594688affc04e5cbed36e8015700f0c76502c974eba4d6befd1523e7337a80",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L552",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_le_two",
      "line": 676,
      "title": "The decomposed two-generator sequence has second difference at most two",
      "intuition": "Because `H_seq` is the sum of two principal-type sequences and each has second difference at most `1`, its second difference is at most `2`. This confines the positive-curvature analysis to the cases `1` and `2`.",
      "source_statement": "lemma diff2_H_seq_le_two {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  diff2 (H_seq (m1:=m1) (m2:=m2) x) d ≤ 2 := by\n…",
      "source_statement_sha256": "4ebd70df73367294913cb624d620e950212dcc2a5f91402207eae9c941de5073",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L676",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "log_concave_core",
      "line": 504,
      "title": "Reduce log-concavity to the only two positive second-difference cases",
      "intuition": "The algebraic identity for the log-concavity deficit expresses it as `y²-x·z`, where `x` is a nonnegative previous value, `y` is the first difference, and `z` is a second difference bounded above by `2`. Nonpositive `z` is automatic; hypotheses for `z=1` and `z=2` close the remaining cases.",
      "source_statement": "lemma log_concave_core (x y z : ℤ) (hx : x ≥ 0) (hz : z ≤ 2) (hy : z = 1 → y^2 ≥ x) (hy2 : z = 2 → y^2 ≥ 2*x) :\n  y^2 ≥ x * z := by\n…",
      "source_statement_sha256": "4188ed751938f478925e3cdbf57cb2020e4c033cfcca5054cbae013ed1533d7a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L504",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_seq_diff2_reduction",
      "line": 919,
      "title": "Shift a critical second-difference-one bound back one degree",
      "intuition": "When the second difference at `d+1` equals `1`, an upper bound at `d+1` by the square of its first difference implies the required bound at `d-1`. This is the formal version of the companion's shift reduction (13).",
      "source_statement": "lemma H_seq_diff2_reduction {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0) :\n  diff2 (H_seq (m1:=m1) (m2:=m2) x) (d + 1) = 1 →\n…",
      "source_statement_sha256": "4246b56b1d3eb2dfe2cb73f6ac03ba2cb8e6f0f8f55b8901e621e2746e48f1bd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L919",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "sum_log_concave_helper",
      "line": 2090,
      "title": "The principal-plus-shifted-residual sequence is log-concave",
      "intuition": "Under the exceptional-coordinate orientation and equal-degree hypotheses, the complete finite-difference case analysis proves log-concavity of the decomposed sequence through the common degree of `m₁` and `m₂`.",
      "source_statement": "lemma sum_log_concave_helper {m1 m2 : Monomial 3} (x : Fin 3)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2) :\n    IsLogConcaveUpTo (totalDeg m1) (fun d =>\n      h_seq m1 d + if d ≥ m1 x + 1 then h_seq (reduced_monomial m1 m2 x) (d - (m1 x + 1)) else 0) := by\n…",
      "source_statement_sha256": "9c87d348bba22101295ac789b52324ebc381bf4fe0e667b5964f5c1d4462c8b9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L2090",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "pureOSeq_eq_sum",
      "line": 2120,
      "title": "Identify the original pure O-sequence with the decomposed sequence",
      "intuition": "If the two named monomials are exactly the maximal elements and satisfy the chosen coordinate orientation, the pure O-sequence of the order ideal equals the principal-plus-shifted-residual formula at every degree.",
      "source_statement": "lemma pureOSeq_eq_sum {P : PureOrderIdeal 3} {m1 m2 : Monomial 3} (x : Fin 3)\n    (h_max_eq : maximalMonomials P.ideal = {m1, m2})\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) :\n    pureOSequence P.ideal = fun d =>\n      h_seq m1 d + if d ≥ m1 x + 1 then h_seq (reduced_monomial m1 m2 x) (d - (m1 x + 1)) else 0 := by\n…",
      "source_statement_sha256": "b272932765cbab21e982458791e6ab7facbe4ddb2b0a1b01ab74b96dde3ea150",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L2120",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "L_seq",
      "line": 330,
      "title": "Count a diagonal slice of a two-dimensional exponent box",
      "intuition": "`L_seq(A,B,d)` (Lean: `L_seq A B d`) counts pairs of nonnegative exponents with first coordinate below `A`, second coordinate below `B`, and total degree `d`. Lean stores only the first coordinate because the second is forced to be `d-x`.",
      "source_statement": "def L_seq (A B d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f38daaaa06110b925a3f8a6790e3bb45e6796922effbc60d4d66bd84a8595b16",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L330",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "delta_L",
      "line": 333,
      "title": "The first difference of a box slice is three threshold indicators",
      "intuition": "For positive `d`, the change in the diagonal slice count is one contribution from crossing each side length, corrected once when the combined length is crossed.",
      "source_statement": "lemma delta_L (A B d : ℕ) (hd : d > 0) :\n  (L_seq A B d : ℤ) - (L_seq A B (d - 1) : ℤ) =\n  (if d < A then 1 else 0) + (if d < B then 1 else 0) - (if d < A + B then 1 else 0) := by\n…",
      "source_statement_sha256": "ac68e12351ef5115b8dc2fb6b4890e91876ba0b7761b55ff1d722951a80f93fb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L333",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_calc",
      "line": 339,
      "title": "Assemble a three-dimensional degree slice from two-dimensional slices",
      "intuition": "`H_calc(A,B,C,d)` sums the two-coordinate count `L_seq(A,B,k)` over possible partial degrees `k`, retaining only those for which the remaining exponent `d-k` fits below `C`. Lean writes these applications with spaces.",
      "source_statement": "def H_calc (A B C d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "220339e40dc5ca3c96b97fd73e9dffb4503b8287efddb9f2390f8ac25c3e6481",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L339",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_eq_H_calc",
      "line": 342,
      "title": "The box-count formula equals the actual monomial divisor count",
      "intuition": "For a three-variable monomial `m`, the number of divisors of degree `d` is exactly `H_calc (m 0+1) (m 1+1) (m 2+1) d`.",
      "source_statement": "lemma H_eq_H_calc (m : Monomial 3) (d : ℕ) :\n  ((divisorsFinset m).filter (fun x => totalDeg x = d)).card =\n  H_calc (m 0 + 1) (m 1 + 1) (m 2 + 1) d := by\n…",
      "source_statement_sha256": "a5a31b5416f8dcc7bbae6aeb3542c6734bc3c5b8f5270b088961095ec56f3a2b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L342",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "delta_H_calc",
      "line": 355,
      "title": "First difference of the three-dimensional box count",
      "intuition": "At a positive degree, the change in `H_calc` is the new two-dimensional slice at `d`, minus the slice that falls out after shifting by the third side length `C`.",
      "source_statement": "lemma delta_H_calc (A B C d : ℕ) (hd : d > 0) :\n  (H_calc A B C d : ℤ) - (H_calc A B C (d - 1) : ℤ) =\n  (L_seq A B d : ℤ) - (if d ≥ C then (L_seq A B (d - C) : ℤ) else 0) := by\n…",
      "source_statement_sha256": "4797fe2f6d3f4bba5ed216197c05987e5a960de6c10c5f21b1502356261e4110",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L355",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "Z_calc",
      "line": 364,
      "title": "The seven-threshold formula for discrete curvature",
      "intuition": "`Z_calc(A,B,C,d)` (Lean: `Z_calc A B C d`) combines three single-side, three pair-sum, and one triple-sum indicators. Its value is the exact second difference of a three-dimensional monomial box count.",
      "source_statement": "def Z_calc (A B C d : ℕ) : ℤ :=\n…",
      "source_statement_sha256": "91aae5e9b3e17e0ece1e61cf6bd1e56efb8c9c0e110510699620e3310c1246e7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L364",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_calc_eq_1",
      "line": 379,
      "title": "Verify the curvature formula at the first positive degree",
      "intuition": "The general predecessor algebra has a boundary at `d=1`; this lemma checks directly that `diff2 H_calc 1` still equals `Z_calc A B C 1`.",
      "source_statement": "lemma diff2_H_calc_eq_1 (A B C : ℕ) :\n  diff2 (fun k => (H_calc A B C k : ℤ)) 1 = Z_calc A B C 1 := by\n…",
      "source_statement_sha256": "299e7546d33295763c983e1908a3d1d5b9aa855f1867e2bc984f768ffdf54e50",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L379",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_calc_d_unfold",
      "line": 389,
      "title": "Expand a positive second difference into two neighbouring slice differences",
      "intuition": "For `d>1`, this lemma substitutes the first-difference formula at `d` and `d-1`, exposing the subtraction whose simplification produces `Z_calc`.",
      "source_statement": "lemma diff2_H_calc_d_unfold (A B C d : ℕ) (hd : d > 1) :\n  diff2 (fun k => (H_calc A B C k : ℤ)) d =\n  (L_seq A B d : ℤ) - (if d ≥ C then (L_seq A B (d - C) : ℤ) else 0) -\n  ((L_seq A B (d - 1) : ℤ) - (if d - 1 ≥ C then (L_seq A B (d - 1 - C) : ℤ) else 0)) := by\n…",
      "source_statement_sha256": "8ffb8e51b84f76644e34f880fed6f4bc2b18461a627a09f28191d3f6527b32de",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L389",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_calc_L2",
      "line": 401,
      "title": "Normalize the shifted slice difference across its threshold",
      "intuition": "The correction term in `delta_H_calc` changes form depending on whether `d` lies above, at, or below `C`; this lemma states those three cases explicitly.",
      "source_statement": "lemma diff2_H_calc_L2 (A B C d : ℕ) (hd : d > 1) :\n  (if d ≥ C then (L_seq A B (d - C) : ℤ) else 0) - (if d - 1 ≥ C then (L_seq A B (d - 1 - C) : ℤ) else 0) =\n    if d > C then (L_seq A B (d - C) : ℤ) - (L_seq A B (d - 1 - C) : ℤ) else if d = C then (L_seq A B 0 : ℤ) else 0 := by\n…",
      "source_statement_sha256": "26b851ce1a6f6c6655ee5de62d6e624597939376e30d49bf2a24534ad7cfd988",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L401",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_calc_eq_d",
      "line": 425,
      "title": "Derive the seven-indicator curvature formula above the boundary",
      "intuition": "For `d>1`, the expanded second difference of `H_calc` simplifies exactly to `Z_calc A B C d`.",
      "source_statement": "lemma diff2_H_calc_eq_d (A B C d : ℕ) (hd : d > 1) :\n  diff2 (fun k => (H_calc A B C k : ℤ)) d = Z_calc A B C d := by\n…",
      "source_statement_sha256": "360bc1fe92c875998192a626a30b1eafc45b94e9490f8f57191eae218bc92241",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L425",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_calc_eq",
      "line": 459,
      "title": "The box count has the manuscript's curvature formula at every positive degree",
      "intuition": "`diff2 H_calc d = Z_calc A B C d` for all positive `d`, joining the direct `d=1` boundary proof with the uniform argument for `d>1`.",
      "source_statement": "lemma diff2_H_calc_eq (A B C d : ℕ) (hd : d > 0) :\n  diff2 (fun k => (H_calc A B C k : ℤ)) d = Z_calc A B C d := by\n…",
      "source_statement_sha256": "0ce055fdd4d7473a1eb12ebca20f241215f987bf735146cf0d34ff03ca1e71e9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L459",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "Z_calc_le_one",
      "line": 467,
      "title": "The curvature indicator expression never exceeds one",
      "intuition": "Regardless of the side lengths or degree, the inclusion-exclusion integer `Z_calc A B C d` is at most `1`.",
      "source_statement": "lemma Z_calc_le_one (A B C d : ℕ) : Z_calc A B C d ≤ 1 := by\n…",
      "source_statement_sha256": "0263e0e41f63a123f70fdf1fe0dce37a9a35584c29750e519663ee76dec9b3e2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L467",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_eq_func",
      "line": 471,
      "title": "Pointwise equal sequences have equal second differences",
      "intuition": "If two integer-valued sequences agree at every natural index, their `diff2` values agree at any selected degree.",
      "source_statement": "lemma diff2_eq_func {h1 h2 : ℕ → ℤ} (heq : ∀ d, h1 d = h2 d) (d : ℕ) :\n  diff2 h1 d = diff2 h2 d := by\n…",
      "source_statement_sha256": "632db0df42eb7059f69a6f1ce5ba6df5b29f5f2e9909c3632d0e1f682fa3b219",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L471",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff_eq",
      "line": 518,
      "title": "Reconstruct a sequence value from its preceding value and first difference",
      "intuition": "At a positive degree, `h(d)` is exactly `h(d-1)+diff h d`; this is the defining first-difference equation rearranged for later algebra.",
      "source_statement": "lemma diff_eq (h : ℕ → ℤ) (d : ℕ) (hd : d > 0) :\n  h d = h (d - 1) + diff h d := by\n…",
      "source_statement_sha256": "73d27d72882f2f279c7010ffb7782797303a7fbc4b8551212f0e1060341c3471",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L518",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_eq",
      "line": 525,
      "title": "Advance the first difference using the second difference",
      "intuition": "For positive `d`, the next first difference is the current first difference plus the second difference at `d+1`.",
      "source_statement": "lemma diff2_eq (h : ℕ → ℤ) (d : ℕ) (hd : d > 0) :\n  diff h (d + 1) = diff h d + diff2 h (d + 1) := by\n…",
      "source_statement_sha256": "595a3495c85d36f492ca6fe1215fe151fbb36322f7dbc9474579b3fee8b0f4b6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L525",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "shifted_h_seq",
      "line": 534,
      "title": "Shift a principal divisor sequence behind a degree threshold",
      "intuition": "`shifted_h_seq(m,c,d)` (Lean: `shifted_h_seq m c d`) is zero below `c` and otherwise reads the divisor-count sequence of `m` at degree `d-c`.",
      "source_statement": "noncomputable def shifted_h_seq (m : Monomial 3) (c : ℕ) (d : ℕ) : ℤ :=\n…",
      "source_statement_sha256": "e3d1cec57515f9ec6b172dd6e0df8f9a5528a9db7c45cd8b994aa2851ebe9291",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L534",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff_add",
      "line": 537,
      "title": "First differences distribute over pointwise addition",
      "intuition": "Taking the first discrete difference of `h₁+h₂` gives `diff h₁ + diff h₂` at every degree, including the special boundary value at zero.",
      "source_statement": "lemma diff_add (h1 h2 : ℕ → ℤ) (d : ℕ) :\n  diff (fun k => h1 k + h2 k) d = diff h1 d + diff h2 d := by\n…",
      "source_statement_sha256": "2ecf5bdb2941e424dd5df243b2231e36033e61e7be36000908fc95d20deabe44",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L537",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_add",
      "line": 544,
      "title": "Second differences distribute over pointwise addition",
      "intuition": "The discrete curvature of a sum is the sum of the two discrete curvatures, with the identity valid at zero as well as positive degrees.",
      "source_statement": "lemma diff2_add (h1 h2 : ℕ → ℤ) (d : ℕ) :\n  diff2 (fun k => h1 k + h2 k) d = diff2 h1 d + diff2 h2 d := by\n…",
      "source_statement_sha256": "f9e81a4bb4c988c689838b87e3423834b7adc8e4476c538b4f06abf924e15c24",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L544",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_seq_eq_add",
      "line": 555,
      "title": "Expose the decomposed Hilbert sequence as an explicit pointwise sum",
      "intuition": "`H_seq` is definitionally the principal sequence of `m₁` plus the shifted sequence of the reduced monomial; this lemma presents that structure in the form expected by the finite-difference linearity lemmas.",
      "source_statement": "lemma H_seq_eq_add {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  H_seq (m1:=m1) (m2:=m2) x d = (fun k => (h_seq m1 k : ℤ)) d + shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1) d := by\n…",
      "source_statement_sha256": "fc349d2d440d62b7b1817086f3be9d7be10cdae47bcac6e746b598334b4dad7b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L555",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_h_seq_eq_Z_calc",
      "line": 565,
      "title": "The actual monomial divisor sequence has the seven-indicator curvature",
      "intuition": "At every positive degree, the second difference of `h_seq(m)` equals `Z_calc` with side lengths one larger than the three exponents of `m`; Lean writes the partial application as `h_seq m`.",
      "source_statement": "lemma diff2_h_seq_eq_Z_calc (m : Monomial 3) (d : ℕ) (hd : d > 0) :\n  diff2 (fun k => (h_seq m k : ℤ)) d = Z_calc (m 0 + 1) (m 1 + 1) (m 2 + 1) d := by\n…",
      "source_statement_sha256": "46218bc98a80927e39931c8b6afc891e766ae6c90313737d72947c799d49b250",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L565",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "Z_calc_eq_zero_cases",
      "line": 574,
      "title": "Classify where the curvature indicator can vanish",
      "intuition": "If `Z_calc A B C d` is zero, then `d` has crossed at least one single-side threshold while remaining below the opposite pair sum, or it has crossed the total sum.",
      "source_statement": "lemma Z_calc_eq_zero_cases (A B C d : ℕ) :\n  Z_calc A B C d = 0 → (d ≥ A ∧ d < B + C) ∨ (d ≥ B ∧ d < A + C) ∨ (d ≥ C ∧ d < A + B) ∨ (d ≥ A + B + C) := by\n…",
      "source_statement_sha256": "3c9ec62a1f12f8103a7c509f79a879b84a70aeb24d8a4fa63b45b54dba7fbbc3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L574",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "Z_calc_eq_one_cases",
      "line": 579,
      "title": "Curvature one occurs only in the lower or upper box region",
      "intuition": "When the indicator expression equals one, either `d` lies below all three side lengths or it lies above all three pair sums but below the total sum.",
      "source_statement": "lemma Z_calc_eq_one_cases (A B C d : ℕ) :\n  Z_calc A B C d = 1 → (d < A ∧ d < B ∧ d < C) ∨ (d ≥ A + B ∧ d ≥ B + C ∧ d ≥ C + A ∧ d < A + B + C) := by\n…",
      "source_statement_sha256": "78e1f2c690e3c8734792aae91d0775edb0f73cd56460a3c0971eec44a0d14cfa",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L579",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_h_seq_eq_one_implies_pos",
      "line": 584,
      "title": "Positive-degree curvature one forces a lower-or-upper region certificate",
      "intuition": "If the monomial divisor count has second difference exactly one at a positive degree, Lean returns the coordinate inequalities describing the manuscript's two permitted regions.",
      "source_statement": "lemma diff2_h_seq_eq_one_implies_pos (m : Monomial 3) (d : ℕ) (hd : d > 0) :\n  diff2 (fun k => (h_seq m k : ℤ)) d = 1 → (d ≤ m 0 ∧ d ≤ m 1 ∧ d ≤ m 2) ∨ (d ≥ m 0 + m 1 + 2 ∧ d ≥ m 1 + m 2 + 2 ∧ d ≥ m 0 + m 2 + 2 ∧ d ≤ m 0 + m 1 + m 2 + 2) := by\n…",
      "source_statement_sha256": "82639d515209e2626da0b55749d6a0a3bdaf49c1cbc90c328a494b5ff942c20d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L584",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_h_seq_eq_one_implies",
      "line": 603,
      "title": "Curvature one has the same two-region classification at every degree",
      "intuition": "This wrapper extends the positive-degree classification to degree zero, yielding one uniform lower-or-upper alternative for all natural indices.",
      "source_statement": "lemma diff2_h_seq_eq_one_implies (m : Monomial 3) (d : ℕ) :\n  diff2 (fun k => (h_seq m k : ℤ)) d = 1 → (d ≤ m 0 ∧ d ≤ m 1 ∧ d ≤ m 2) ∨ (d ≥ m 0 + m 1 + 2 ∧ d ≥ m 1 + m 2 + 2 ∧ d ≥ m 0 + m 2 + 2 ∧ d ≤ m 0 + m 1 + m 2 + 2) := by\n…",
      "source_statement_sha256": "fd1a8f09ab198f02ba4d0846afe33fd3f25aaf7c7e38de90041c96ae526b4ec4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L603",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_shifted_h_seq_eq_zero_lt",
      "line": 614,
      "title": "A shifted divisor sequence has zero curvature before it begins",
      "intuition": "When `d<c`, the shifted residual sequence is still zero at `d` and at the predecessor indices used by `diff2`, so its second difference is exactly zero.",
      "source_statement": "lemma diff2_shifted_h_seq_eq_zero_lt (m : Monomial 3) (c d : ℕ) (h : d < c) :\n  diff2 (shifted_h_seq m c) d = 0 := by\n…",
      "source_statement_sha256": "034ed69a9cf516e7644cb915f36f29535872998dfd38bc610b0c91ab659e3024",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L614",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_shifted_h_seq_le_one_lt",
      "line": 619,
      "title": "The shifted summand obeys the curvature bound below its threshold",
      "intuition": "Before the residual term turns on, its second difference is zero and therefore certainly at most one.",
      "source_statement": "lemma diff2_shifted_h_seq_le_one_lt (m : Monomial 3) (c d : ℕ) (h : d < c) :\n  diff2 (shifted_h_seq m c) d ≤ 1 := by\n…",
      "source_statement_sha256": "015546bfe634c5c5479eab6873a758c405fdd0f791f7c77740248823600e18e5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L619",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_shifted_h_seq_eq_ge",
      "line": 624,
      "title": "Above the threshold, shifting only translates the curvature index",
      "intuition": "Once `d≥c`, the second difference of `shifted_h_seq(m,c)` at `d` equals the second difference of the original divisor sequence at `d-c`; Lean writes the partial application as `shifted_h_seq m c`.",
      "source_statement": "lemma diff2_shifted_h_seq_eq_ge (m : Monomial 3) (c d : ℕ) (h : d ≥ c) :\n  diff2 (shifted_h_seq m c) d = diff2 (fun k => (h_seq m k : ℤ)) (d - c) := by\n…",
      "source_statement_sha256": "b52ac4b27858d4472f31c6d41d79916d407e83df1a1da00014a6754e21a5d691",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L624",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_shifted_h_seq_le_one_ge",
      "line": 630,
      "title": "The shifted summand obeys the curvature bound after activation",
      "intuition": "At and above the shift threshold, the residual curvature is just the original monomial curvature at a translated index, which is already bounded by one.",
      "source_statement": "lemma diff2_shifted_h_seq_le_one_ge (m : Monomial 3) (c d : ℕ) (h : d ≥ c) :\n  diff2 (shifted_h_seq m c) d ≤ 1 := by\n…",
      "source_statement_sha256": "ca0b019b3b39a875e09dddfdc6b9d70b2b3a453a6408838b9f54538eb3dce873",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L630",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_shifted_h_seq_le_one",
      "line": 636,
      "title": "Every shifted principal sequence has second difference at most one",
      "intuition": "The curvature bound for a shifted residual sequence holds at every natural degree, whether the degree lies before or after the activation threshold.",
      "source_statement": "lemma diff2_shifted_h_seq_le_one (m : Monomial 3) (c d : ℕ) :\n  diff2 (shifted_h_seq m c) d ≤ 1 := by\n…",
      "source_statement_sha256": "0afa4291b1cf6fe777454df6e521af1f8a9d9eadbd260dabe62503a04bdfa080",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L636",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_shifted_h_seq_eq_one_implies",
      "line": 643,
      "title": "Curvature one proves that the shifted term has activated",
      "intuition": "If the shifted sequence has second difference exactly one at `d`, then `d` cannot lie below `c`; it must satisfy `d≥c`.",
      "source_statement": "lemma diff2_shifted_h_seq_eq_one_implies (m : Monomial 3) (c d : ℕ) :\n  diff2 (shifted_h_seq m c) d = 1 → d ≥ c := by\n…",
      "source_statement_sha256": "11958bb5a4eafbe4c6f3f4a3b2ade4fd01b39bb5c7400fadb553a320b5f4ec78",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L643",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_shifted_h_seq_eq_one_implies_eq",
      "line": 651,
      "title": "Transfer residual curvature one to the unshifted reduced monomial",
      "intuition": "Curvature one for the shifted residual at degree `d` becomes curvature one for the reduced monomial's divisor sequence at degree `d-c`.",
      "source_statement": "lemma diff2_shifted_h_seq_eq_one_implies_eq (m : Monomial 3) (c d : ℕ) :\n  diff2 (shifted_h_seq m c) d = 1 → diff2 (fun k => (h_seq m k : ℤ)) (d - c) = 1 := by\n…",
      "source_statement_sha256": "3a7213f8fcab32d918671a0f7c4fc8dab2c01f1610d359596f1a9d3618d42ba9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L651",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_seq_eval_add",
      "line": 658,
      "title": "Evaluate the decomposed sequence as principal plus shifted residual",
      "intuition": "At a concrete degree `d`, `H_seq` is the integer sum of the principal divisor count and the shifted reduced-monomial count.",
      "source_statement": "lemma H_seq_eval_add {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  H_seq (m1:=m1) (m2:=m2) x d = (h_seq m1 d : ℤ) + shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1) d := by\n…",
      "source_statement_sha256": "f8874d1e555534c677c8f8208c9d7b8b607d55d65883b2b4fba134a9c372259e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L658",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff_H_seq_eval_add",
      "line": 662,
      "title": "Split the first difference of the decomposed sequence",
      "intuition": "The first difference of `H_seq` is the sum of the first differences of its principal and shifted-residual components.",
      "source_statement": "lemma diff_H_seq_eval_add {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  diff (H_seq (m1:=m1) (m2:=m2) x) d = diff (fun k => (h_seq m1 k : ℤ)) d + diff (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) d := by\n…",
      "source_statement_sha256": "0da8b228502d0c0985c63e2ade6129239c0171b27ce617126a98592993fb8bfb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L662",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_add",
      "line": 669,
      "title": "Split the second difference of the decomposed sequence",
      "intuition": "The discrete curvature of `H_seq` is exactly the sum of the curvatures of its principal and shifted-residual parts.",
      "source_statement": "lemma diff2_H_seq_eq_add {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  diff2 (H_seq (m1:=m1) (m2:=m2) x) d = diff2 (fun k => (h_seq m1 k : ℤ)) d + diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) d := by\n…",
      "source_statement_sha256": "c29ea853f8b623a2dc4e879fe43e0a2b5e877d319c6cbcc0c414c1fe9ff33deb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L669",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "L_seq_le",
      "line": 685,
      "title": "A two-variable degree slice has at most d+1 points",
      "intuition": "`L_seq(A,B,d)` filters possible first exponents from the `d+1` integers between zero and `d`, so its cardinality cannot exceed `d+1`; Lean writes this as `L_seq A B d`.",
      "source_statement": "lemma L_seq_le (A B d : ℕ) : L_seq A B d ≤ d + 1 := by\n…",
      "source_statement_sha256": "d0fe8780e7dbb443e3389b99414cae03645c22218c8724b035ca32744d08328a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L685",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq_le_binom",
      "line": 689,
      "title": "A principal three-variable degree slice is bounded by all weak compositions",
      "intuition": "The divisors of a fixed monomial at degree `d` form a subset of all triples of nonnegative integers summing to `d`, of which there are `(d+1)(d+2)/2`.",
      "source_statement": "lemma h_seq_le_binom (m : Monomial 3) (d : ℕ) :\n  h_seq m d ≤ (d + 1) * (d + 2) / 2 := by\n…",
      "source_statement_sha256": "0a28219c72f941cbbf1176b89a51b1189dd6591caa70abcd732682bfe0b8abc1",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L689",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq_pure_lower",
      "line": 709,
      "title": "Exact quadratic formula in the lower box region",
      "intuition": "If `d` does not exceed any exponent of `m`, none of the divisor-box ceilings are active, so twice the degree count is `(d+1)(d+2)`.",
      "source_statement": "lemma h_seq_pure_lower (m : Monomial 3) (d : ℕ) (h : d ≤ m 0 ∧ d ≤ m 1 ∧ d ≤ m 2) :\n  (h_seq m d : ℤ) * 2 = (d + 1) * (d + 2) := by\n…",
      "source_statement_sha256": "3b0e11a788c6688b7f3cca24e73a810c1d047236d98c07c98cb503d043a115a6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L709",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq_pure_lower_diff",
      "line": 728,
      "title": "First difference in the lower region is d+2",
      "intuition": "When degree `d+1` remains below every exponent ceiling, subtracting consecutive lower-region quadratic formulas gives first difference `d+2`.",
      "source_statement": "lemma h_seq_pure_lower_diff (m : Monomial 3) (d : ℕ) (h : d + 1 ≤ m 0 ∧ d + 1 ≤ m 1 ∧ d + 1 ≤ m 2) :\n  diff (fun k => (h_seq m k : ℤ)) (d + 1) = d + 2 := by\n…",
      "source_statement_sha256": "a52c858afe6edfd8d593db396cc5c6463cbc69f325da1efc0d948218be5cd20f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L728",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq_symm",
      "line": 748,
      "title": "A monomial divisor sequence is symmetric about half its total degree",
      "intuition": "Divisors of `m` at degree `d` are in bijection with complementary divisors `m-u` at degree `|m|-d`.",
      "source_statement": "lemma h_seq_symm (m : Monomial 3) (d : ℕ) (hd : d ≤ m 0 + m 1 + m 2) :\n  h_seq m d = h_seq m (m 0 + m 1 + m 2 - d) := by\n…",
      "source_statement_sha256": "4ffcf1652cf3b3a980caff8b1ff11bd34ff1d53a2cde6091665467e53776aca5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L748",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq_upper_bound_int",
      "line": 763,
      "title": "Universal complementary quadratic upper bound",
      "intuition": "Below the total degree, twice `h_seq(m,d)` is at most `(|m|-d+1)(|m|-d+2)`, the number of unrestricted complementary triples; Lean writes the application as `h_seq m d`.",
      "source_statement": "lemma h_seq_upper_bound_int (m : Monomial 3) (d : ℕ) (h_le : d ≤ m 0 + m 1 + m 2) :\n  (h_seq m d : ℤ) * 2 ≤ (m 0 + m 1 + m 2 - d + 1 : ℤ) * (m 0 + m 1 + m 2 - d + 2 : ℤ) := by\n…",
      "source_statement_sha256": "f795b8f6f495e47a9968eff74b13b938a0afb144f7a7f0ff776302821030842f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L763",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq_pure_upper",
      "line": 790,
      "title": "Exact complementary quadratic formula in the upper box region",
      "intuition": "Once `d` lies beyond every pairwise exponent sum, the complementary degree is below every exponent ceiling, giving the exact upper-tail formula for the divisor count.",
      "source_statement": "lemma h_seq_pure_upper (m : Monomial 3) (d : ℕ) (h : d ≥ m 0 + m 1 ∧ d ≥ m 1 + m 2 ∧ d ≥ m 0 + m 2 ∧ d ≤ m 0 + m 1 + m 2 + 1) :\n  (h_seq m d : ℤ) * 2 = (m 0 + m 1 + m 2 + 1 - d : ℤ) * (m 0 + m 1 + m 2 + 2 - d : ℤ) := by\n…",
      "source_statement_sha256": "c10ada38b68c0fbb6169974f7cb3123e4354cd1af9271c2f535b5dfd69ac889d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L790",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h_seq_pure_upper_diff",
      "line": 813,
      "title": "First difference in the upper region is negative complementary distance",
      "intuition": "In the upper box region, subtracting the exact complementary quadratic formulas at adjacent degrees gives `Δh(d)=-(|m|+2-d)`.",
      "source_statement": "lemma h_seq_pure_upper_diff (m : Monomial 3) (d : ℕ) (h : d ≥ m 0 + m 1 + 1 ∧ d ≥ m 1 + m 2 + 1 ∧ d ≥ m 0 + m 2 + 1 ∧ d ≤ m 0 + m 1 + m 2 + 1) :\n  diff (fun k => (h_seq m k : ℤ)) d = - (m 0 + m 1 + m 2 + 2 - d : ℤ) := by\n…",
      "source_statement_sha256": "e2dad321704e3b45052de335db898152188c6462e0225ae2b3d6b36920879e35",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L813",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "int_add_eq_two_of_le_one",
      "line": 832,
      "title": "A sum of two one-bounded integers reaches two only as one plus one",
      "intuition": "If `a≤1`, `b≤1`, and `a+b=2`, then both summands must equal one.",
      "source_statement": "lemma int_add_eq_two_of_le_one (a b : ℤ) (ha : a ≤ 1) (hb : b ≤ 1) (hab : a + b = 2) : a = 1 ∧ b = 1 := by omega",
      "source_statement_sha256": "59d63dffa8e2d31b777a068ef2f2e29910f317d606b0e2e031ceedef090f2d15",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L832",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_two_implies_both_one",
      "line": 834,
      "title": "Curvature two means both component curvatures equal one",
      "intuition": "If the decomposed sequence has second difference two, its principal and shifted-residual summands must each contribute their maximum possible curvature one.",
      "source_statement": "lemma diff2_H_seq_eq_two_implies_both_one {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  diff2 (H_seq (m1:=m1) (m2:=m2) x) d = 2 →\n…",
      "source_statement_sha256": "98ef67499e046ff6974aba9e2392409012e8a8d34e0eaac611eb8c3954e04855",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L834",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "int_add_eq_one_of_le_one_cases",
      "line": 843,
      "title": "A sum of two one-bounded integers equals one in exactly two zero-one cases",
      "intuition": "Under upper bounds `a≤1` and `b≤1`, an integer sum of one forces either `(a,b)=(1,0)` or `(0,1)`.",
      "source_statement": "lemma int_add_eq_one_of_le_one_cases (a b : ℤ) (ha : a ≤ 1) (hb : b ≤ 1) (hab : a + b = 1) :\n  (a = 1 ∧ b = 0) ∨ (a = 0 ∧ b = 1) := by omega",
      "source_statement_sha256": "25928a72ec89f42c248a27854f9ca3ba333541b1240c411caa2b93a05db6085d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L843",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies_cases",
      "line": 846,
      "title": "Curvature one splits into the manuscript's principal or residual case",
      "intuition": "When `H=f+g` has second difference one, exactly one component has curvature one and the other has curvature zero.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies_cases {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  diff2 (H_seq (m1:=m1) (m2:=m2) x) d = 1 →\n…",
      "source_statement_sha256": "6f8f95f89ce0d197a87eab213a3791496282069334db016178bda63de58297e3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L846",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "log_concave_algebra_lower_eval",
      "line": 856,
      "title": "Lower-region quadratic counts satisfy the needed square bound",
      "intuition": "For an active shift `c≥1` and `d≥c-1`, the sum of the two lower-region quadratic count expressions is bounded by the square of their combined first differences.",
      "source_statement": "lemma log_concave_algebra_lower_eval (d c : ℤ) (hc : c ≥ 1) (hd : d ≥ c - 1) :\n  (d) * (d + 1) + (d - c) * (d - c + 1) ≤ (d + 1 + d - c + 1) ^ 2 := by\n…",
      "source_statement_sha256": "104ec7e237ceff595c324ce4730e2d3e09b8b190c453f7e7e470be490aaaa43f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L856",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "log_concave_weak_lower",
      "line": 869,
      "title": "A lower-region principal term absorbs a bounded residual term",
      "intuition": "If the residual value has the companion's quadratic upper bound and its first difference is nonnegative, the combined value is controlled by twice the square of the combined first difference.",
      "source_statement": "lemma log_concave_weak_lower (d c S dS : ℤ) (hS : 2 * S ≤ (d - c) * (d - c + 1)) (hdS : dS ≥ 0) (hc : c ≥ 1) (hd : d ≥ c) :\n  d * (d + 1) + 2 * S ≤ 2 * (d + 1 + dS)^2 := by\n…",
      "source_statement_sha256": "8265de491369519edea5fdb13fb3a4cabe7630fb277927f2ae96de9f9ae713c4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L869",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "log_concave_algebra_upper_two",
      "line": 877,
      "title": "Two upper-tail quadratics are dominated by the squared combined slope",
      "intuition": "For positive complementary distances `k₁,k₂`, the two quadratic upper-tail counts fit beneath twice the square of the sum of their negative first differences.",
      "source_statement": "lemma log_concave_algebra_upper_two (k1 k2 : ℤ) (h1 : k1 ≥ 1) (h2 : k2 ≥ 1) :\n  k1 * (k1 + 1) + k2 * (k2 + 1) ≤ 2 * (-k1 - k2)^2 := by\n…",
      "source_statement_sha256": "46d7907a33ba7096242557679881e2b133332f6b308e8838279c894b0ec639a0",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L877",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "log_concave_algebra_upper_eval",
      "line": 887,
      "title": "The exact upper-upper region inequality",
      "intuition": "When the two upper-region offsets satisfy `U₁,U₂≥-1`, their complementary quadratic counts are bounded by the square of the sum of their negative slopes.",
      "source_statement": "lemma log_concave_algebra_upper_eval (U1 U2 : ℤ) (h1 : U1 ≥ -1) (h2 : U2 ≥ -1) :\n  (U1 + 2) * (U1 + 3) + (U2 + 2) * (U2 + 3) ≤ (- (U1 + 2) - (U2 + 2)) ^ 2 := by\n…",
      "source_statement_sha256": "5b5601f900627a8aae30ac74a7f7142abfb14b658f24b27e372534196526c04f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L887",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_eq_one_reduction",
      "line": 895,
      "title": "Shift a curvature-one square bound back by one degree",
      "intuition": "If neighbouring first differences differ by one and the earlier sequence value is reconstructed from the later one, a square bound at the later degree implies the required square bound two indices earlier.",
      "source_statement": "lemma diff2_eq_one_reduction (H_d_minus_1 H_d_plus_1 diff_H_d diff_H_d_plus_1 : ℤ) :\n  diff_H_d_plus_1 - diff_H_d = 1 →\n  H_d_plus_1 - diff_H_d_plus_1 - diff_H_d = H_d_minus_1 →\n  H_d_plus_1 ≤ diff_H_d_plus_1 ^ 2 →\n  H_d_minus_1 ≤ diff_H_d ^ 2 := by\n…",
      "source_statement_sha256": "f4a39361a730426c44aae018114245f3a33875dff368d1b7aefb7b588a5355e9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L895",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_seq_identity",
      "line": 905,
      "title": "Reconstruct H(d−1) from H(d+1) and two first differences",
      "intuition": "For positive `d`, the earlier value equals the later value minus the first differences at `d+1` and `d`.",
      "source_statement": "lemma H_seq_identity {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0) :\n  H_seq (m1:=m1) (m2:=m2) x (d + 1) - diff (H_seq (m1:=m1) (m2:=m2) x) (d + 1) - diff (H_seq (m1:=m1) (m2:=m2) x) d = H_seq (m1:=m1) (m2:=m2) x (d - 1) := by\n…",
      "source_statement_sha256": "f7d3855918d1b0106002465c9410ae6cf82869f0aae84935828439c981ba3a4b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L905",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "H_seq_diff_eq",
      "line": 929,
      "title": "The first difference of H is the sum of the two component differences",
      "intuition": "This theorem packages the decomposition `H=f+g` at the level of first differences, so later case proofs can reason about the principal and residual slopes separately.",
      "source_statement": "lemma H_seq_diff_eq {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) :\n  diff (H_seq (m1:=m1) (m2:=m2) x) d = diff (fun k => (h_seq m1 k : ℤ)) d + diff (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) d := by\n…",
      "source_statement_sha256": "c3335eb03c1170bb0303fbc75beb197528ab84d696aeaf1b7adb0923c4ae0116",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L929",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies_inl_low",
      "line": 950,
      "title": "Curvature-one case A closes immediately in the principal lower region",
      "intuition": "When the principal summand has curvature one in its lower region, the shift has not activated yet, so the desired square bound reduces to the exact lower quadratic formula.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies_inl_low {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 1)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 0)\n    (h1_low : d + 1 ≤ m1 0 ∧ d + 1 ≤ m1 1 ∧ d + 1 ≤ m1 2) :\n  H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "a463ec26caf3bfc5b54bf1920172f881fce46173b9e0be313ad4883659762252",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L950",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "inl_high_H_eval_bounds",
      "line": 1326,
      "title": "Curvature-one case A satisfies the square bound in the principal upper region",
      "intuition": "The principal upper-tail value is exact, while the residual value has a matching quadratic envelope and nonpositive slope; together they fit under the square of the combined slope.",
      "source_statement": "lemma inl_high_H_eval_bounds {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1_high : d + 1 ≥ m1 0 + m1 1 + 2 ∧ d + 1 ≥ m1 1 + m1 2 + 2 ∧ d + 1 ≥ m1 0 + m1 2 + 2 ∧ d + 1 ≤ m1 0 + m1 1 + m1 2 + 2) :\n  H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "49299172fd8a343e0abc183ef7964dc39a7b4fcf6e20f7c5d38275d2d2ff1227",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1326",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies_inl_high",
      "line": 1373,
      "title": "Case A upper region is discharged by the assembled evaluation bound",
      "intuition": "This wrapper identifies the upper-region branch of the principal-curvature case with the already proved `inl_high_H_eval_bounds` estimate.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies_inl_high {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 1)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 0)\n    (h1_high : d + 1 ≥ m1 0 + m1 1 + 2 ∧ d + 1 ≥ m1 1 + m1 2 + 2 ∧ d + 1 ≥ m1 0 + m1 2 + 2 ∧ d + 1 ≤ m1 0 + m1 1 + m1 2 + 2) :\n  H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "8042df27efd452067096ab44a20acf072f223566957b9ab30d93713daa52ed65",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1373",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies_inl",
      "line": 1381,
      "title": "Case A is complete after splitting the principal sequence into lower and upper regions",
      "intuition": "If the principal component contributes curvature one and the residual contributes zero, the monomial curvature classifier leaves exactly the lower and upper cases already proved.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies_inl {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 1)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 0) :\n  H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "f73b5174aa30836a4d2ff1fb3d885a1489669dfbf49f955d769704bee418006e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1381",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "shifted_h_seq_eval_lower",
      "line": 1393,
      "title": "Exact value of the shifted residual in its lower region",
      "intuition": "Once the shift is active and the translated degree lies below every exponent ceiling, twice the residual value is the usual lower-region quadratic at `d-c`.",
      "source_statement": "lemma shifted_h_seq_eval_lower (m : Monomial 3) (c d : ℕ) (hc : d ≥ c) (h_low : d - c ≤ m 0 ∧ d - c ≤ m 1 ∧ d - c ≤ m 2) :\n  shifted_h_seq m c d * 2 = (d - c + 1 : ℤ) * (d - c + 2 : ℤ) := by\n…",
      "source_statement_sha256": "41d781d5e5f01bf72ff4523f3a6ea156a9be433e28ef73d1b61e1deec324954e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1393",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff_shifted_h_seq_eval_lower",
      "line": 1399,
      "title": "Exact first difference of the shifted residual in its lower region",
      "intuition": "In the active lower region, the residual slope is `d-c+1`, including the activation boundary where the preceding shifted value is still zero.",
      "source_statement": "lemma diff_shifted_h_seq_eval_lower (m : Monomial 3) (c d : ℕ) (hd : d > 0) (hc : d ≥ c) (h_low : d - c ≤ m 0 ∧ d - c ≤ m 1 ∧ d - c ≤ m 2) :\n  diff (shifted_h_seq m c) d = (d - c + 1 : ℤ) := by\n…",
      "source_statement_sha256": "72d6872fabb7f249b14e85de1035f70ad5fd18854c10e572a3a988e3f8a9701e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1399",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "inr_low_H_eval_bounds",
      "line": 1663,
      "title": "Case B lower region supplies the four value-and-slope bounds",
      "intuition": "For a lower-region residual curvature-one event, this lemma assembles the principal growth bound, a lower bound on its slope, and exact residual value and slope formulas.",
      "source_statement": "lemma inr_low_H_eval_bounds {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (hz : Z_calc (m1 0 + 1) (m1 1 + 1) (m1 2 + 1) (d + 1) = 0) (hd_c : d + 1 ≥ m1 x + 1)\n    (h2_low : d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 1 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 2) :\n  (h_seq m1 (d + 1) : ℤ) ≤ (diff (fun k => (h_seq m1 k : ℤ)) (d + 1)) * (d + 2 : ℤ) ∧\n  diff (fun k => (h_seq m1 k : ℤ)) (d + 1) ≥ m1 x + 1 ∧\n  shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1) (d + 1) * 2 = (d + 1 - (m1 x + 1) + 1 : ℤ) * (d + 1 - (m1 x + 1) + 2 : ℤ) ∧\n  diff (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = (d + 1 - (m1 x + 1) + 1 : ℤ) := by\n…",
      "source_statement_sha256": "a7d609ee3e949fe10d5661c4b77822458e890ce56e4eda68a2c4f34588a65092",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1663",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "inr_low_reduction_goal_helper",
      "line": 1692,
      "title": "Case B lower region proves the later-degree square bound",
      "intuition": "The principal estimate and exact lower residual formula combine into `H(d+1)≤(ΔH(d+1))²`, which is the premise needed by the one-step reduction.",
      "source_statement": "lemma inr_low_reduction_goal_helper {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 0)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 1)\n    (h2_low : d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 1 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 2) :\n    H_seq (m1:=m1) (m2:=m2) x (d + 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) (d + 1))^2 := by\n…",
      "source_statement_sha256": "be1e2b8c0622f25ff130e5a9ab35441d55e127d361bdd8df031bfbcf5daa423d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1692",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies_inr_low",
      "line": 1739,
      "title": "Case B lower region transfers its later-degree bound back to d−1",
      "intuition": "After proving the square bound at `d+1`, the curvature-one reduction gives the exact earlier-degree inequality required for log-concavity.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies_inr_low {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 0)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 1)\n    (h2_low : d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 1 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 2) :\n  H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "5a153e0390f9ab5f4d2be8ed7e74f9813cf9269a171616e59a95eba2fa49524b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1739",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "inr_high_H_eval_bounds",
      "line": 1783,
      "title": "Case B upper region supplies the residual upper-tail formulas",
      "intuition": "When the residual component has curvature one in its upper region, both its value and slope are controlled by the complementary offset while the principal slope is nonpositive.",
      "source_statement": "lemma inr_high_H_eval_bounds {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (hz : Z_calc (m1 0 + 1) (m1 1 + 1) (m1 2 + 1) (d + 1) = 0) (hd_c : d + 1 ≥ m1 x + 1)\n    (h2_high : d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + 2 ∧ d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2 ∧ d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 2 + 2 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2) :\n  (h_seq m1 (d + 1) : ℤ) * 2 ≤ (m1 0 + m1 1 + m1 2 + 1 - (d + 1 : ℤ)) * (m1 0 + m1 1 + m1 2 + 2 - (d + 1 : ℤ)) ∧\n  diff (fun k => (h_seq m1 k : ℤ)) (d + 1) ≤ 0 ∧\n  shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1) (d + 1) * 2 = (m1 0 + m1 1 + m1 2 + 1 - (d + 1 : ℤ)) * (m1 0 + m1 1 + m1 2 + 2 - (d + 1 : ℤ)) ∧\n  diff (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = (d + 1 : ℤ) - (m1 0 + m1 1 + m1 2 + 2) ∧\n  (d + 1 : ℤ) ≤ m1 0 + m1 1 + m1 2 + 2 := by\n…",
      "source_statement_sha256": "20a48b532bdafd8cdba29f35c76ddb9a3df639d6f26ec7968247dd3fb0fb67c9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1783",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "inr_high_reduction_goal_helper",
      "line": 1840,
      "title": "Case B upper region proves the later-degree square bound",
      "intuition": "The two upper-tail envelopes fit beneath the square of the combined negative slope, establishing the later-degree hypothesis for the curvature-one reduction.",
      "source_statement": "lemma inr_high_reduction_goal_helper {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 0)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 1)\n    (h2_high : d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + 2 ∧ d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2 ∧ d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 2 + 2 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2) :\n  H_seq (m1:=m1) (m2:=m2) x (d + 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) (d + 1))^2 := by\n…",
      "source_statement_sha256": "dbff4295ca443820dec54a4f5e2e9be4bea5dbccb02a190607b76a7ffccdc1e9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1840",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies_inr_high",
      "line": 1895,
      "title": "Case B upper region transfers its later-degree bound back to d−1",
      "intuition": "This is the upper-region analogue of the lower residual case: prove the later square bound, then apply the same exact curvature-one reduction.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies_inr_high {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 0)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 1)\n    (h2_high : d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + 2 ∧ d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2 ∧ d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 2 + 2 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2) :\n  H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "2c5054dcd45e60e9e4746e50cfd132ecab1d19a0564e369c72345b44fda49ce2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1895",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies_inr",
      "line": 1905,
      "title": "Case B is complete after classifying the translated residual sequence",
      "intuition": "Residual curvature one first proves that the shift is active, then the unshifted reduced monomial lies in either its lower or upper critical region.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies_inr {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1 : diff2 (fun k => (h_seq m1 k : ℤ)) (d + 1) = 0)\n    (h2 : diff2 (shifted_h_seq (reduced_monomial m1 m2 x) (m1 x + 1)) (d + 1) = 1) :\n  H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "b680a43dfcdb95456127206a3f5b55f21625f507eaef8eee465ca505ec04b422",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1905",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_one_implies",
      "line": 1917,
      "title": "The full curvature-one implication is the union of cases A and B",
      "intuition": "Whenever `H` has second difference one, exactly one component contributes curvature one; both component choices have now been proved in both critical regions.",
      "source_statement": "lemma diff2_H_seq_eq_one_implies {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2) :\n  diff2 (H_seq (m1:=m1) (m2:=m2) x) (d + 1) = 1 →\n…",
      "source_statement_sha256": "2e255c65c9aaec8fab9bd1f01d31502bacbb2fa62a4c0d340ec6a3c8db872e08",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1917",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h1_high_h2_low_contradiction",
      "line": 1941,
      "title": "In the curvature-two case, principal-upper and residual-lower regions are incompatible",
      "intuition": "Equal total degrees and the distinguished-coordinate shift make the residual lower inequalities contradict one of the principal upper inequalities.",
      "source_statement": "lemma h1_high_h2_low_contradiction {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ)\n  (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n  (h1_high : d + 1 ≥ m1 0 + m1 1 + 2 ∧ d + 1 ≥ m1 1 + m1 2 + 2 ∧ d + 1 ≥ m1 0 + m1 2 + 2 ∧ d + 1 ≤ m1 0 + m1 1 + m1 2 + 2)\n  (h2_low : d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 1 ∧ d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 2) :\n  False := by\n…",
      "source_statement_sha256": "3d1af607e31a02290dd7e131b3bd48783387dc3d16e27d45f0e7fa97e948e44a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1941",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h1_low_h2_high_contradiction",
      "line": 1979,
      "title": "In the curvature-two case, a principal lower region is impossible after shift activation",
      "intuition": "Residual curvature one forces `d+1≥c=m1(x)+1`, while a principal lower region requires `d+1≤m1(x)`.",
      "source_statement": "lemma h1_low_h2_high_contradiction {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ)\n  (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n  (h1_low : d + 1 ≤ m1 0 ∧ d + 1 ≤ m1 1 ∧ d + 1 ≤ m1 2)\n  (h2_high : d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + 2 ∧\n    d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2 ∧\n    d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 2 + 2 ∧\n    d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2) :\n  False := by match x with|0|2|1=>omega",
      "source_statement_sha256": "e8bd26ddb6813e7d3a8a30b05ff9d5bfe03ad8b579566750642091e465c358c6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1979",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_two_implies_inr_inr",
      "line": 1988,
      "title": "The curvature-two upper-upper case satisfies the doubled square bound",
      "intuition": "Once both components are in their upper regions, their exact complementary quadratics are bounded by the square of the sum of their negative first differences.",
      "source_statement": "lemma diff2_H_seq_eq_two_implies_inr_inr {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2)\n    (h1_high : d + 1 ≥ m1 0 + m1 1 + 2 ∧ d + 1 ≥ m1 1 + m1 2 + 2 ∧ d + 1 ≥ m1 0 + m1 2 + 2 ∧ d + 1 ≤ m1 0 + m1 1 + m1 2 + 2)\n    (h2_high : d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + 2 ∧\n      d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2 ∧\n      d + 1 - (m1 x + 1) ≥ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 2 + 2 ∧\n      d + 1 - (m1 x + 1) ≤ (reduced_monomial m1 m2 x) 0 + (reduced_monomial m1 m2 x) 1 + (reduced_monomial m1 m2 x) 2 + 2) :\n    2 * H_seq (m1:=m1) (m2:=m2) x (d - 1) ≤ (diff (H_seq (m1:=m1) (m2:=m2) x) d)^2 := by\n…",
      "source_statement_sha256": "7b3442b7ba4578e9e2cbd289418a9b34ee4b345c8066420ab3205ee7b4804f65",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L1988",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "h1_low_contradiction",
      "line": 2051,
      "title": "Shift activation alone rules out the principal lower region",
      "intuition": "If `d+1` reaches the shift threshold `m1(x)+1`, it cannot simultaneously remain at most `m1(x)` as a lower-region hypothesis demands.",
      "source_statement": "lemma h1_low_contradiction {m1 : Monomial 3} (x : Fin 3) (d : ℕ)\n  (h1_low : d + 1 ≤ m1 0 ∧ d + 1 ≤ m1 1 ∧ d + 1 ≤ m1 2)\n  (h_ge : d + 1 ≥ m1 x + 1) : False := by\n…",
      "source_statement_sha256": "ddaf40106f8d316c46ca41b3b81c000a9f4df1b52bd8e5a2b78baed245f32c25",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L2051",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean",
      "declaration": "diff2_H_seq_eq_two_implies",
      "line": 2074,
      "title": "The full curvature-two implication leaves only the upper-upper case",
      "intuition": "Total curvature two forces both summands to have curvature one; activation rules out a lower principal region, and the remaining mixed region is contradictory.",
      "source_statement": "lemma diff2_H_seq_eq_two_implies {m1 m2 : Monomial 3} (x : Fin 3) (d : ℕ) (hd : d > 0)\n    (h_gt : m2 x > m1 x) (h_le : ∀ y, y ≠ x → m2 y ≤ m1 y) (h_deg : totalDeg m1 = totalDeg m2) :\n  diff2 (H_seq (m1:=m1) (m2:=m2) x) (d + 1) = 2 →\n…",
      "source_statement_sha256": "7d15921465b72edceb7f94aaa6c9b68c6b1f7ad63ba910fd6cf5c9540f339231",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_5.lean#L2074",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.IsGood",
      "line": 48,
      "title": "Property P: no member divides the sum of two larger distinct members",
      "intuition": "`IsGood A` means that `A` is infinite and whenever `a,b,c ∈ A`, `a ∣ b+c`, `a<b`, and `a<c`, the two larger terms must coincide. Equivalently, no three distinct members realize the forbidden divisibility pattern.",
      "source_statement": "abbrev IsGood (A : Set ℕ) : Prop := A.Infinite ∧\n…",
      "source_statement_sha256": "40e146e3b1209cfb96e5971f181be3dbaac6d1e8a89c4cccf57d357ae7d68e7c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L48",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q",
      "line": 95,
      "title": "A recursively chosen increasing sequence of primes",
      "intuition": "`q 0 = 5`; each `q (m+1)` is a prime chosen strictly above `q m` but no larger than `2*q m+2`.",
      "source_statement": "noncomputable def q : ℕ → ℕ\n| 0 => 5\n| m + 1 => Classical.choose (exists_prime_bounded (q m))",
      "source_statement_sha256": "68f5af535ddb3bc3af39b332bb681f1ed6431ff4c539c2555d9434a201d65fbb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.K",
      "line": 117,
      "title": "The accumulated modulus from all earlier primes",
      "intuition": "`K 0 = 729 = 3^6`, and `K (m+1) = K m * q m`.",
      "source_statement": "noncomputable def K : ℕ → ℕ\n| 0 => 729\n| m + 1 => K m * q m",
      "source_statement_sha256": "17f8601510b7c06116a59e6b12de5a64229980b4e1a890870048cf37336c6c91",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L117",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.P",
      "line": 121,
      "title": "The full period for block $m$",
      "intuition": "`P m = K m * q m`, combining the accumulated earlier-prime modulus with the block's new prime.",
      "source_statement": "noncomputable def P (m : ℕ) : ℕ := K m * q m",
      "source_statement_sha256": "7f656be75a3679c3d747d6be601ea368ac060ea583c9eb6e93e4c577eb691091",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L121",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.M",
      "line": 123,
      "title": "The rapidly growing scale separating the blocks",
      "intuition": "`M m` is the eighth power of `P m`.",
      "source_statement": "noncomputable def M (m : ℕ) : ℕ := (P m)^8",
      "source_statement_sha256": "bcf6b352798f8c9f5eaf3fd8f965d23d2965e311cbe76aa01dc5005c25afdf09",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L123",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.MyGoodSet",
      "line": 125,
      "title": "The Lean proof's congruence-block witness",
      "intuition": "`MyGoodSet` contains a natural number `n` when some block index `m` places it near `M m`, below a controlled progression cutoff, with `n ≡ 1 (mod K m)` and `n ≡ 0 (mod q m)`.",
      "source_statement": "def MyGoodSet : Set ℕ :=\n…",
      "source_statement_sha256": "11aa35062e4fd257dae70bf065bef91f4f96fb6f54cd949995514d082182974b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L125",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.M_increasing",
      "line": 147,
      "title": "Consecutive block scales are more than a factor of two apart",
      "intuition": "`M_increasing m` proves `2*M m < M (m+1)`.",
      "source_statement": "lemma M_increasing (m : ℕ) : 2 * M m < M (m + 1) := by\n…",
      "source_statement_sha256": "521574d7594c1875f7f1ae0363fc9946b85769f4bc281518dfda585638e416e7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L147",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.m_le_of_lt",
      "line": 172,
      "title": "Numerical order respects block order",
      "intuition": "If `a` lies in block `ma`, `b` lies in block `mb`, and `a<b`, then `ma≤mb`.",
      "source_statement": "lemma m_le_of_lt {a b ma mb : ℕ} (ha : M ma ≤ a) (_ : a ≤ 2 * M ma)\n  (_ : M mb ≤ b) (hb2 : b ≤ 2 * M mb) (hab : a < b) : ma ≤ mb := by\n…",
      "source_statement_sha256": "7d002e1ee292d6559089625aee98fcba2260d4f07717d685f7b5cafb1304afbf",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L172",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.mod_q_of_lt",
      "line": 192,
      "title": "A later block is $1$ modulo every earlier block prime",
      "intuition": "If `b ≡ 1 (mod K mb)` and `ma<mb`, then `b ≡ 1 (mod q ma)`.",
      "source_statement": "lemma mod_q_of_lt {b mb ma : ℕ} (hb : b % K mb = 1) (hma : ma < mb) : b % q ma = 1 := by\n…",
      "source_statement_sha256": "e6bdce19e1f3cf4c3352ff45ac43c28853740d240e72f675a996a01daae4ca18",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L192",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.K_q_coprime",
      "line": 261,
      "title": "The accumulated modulus and the new prime are coprime",
      "intuition": "`K_q_coprime m` proves `Nat.Coprime (K m) (q m)`.",
      "source_statement": "lemma K_q_coprime (m : ℕ) : Nat.Coprime (K m) (q m) := by\n…",
      "source_statement_sha256": "dcd3a90208785f5542673ad46d2ae0ce809fec88021c8dcb665d8fa7b761c76b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L261",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.r_val",
      "line": 298,
      "title": "A Chinese-remainder solution for the two block residues",
      "intuition": "`r_val m` selects the natural-number representative produced by the Chinese remainder theorem for residue `1` modulo `K m` and residue `0` modulo `q m`.",
      "source_statement": "noncomputable def r_val (m : ℕ) : ℕ := (Nat.chineseRemainder (K_q_coprime m) 1 0).val",
      "source_statement_sha256": "4c564658f13adf1bdea953a0755b0f605496ed901fc0c93ccdf67aaccedee2ca",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L298",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.x_val",
      "line": 300,
      "title": "The canonical block offset modulo the full period",
      "intuition": "`x_val m` reduces `r_val m` modulo `P m = K m*q m`.",
      "source_statement": "noncomputable def x_val (m : ℕ) : ℕ := (r_val m) % P m",
      "source_statement_sha256": "4fbabd403dd9e83c1419b2ca7e51812bf32e1d4f4b5bae9a101aec295b95fb0d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L300",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.x_val_mod_K",
      "line": 304,
      "title": "The canonical offset is $1$ modulo the accumulated modulus",
      "intuition": "`x_val_mod_K m` establishes `x_val m % K m = 1`.",
      "source_statement": "lemma x_val_mod_K (m : ℕ) : (x_val m) % K m = 1 := by\n…",
      "source_statement_sha256": "6a837ef87158dbc78dd966fa4c2b1896cda74fa909290cbc80550ab6621cf976",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L304",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.x_val_mod_q",
      "line": 314,
      "title": "The canonical offset is $0$ modulo the block's new prime",
      "intuition": "`x_val_mod_q m` establishes `x_val m % q m = 0`.",
      "source_statement": "lemma x_val_mod_q (m : ℕ) : (x_val m) % q m = 0 := by\n…",
      "source_statement_sha256": "097d16ca024bcc9832ce0b96a232e832d31a211562ce079756e6556cd126fabb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L314",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.my_good_set_infinite",
      "line": 398,
      "title": "The congruence-block witness is infinite",
      "intuition": "Beyond every natural bound `N`, the proof constructs a member of `MyGoodSet` in block `N+1`.",
      "source_statement": "lemma my_good_set_infinite : MyGoodSet.Infinite := by\n…",
      "source_statement_sha256": "37fd0ab42847c1d26c5ca830ab1d88753fb67072be762832140ee9f278f69637",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L398",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.my_good_set_is_good",
      "line": 409,
      "title": "The congruences exclude every forbidden divisibility triple",
      "intuition": "`my_good_set_is_good` proves that `MyGoodSet` is infinite and satisfies property P.",
      "source_statement": "lemma my_good_set_is_good : IsGood MyGoodSet := by\n…",
      "source_statement_sha256": "d6ea40c13cd26f904bd888f9ede0be9732ecdbe8d69737bd161d31d48214d511",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L409",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.f_val",
      "line": 519,
      "title": "Enumerate the arithmetic progression inside one block",
      "intuition": "`f_val m i = M m + x_val m + i*P m` is the `i`th candidate in block `m`.",
      "source_statement": "noncomputable def f_val (m i : ℕ) : ℕ := M m + x_val m + i * P m",
      "source_statement_sha256": "73b5cc124d9c0ee295aa844477dabc10951da4df2f400290e96207ae5166152c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L519",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_pow_four_bound",
      "line": 563,
      "title": "The next prime grows slowly relative to the current period",
      "intuition": "The inequality `q (m+1)^4 ≤ P m^3` supplies the quantitative room needed to fit many progression points into block `m`.",
      "source_statement": "lemma q_pow_four_bound (m : ℕ) : q (m + 1) ^ 4 ≤ P m ^ 3 := by\n…",
      "source_statement_sha256": "675bc5f8d969b4dca2a5e8ead436f85abd9552316e83f7b82fd3890e5f536bf7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L563",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.f_val_in_good_set",
      "line": 631,
      "title": "Every bounded progression index produces a valid block member",
      "intuition": "For `i < P (m+1)^4/2`, the point `f_val m i` belongs to `MyGoodSet ∩ [M m,2*M m]`.",
      "source_statement": "lemma f_val_in_good_set (m i : ℕ) (hi : i < (P (m + 1))^4 / 2) : f_val m i ∈ MyGoodSet ∩ Set.Icc (M m) (2 * M m) := by\n…",
      "source_statement_sha256": "3d7f2d2b5fdaa88e65f9a5335e37c1917de07fb2c2585cdaaf634c702f4f809d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L631",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.count_interval",
      "line": 723,
      "title": "Each block contains at least half a fourth power of points",
      "intuition": "Block `m` contains at least `P (m+1)^4/2` members of `MyGoodSet` between `M m` and `2*M m`.",
      "source_statement": "lemma count_interval (m : ℕ) : P (m + 1) ^ 4 / 2 ≤ (MyGoodSet ∩ Set.Icc (M m) (2 * M m)).ncard := by\n…",
      "source_statement_sha256": "1ab357dc475c3c7ae7457e8fd70c22a43e2318434e0cddfeb942f882a0d7721f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L723",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_ineq",
      "line": 831,
      "title": "The next scale is controlled by the square of the block count",
      "intuition": "`density_ineq m` proves `2*M (m+1) ≤ 9*(P (m+1)^4/2)^2`.",
      "source_statement": "lemma density_ineq (m : ℕ) : 2 * M (m + 1) ≤ 9 * (P (m + 1) ^ 4 / 2) ^ 2 := by\n…",
      "source_statement_sha256": "1812c0bb67d2f969c67942420d589f43f323ea8bf528f499a5af7bd2609fa066",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L831",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.exists_M_interval",
      "line": 847,
      "title": "Locate every sufficiently large $N$ between consecutive block scales",
      "intuition": "For `N≥M 7`, there is an `m≥7` with `M m≤N<M (m+1)`.",
      "source_statement": "lemma exists_M_interval (N : ℕ) (hN : M 7 ≤ N) : ∃ m ≥ 7, M m ≤ N ∧ N < M (m + 1) := by\n…",
      "source_statement_sha256": "30adb548b56f2880d5d0fa14e03e93e30015f6994402f60226f9ee5e1c3acd13",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L847",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_bound_N",
      "line": 901,
      "title": "A uniform lower bound for every sufficiently large cutoff",
      "intuition": "For every `N≥M 7`, the normalized count `|MyGoodSet∩[1,N]|/√N` is at least `1/3`.",
      "source_statement": "lemma density_bound_N (N : ℕ) (hN : M 7 ≤ N) : (1 / 3 : ℝ) ≤ (MyGoodSet ∩ Set.Icc 1 N).ncard / (N : ℝ).sqrt := by\n…",
      "source_statement_sha256": "3b64643b5f7f98b6b3c33b111810e241ca0314662602bbf7ff82c31b18218a1e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L901",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.g_val",
      "line": 1065,
      "title": "Encode a good-set member by its block and progression cell",
      "intuition": "`g_val n` pairs the block index containing `n` with the quotient `(n-M k)/P k` inside that block.",
      "source_statement": "noncomputable def g_val (n : ℕ) : ℕ × ℕ :=\n…",
      "source_statement_sha256": "e14ee3a9c89b66a4c3ac0b38d39904512deb225689a61ca97c643e7efa0adb83",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1065",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.g_val_inj",
      "line": 1068,
      "title": "The block-and-cell coordinate is injective on the witness set",
      "intuition": "Two members of `MyGoodSet` with the same `g_val` must be the same natural number.",
      "source_statement": "lemma g_val_inj : Set.InjOn g_val MyGoodSet := by\n…",
      "source_statement_sha256": "096a5e1a3de44210c8caa7027a4c1833b807b68f5e61aabaac34503e36279974",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1068",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.target_set",
      "line": 1090,
      "title": "A finite coordinate box for all earlier blocks",
      "intuition": "`target_set m` contains pairs `(k,i)` with `k<m` and `0≤i≤P (k+1)^4/2`.",
      "source_statement": "noncomputable def target_set (m : ℕ) : Finset (ℕ × ℕ) :=\n…",
      "source_statement_sha256": "858d34c3698e461a490135a8a5c74d376c0ba893e36be98289b4f9ba2a4ffaf9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1090",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.count_upper_bound",
      "line": 1161,
      "title": "The witness count at a block scale is at most twice a fourth power",
      "intuition": "Up to `M m`, `MyGoodSet` has at most `2*P m^4` members.",
      "source_statement": "lemma count_upper_bound (m : ℕ) : (MyGoodSet ∩ Set.Icc 1 (M m)).ncard ≤ 2 * P m^4 := by\n…",
      "source_statement_sha256": "bfffe655833f15c4c76b99e66e6419ec296d503198aada214cb84b07cbe557dd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1161",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.exists_f_le_two",
      "line": 1179,
      "title": "The normalized count is at most two infinitely often",
      "intuition": "Along the unbounded subsequence `N=M m`, the ratio `|MyGoodSet∩[1,N]|/√N` is at most `2`.",
      "source_statement": "lemma exists_f_le_two : ∃ᶠ N in Filter.atTop, (MyGoodSet ∩ Set.Icc 1 N).ncard / (N : ℝ).sqrt ≤ 2 := by\n…",
      "source_statement_sha256": "f10276f33d2531f191f6abf6d4f07d5e048bf2f35aa189ce23add6244041e886",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1179",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.my_good_set_density",
      "line": 1214,
      "title": "The normalized counting function has positive lower limit",
      "intuition": "The liminf of `|MyGoodSet∩[1,N]|/√N` is at least `1/3`, hence strictly positive.",
      "source_statement": "lemma my_good_set_density : (0 : ℝ) < Filter.atTop.liminf (fun N => (MyGoodSet ∩ Set.Icc 1 N).ncard / (N : ℝ).sqrt) := by\n…",
      "source_statement_sha256": "46c8089b9dcf346759f8a6c209feed8cd739ee1c927613493733258dc1960c5c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1214",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.exists_prime_bounded",
      "line": 89,
      "title": "Find a controlled next prime",
      "intuition": "For every natural number `n`, there is a prime `p` with `n < p ≤ 2*n+2`.",
      "source_statement": "lemma exists_prime_bounded (n : ℕ) : ∃ p, n < p ∧ p ≤ 2 * n + 2 ∧ p.Prime := by\n…",
      "source_statement_sha256": "9f7c6b95aec43465df1a01e73f549c69016b3196d712200c636853df55624b78",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L89",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_prime",
      "line": 99,
      "title": "Every chosen modulus is prime",
      "intuition": "Each value `q m` is prime.",
      "source_statement": "lemma q_prime (m : ℕ) : (q m).Prime := by\n…",
      "source_statement_sha256": "85f996f1bd4d4f18d9701aec20ecf6f0576e2223021e15cda7ec9f17b5f90c6c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L99",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_gt",
      "line": 106,
      "title": "Successive chosen primes strictly increase",
      "intuition": "`q m < q (m+1)` for every block index `m`.",
      "source_statement": "lemma q_gt (m : ℕ) : q m < q (m + 1) := by\n…",
      "source_statement_sha256": "2a779cd0a2da53d61bd100072002465c3c130b7eaad970be29d26a7a2a718b2a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_ge_5",
      "line": 110,
      "title": "Every chosen prime is at least five",
      "intuition": "The increasing sequence `q` never falls below its initial value `5`.",
      "source_statement": "lemma q_ge_5 (m : ℕ) : 5 ≤ q m := by\n…",
      "source_statement_sha256": "6d333ea4b5ecd1abf607a10f9307d0297cb9768674bf2656ad53630d942e82f4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.K_pos",
      "line": 129,
      "title": "The accumulated modulus is positive",
      "intuition": "`K m` is positive for every block index.",
      "source_statement": "lemma K_pos (m : ℕ) : 0 < K m := by\n…",
      "source_statement_sha256": "6f977bb80d9b276dde01890cf0f08bc5a7bb230b0e79b39bf2c35ecf2081bc9f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L129",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.P_pos",
      "line": 136,
      "title": "The joint period is positive",
      "intuition": "`P m = K m*q m` is always positive.",
      "source_statement": "lemma P_pos (m : ℕ) : 0 < P m := Nat.mul_pos (K_pos m) (by linarith [q_ge_5 m])",
      "source_statement_sha256": "7db74b95aaf0e8e57046d78fb76a3755babd92d451ae1beab21e2668d2aba185",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L136",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.M_succ",
      "line": 138,
      "title": "Successive block scales differ by an eighth power",
      "intuition": "`M (m+1) = M m * q (m+1)^8`.",
      "source_statement": "lemma M_succ (m : ℕ) : M (m + 1) = M m * (q (m + 1))^8 := by\n…",
      "source_statement_sha256": "6fd12971496ad7ead722ba1b36b947a5ea79672403d200810796873062f7a673",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L138",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.pow_eight_le",
      "line": 144,
      "title": "Eighth powers preserve order on naturals",
      "intuition": "If `a≤b`, then `a^8≤b^8`.",
      "source_statement": "lemma pow_eight_le {a b : ℕ} (h : a ≤ b) : a^8 ≤ b^8 := by\n…",
      "source_statement_sha256": "729d206501b9a5661de07966c8d7a7bd18fb5e1e921636ad55dced6f2accd385",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L144",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.M_monotone",
      "line": 159,
      "title": "Block scales are monotone",
      "intuition": "If `m1≤m2`, then `M m1≤M m2`.",
      "source_statement": "lemma M_monotone {m1 m2 : ℕ} (h : m1 ≤ m2) : M m1 ≤ M m2 := by\n…",
      "source_statement_sha256": "1c0fe9ec5150828bd3f3ce408afd807caaa8243c73c4aff6202296dd13a10a06",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L159",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_dvd_K",
      "line": 181,
      "title": "Every earlier prime is absorbed into later accumulated moduli",
      "intuition": "When `ma<mb`, the prime `q ma` divides `K mb`.",
      "source_statement": "lemma q_dvd_K {ma mb : ℕ} (hma : ma < mb) : q ma ∣ K mb := by\n…",
      "source_statement_sha256": "6cb34396f17dff0936603bb74837a4db75d78a4c3f3dff7161d248384759ea90",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L181",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_div_a",
      "line": 214,
      "title": "Zero remainder gives divisibility by the block prime",
      "intuition": "If `a mod q ma = 0`, then `q ma` divides `a`.",
      "source_statement": "lemma q_div_a {a ma : ℕ} (ha : a % q ma = 0) : q ma ∣ a := Nat.dvd_of_mod_eq_zero ha",
      "source_statement_sha256": "da4cae5426ee98eef61d8f23c0752d688217a25f8745e8a4ce9d453b98e3ed21",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L214",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.K_mod_3",
      "line": 216,
      "title": "Every accumulated modulus is divisible by three",
      "intuition": "`K m mod 3 = 0` for every `m`.",
      "source_statement": "lemma K_mod_3 (m : ℕ) : K m % 3 = 0 := by\n…",
      "source_statement_sha256": "00d888497f358a8b4612ca67fd286cfecd6526fbfdc14ee5646ea0d2b8a3201b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L216",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.mod_3_eq_1",
      "line": 226,
      "title": "The block residue forces residue one modulo three",
      "intuition": "If `n mod K m = 1`, then `n mod 3 = 1`.",
      "source_statement": "lemma mod_3_eq_1 {n m : ℕ} (hn : n % K m = 1) : n % 3 = 1 := by\n…",
      "source_statement_sha256": "d979c53d07120e8624f338685e649d57d302b23809c28dc1b09c5dd1e69ee85c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L226",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.prime_dvd_K",
      "line": 237,
      "title": "Prime factors of an accumulated modulus precede the current prime",
      "intuition": "If a prime `p` divides `K m`, then `p<q m`.",
      "source_statement": "lemma prime_dvd_K {p m : ℕ} (hp : p.Prime) (h : p ∣ K m) : p < q m := by\n…",
      "source_statement_sha256": "226faa8ee3d1f78ff01a468b6290a9f12c12fa68d97207362f848518ac639aa0",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L237",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.K_ge_3",
      "line": 272,
      "title": "The accumulated modulus is never smaller than three",
      "intuition": "`3≤K m` for every block index.",
      "source_statement": "lemma K_ge_3 (m : ℕ) : 3 ≤ K m := by\n…",
      "source_statement_sha256": "42158e2f45e282b11a1ba185edcb63c5ebf0ccf5543960cbd0c88989904e26a7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L272",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.P_dvd_M",
      "line": 282,
      "title": "The joint period divides the block scale",
      "intuition": "`P m` divides `M m=P m^8`.",
      "source_statement": "lemma P_dvd_M (m : ℕ) : P m ∣ M m := by\n…",
      "source_statement_sha256": "9575f174a3bfd4acaaad32b21dc958f55e7f029622767015e54195bb24abab1e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L282",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.K_dvd_P",
      "line": 288,
      "title": "The accumulated modulus divides the joint period",
      "intuition": "`K m` divides `P m` because `P m=K m*q m`.",
      "source_statement": "lemma K_dvd_P (m : ℕ) : K m ∣ P m := ⟨q m, rfl⟩",
      "source_statement_sha256": "07c966b997a7bf05eaa2d41d89ebda875ecca2e6aff735c158cccdf16f6357ba",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L288",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_dvd_P",
      "line": 290,
      "title": "The current prime divides the joint period",
      "intuition": "`q m` divides `P m` because `P m=K m*q m`.",
      "source_statement": "lemma q_dvd_P (m : ℕ) : q m ∣ P m := ⟨K m, mul_comm (K m) (q m)⟩",
      "source_statement_sha256": "f38ee647836d4590f59f546c126acd09896916eade840342cb71dc731f8d78a5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L290",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.K_dvd_M",
      "line": 292,
      "title": "The accumulated modulus divides the block scale",
      "intuition": "`K m` divides `M m`.",
      "source_statement": "lemma K_dvd_M (m : ℕ) : K m ∣ M m := dvd_trans (K_dvd_P m) (P_dvd_M m)",
      "source_statement_sha256": "78dfa1923cc48d13e906b4a71b96612dc04c80e6a7a486c6c007134455c1cc87",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L292",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_dvd_M",
      "line": 294,
      "title": "The current prime divides the block scale",
      "intuition": "`q m` divides `M m`.",
      "source_statement": "lemma q_dvd_M (m : ℕ) : q m ∣ M m := dvd_trans (q_dvd_P m) (P_dvd_M m)",
      "source_statement_sha256": "690ba3e51262d36d2f97b8914fb8c876c1c1febb3d5a2958a734cac808c4f4e8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L294",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.pow_four_le",
      "line": 296,
      "title": "Fourth powers preserve order on naturals",
      "intuition": "If `a≤b`, then `a^4≤b^4`.",
      "source_statement": "lemma pow_four_le {a b : ℕ} (h : a ≤ b) : a^4 ≤ b^4 := by gcongr",
      "source_statement_sha256": "c551a5b899bb6faaabcc82ba7ddac30c0157db41b9404cc8132b4f073aaa01c2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L296",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.x_val_lt_P",
      "line": 302,
      "title": "Choose the canonical residue below one period",
      "intuition": "`x_val m` is strictly smaller than `P m`.",
      "source_statement": "lemma x_val_lt_P (m : ℕ) : x_val m < P m := Nat.mod_lt _ (P_pos m)",
      "source_statement_sha256": "e26fb222b89e5d257037c2aafe7ce7d75e3abb19173fa29bfcd71fad53ce5a7c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L302",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.exists_good_in_block",
      "line": 323,
      "title": "Every block contains at least one admissible residue",
      "intuition": "For each `m`, some `n` lies in the `m`th block and satisfies `n ≡ 1 (mod K m)` and `n ≡ 0 (mod q m)`.",
      "source_statement": "lemma exists_good_in_block (m : ℕ) : ∃ n, M m ≤ n ∧ n ≤ 2 * M m ∧ n < M m + (P (m + 1)^4 / 2) * P m ∧ n % K m = 1 ∧ n % q m = 0 := by\n…",
      "source_statement_sha256": "9ac5c7d6546e28d7e3cfea81cd081d1fba9d0f3700af04f8894c2c390f1b68a9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L323",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.M_gt_m",
      "line": 383,
      "title": "Block scales eventually dominate their indices immediately",
      "intuition": "Every index satisfies `m<M m`.",
      "source_statement": "lemma M_gt_m (m : ℕ) : m < M m := by\n…",
      "source_statement_sha256": "f394dc11869c961556c3929309e9518fbde286ebd374a7f71fa943a4dc323edb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L383",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.P_le_M",
      "line": 521,
      "title": "One period fits inside its block scale",
      "intuition": "`P m≤M m=P m^8`.",
      "source_statement": "lemma P_le_M (m : ℕ) : P m ≤ M m := by\n…",
      "source_statement_sha256": "389beec0645fcf39d4ec422c67cf3d7b70ab0646003e6b1df53003ff78b5d60e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L521",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_bound_le",
      "line": 530,
      "title": "An exponential upper bound for the chosen primes",
      "intuition": "`q m+2 ≤ 7*2^m`.",
      "source_statement": "lemma q_bound_le (m : ℕ) : q m + 2 ≤ 7 * 2^m := by\n…",
      "source_statement_sha256": "a53cde9230162a250e32b2efae643c94d9867e84816b0068c15b6f49781247d3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L530",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.P_lower_bound",
      "line": 540,
      "title": "An exponential lower bound for the joint periods",
      "intuition": "`3645*5^m ≤ P m`.",
      "source_statement": "lemma P_lower_bound (m : ℕ) : 3645 * 5^m ≤ P m := by\n…",
      "source_statement_sha256": "ec9a0a58964f84e6db6eed4165949f2093726e8cca285e6d6fbbd5af958d813f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L540",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.q_pow_four_bound_helper",
      "line": 552,
      "title": "Compare the explicit exponential envelopes",
      "intuition": "`38416*16^m ≤ 48427561125*125^m`.",
      "source_statement": "lemma q_pow_four_bound_helper (m : ℕ) : 38416 * 16^m ≤ 48427561125 * 125^m := by\n…",
      "source_statement_sha256": "140516d2f115674f206b61ac7c4096798ebca65e93526c260aa7bfe9504a7b7a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L552",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.f_val_bound",
      "line": 596,
      "title": "Progression offsets occupy at most half a block",
      "intuition": "If `i≤P (m+1)^4/2`, then `i*P m≤M m/2`.",
      "source_statement": "lemma f_val_bound (m i : ℕ) (hi : i ≤ (P (m + 1))^4 / 2) : i * P m ≤ M m / 2 := by\n…",
      "source_statement_sha256": "b89bdfb1686135092e5745bd733808ed4ed53c8b8259ea0d32bc66a33a6f8e9e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L596",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.P_le_M_half",
      "line": 613,
      "title": "Even one full period fits inside half a block",
      "intuition": "`P m≤M m/2`.",
      "source_statement": "lemma P_le_M_half (m : ℕ) : P m ≤ M m / 2 := by\n…",
      "source_statement_sha256": "c25df0d42e3e0de22fdb7b1e92f5f3e75db130be580f46c79272d31ed61e13f8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L613",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.f_val_inj",
      "line": 708,
      "title": "Distinct progression indices give distinct block members",
      "intuition": "The map `i ↦ f_val m i` is injective on the permitted index interval.",
      "source_statement": "lemma f_val_inj (m : ℕ) : Set.InjOn (f_val m) (Set.Ico 0 ((P (m + 1))^4 / 2)) := by\n…",
      "source_statement_sha256": "4b8980812af30b572dcd555e8724494ed7d025c4e93ef89cf6078b235f924750",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L708",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.finset_coe_ncard",
      "line": 715,
      "title": "Set cardinality agrees with finite-set cardinality",
      "intuition": "Viewing a finite set `s` as a `Set` does not change its number of elements.",
      "source_statement": "lemma finset_coe_ncard {α : Type*} (s : Finset α) : (s : Set α).ncard = s.card := by\n…",
      "source_statement_sha256": "a3f0ef64ce8e590cccaede449121c3538eb0a898557f2caf7739b9e978a098a3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L715",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.good_set_inter_finite",
      "line": 718,
      "title": "A bounded slice of the witness set is finite",
      "intuition": "`MyGoodSet ∩ [M m,2*M m]` is finite.",
      "source_statement": "lemma good_set_inter_finite (m : ℕ) : (MyGoodSet ∩ Set.Icc (M m) (2 * M m)).Finite := by\n…",
      "source_statement_sha256": "01204f56a2229cabf10d459cbd2a02c3bd6b4a2441a465aacacf8883d1e37426",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L718",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_ineq_x",
      "line": 764,
      "title": "A floor-division estimate beyond ten",
      "intuition": "For `x≥10`, the inequality `8*x≤9*(x/2)^2` holds in natural arithmetic.",
      "source_statement": "lemma density_ineq_x {x : ℕ} (hx : 10 ≤ x) : 8 * x ≤ 9 * (x / 2)^2 := by\n…",
      "source_statement_sha256": "77b67108473d1e45b3be119f3e81ff76f071424986762b808f6c56ffb6d935e3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L764",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_ineq_x_sq",
      "line": 770,
      "title": "A squared floor-division estimate beyond twenty",
      "intuition": "For `x≥20`, the inequality `2*x^2≤9*(x/2)^2` holds.",
      "source_statement": "lemma density_ineq_x_sq {x : ℕ} (hx : 20 ≤ x) : 2 * x^2 ≤ 9 * (x / 2)^2 := by\n…",
      "source_statement_sha256": "a1e3e705c94f2071970d96d153608d7d7724a219a50523b2daf5e00b5c3fbc67",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L770",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_helper_pow",
      "line": 776,
      "title": "The density envelopes separate exponentially",
      "intuition": "`2151296*32^(m+6) ≤ 3375*125^(m+6)`.",
      "source_statement": "lemma density_helper_pow (m : ℕ) : 2151296 * 32^(m + 6) ≤ 3375 * 125^(m + 6) := by\n…",
      "source_statement_sha256": "81229c70dc5eb901a7500597c3e523d92f57278f6273b07759cb417d30431037",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L776",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_ineq_q_helper",
      "line": 793,
      "title": "Compare prime-growth and period-growth envelopes",
      "intuition": "For `m≥6`, `4*(7*2^(m+1))^5 ≤ (15*5^m)^3`.",
      "source_statement": "lemma density_ineq_q_helper (m : ℕ) (hm : 6 ≤ m) : 4 * (7 * 2^(m + 1))^5 ≤ (15 * 5^m)^3 := by\n…",
      "source_statement_sha256": "757db6a3b6b8af6cc9d308b2a7ff64e1898eec0ef13e00b51b3c36b801a4979b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L793",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_ineq_q",
      "line": 821,
      "title": "Bound the fifth power of the next prime",
      "intuition": "For `m≥6`, `4*q (m+1)^5≤P m^3`.",
      "source_statement": "lemma density_ineq_q (m : ℕ) (hm : 6 ≤ m) : 4 * (q (m + 1))^5 ≤ P m ^ 3 := by\n…",
      "source_statement_sha256": "682d1db7771914c4b1359c3fa4aabbf0ccd32c1e3562f4197bd91db104147522",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L821",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.density_ratio_bound",
      "line": 884,
      "title": "Convert a square count bound into a normalized density bound",
      "intuition": "If `C≤count` and `N≤9*C^2` with `N>0`, then `1/3≤count/√N`.",
      "source_statement": "lemma density_ratio_bound (N C count : ℕ) (h_N_pos : 0 < N) (h_count : C ≤ count) (h_N_bound : N ≤ 9 * C^2) : (1 / 3 : ℝ) ≤ (count : ℝ) / (N : ℝ).sqrt := by\n…",
      "source_statement_sha256": "7bb0d665d275580effdf5a2530f0c819c973e7ab00c1bb3bdcc4b4d94abfa66e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L884",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.sum_P_pow_four",
      "line": 972,
      "title": "Earlier fourth-power periods are dominated by the latest one",
      "intuition": "`∑ k<m, P (k+1)^4 ≤ 2*P m^4`.",
      "source_statement": "lemma sum_P_pow_four (m : ℕ) : ∑ k ∈ Finset.range m, P (k + 1)^4 ≤ 2 * P m^4 := by\n…",
      "source_statement_sha256": "575a67620ffc5a29d7c56702288bec3d282bf52df05d358021840e307601a6fc",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L972",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.dvd_sub_of_mod_eq",
      "line": 989,
      "title": "Equal residues make the difference divisible by the modulus",
      "intuition": "If `a` and `b` have the same remainder modulo `c` and `b≤a`, then `c` divides `a-b`.",
      "source_statement": "lemma dvd_sub_of_mod_eq {a b c x : ℕ} (ha : a % c = x) (hb : b % c = x) (hle : b ≤ a) : c ∣ a - b := by\n…",
      "source_statement_sha256": "acff807d5eb4ba2725c2325aeaa42d5f24898544b38ac16417f44265ffa835cf",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L989",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.div_eq_implies_sub_lt",
      "line": 1002,
      "title": "Equal quotient cells are narrower than one period",
      "intuition": "If `a/P=b/P`, `b≤a`, and `P>0`, then `a-b<P`.",
      "source_statement": "lemma div_eq_implies_sub_lt {a b P : ℕ} (h_div : a / P = b / P) (_hle : b ≤ a) (hP : 0 < P) : a - b < P := by\n…",
      "source_statement_sha256": "b780947c90531514f53a48ddcc514b0ad71b02340531831f4ea416d72f704595",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1002",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.good_set_inj",
      "line": 1010,
      "title": "A block and quotient cell contain at most one admissible number",
      "intuition": "Two numbers with the block's required residues and the same quotient `(n-M m)/P m` must be equal.",
      "source_statement": "lemma good_set_inj {n1 n2 m : ℕ}\n  (hn1_ge : M m ≤ n1) (hn1_modK : n1 % K m = 1) (hn1_modq : n1 % q m = 0)\n  (hn2_ge : M m ≤ n2) (hn2_modK : n2 % K m = 1) (hn2_modq : n2 % q m = 0)\n  (h_div : (n1 - M m) / P m = (n2 - M m) / P m) : n1 = n2 := by\n…",
      "source_statement_sha256": "9120d9805b62d1b334b6af0a4f69c70b16c99cf70208f84517cf9f2867822238",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1010",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.block_unique",
      "line": 1040,
      "title": "A number has only one compatible block index",
      "intuition": "If `n∈[M m,2*M m]` and also `M k≤n<M (k+1)`, then `m=k`.",
      "source_statement": "lemma block_unique {n m k : ℕ} (h1 : M m ≤ n) (h2 : n ≤ 2 * M m) (h3 : M k ≤ n) (h4 : n < M (k + 1)) : m = k := by\n…",
      "source_statement_sha256": "0216dfbc6f0d4627b3afdd4e4e8d4839100b7c2960ebde79ecfd0a68789fa151",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1040",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.g_block",
      "line": 1049,
      "title": "Select the scale interval containing a number",
      "intuition": "`g_block n` is the least `k` with `M k≤n<M (k+1)`, when such a block exists; otherwise it is zero.",
      "source_statement": "noncomputable def g_block (n : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "26ab6dc25440f99ee7ecc1554e18f33fb3ba70c42f02b795c2fbf7341859a05f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1049",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.g_block_spec",
      "line": 1052,
      "title": "The selected block really contains the number",
      "intuition": "Whenever some scale interval contains `n`, the selected index satisfies `M (g_block n)≤n<M (g_block n+1)`.",
      "source_statement": "lemma g_block_spec {n : ℕ} (h : ∃ k, M k ≤ n ∧ n < M (k + 1)) :\n  M (g_block n) ≤ n ∧ n < M (g_block n + 1) := by\n…",
      "source_statement_sha256": "60de23b72cbab95dffb88e7abd693f27f4f2f8e414dfdf2d457fe5c68e751e6c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1052",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.my_good_set_has_block",
      "line": 1058,
      "title": "Every witness-set member lies between consecutive scales",
      "intuition": "If `n∈MyGoodSet`, then some `k` satisfies `M k≤n<M (k+1)`.",
      "source_statement": "lemma my_good_set_has_block {n : ℕ} (hn : n ∈ MyGoodSet) : ∃ k, M k ≤ n ∧ n < M (k + 1) := by\n…",
      "source_statement_sha256": "629a1abe660421b6c85520dcf55e94366d6ecec99da13ed9e13cb6da18d09cd8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1058",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.target_set_card",
      "line": 1093,
      "title": "Bound the finite code space for all earlier blocks",
      "intuition": "The encoding set `target_set m` has at most `2*P m^4` pairs.",
      "source_statement": "lemma target_set_card (m : ℕ) : (target_set m).card ≤ 2 * P m^4 := by\n…",
      "source_statement_sha256": "e0fabf963bc7cdca57fb3377c4b4be582f05396f0f144f047749a73a8e142482",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1093",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.g_val_maps_to",
      "line": 1118,
      "title": "Every early witness-set member has a bounded block code",
      "intuition": "If `n∈MyGoodSet∩[1,M m]`, then `g_val n∈target_set m`.",
      "source_statement": "lemma g_val_maps_to (m : ℕ) {n : ℕ} (hn : n ∈ MyGoodSet ∩ Set.Icc 1 (M m)) :\n  g_val n ∈ target_set m := by\n…",
      "source_statement_sha256": "62cc41dd6d4ba6469d231a149b8d643e8b457ab6d5f7017eca499e6ab4a4062c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1118",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.bdd_above_of_frequently_le",
      "line": 1206,
      "title": "A frequent upper bound caps every eventual lower bound",
      "intuition": "If `f N≤c` frequently, then the set of real numbers that are eventually below `f N` is bounded above by `c`.",
      "source_statement": "lemma bdd_above_of_frequently_le {f : ℕ → ℝ} {c : ℝ} (h : ∃ᶠ N in Filter.atTop, f N ≤ c) :\n  BddAbove {a | ∀ᶠ N in Filter.atTop, a ≤ f N} := by\n…",
      "source_statement_sha256": "d7d6395604a1b5b07b18f29399478b8017d3235745ac1829c1030bef7ae40142",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1206",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.i.lean",
      "declaration": "Erdos12.target_theorem_0",
      "line": 1228,
      "title": "Erdős problem 12(i): a dense infinite set avoiding the divisibility pattern",
      "intuition": "There exists an infinite set `A` of natural numbers with no distinct `a,b,c` in `A` satisfying `a < b`, `a < c`, and `a ∣ b+c`, while `|A ∩ [1,N]|/√N` has positive lower limit.",
      "source_statement": "theorem target_theorem_0\n  : answer(True) ↔ ∃ (A : Set ℕ), IsGood A ∧ (0 : ℝ) < Filter.atTop.liminf (fun N => (A ∩ Icc 1 N).ncard / (N : ℝ).sqrt) := by\n…",
      "source_statement_sha256": "9760b9e704c6463cdd5dac8bbeea26603120aa6528709b03e970b813574801a6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.i.lean#L1228",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.ii.lean",
      "declaration": "Erdos12.target_theorem_0",
      "line": 762,
      "title": "Erdős problem 12(ii): no universal power saving for every admissible set",
      "intuition": "The proposed universal exponent `c > 0` is false: one can construct an admissible infinite set whose counting function is at least `N^(1-c)` eventually for every chosen positive `c`. The theorem packages that negative answer as the failure of the asserted uniform sparse-subsequence bound.",
      "source_statement": "theorem target_theorem_0\n  : answer(False) ↔ ∃ c > (0 : ℝ), ∀ (A : Set ℕ), IsGood A → {N : ℕ | (A ∩ Icc 1 N).ncard < (N : ℝ) ^ (1 - c)}.Infinite := by\n…",
      "source_statement_sha256": "721811088140b6df99c3ddbd5969d0e083bfd6a90b648c0718a4b96eeba523d6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.ii.lean#L762",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_12.parts.ii.lean",
      "declaration": "Erdos12.IsGood",
      "line": 48,
      "title": "Property P: no member divides the sum of two larger distinct members",
      "intuition": "`IsGood A` means that `A` is infinite and whenever `a,b,c ∈ A`, `a ∣ b+c`, `a<b`, and `a<c`, the two larger terms must coincide. Equivalently, no three distinct members realize the forbidden divisibility pattern.",
      "source_statement": "abbrev IsGood (A : Set ℕ) : Prop := A.Infinite ∧\n…",
      "source_statement_sha256": "40e146e3b1209cfb96e5971f181be3dbaac6d1e8a89c4cccf57d357ae7d68e7c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_12.parts.ii.lean#L48",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.zero_in_A",
      "line": 91,
      "title": "Zero belongs to the base-3 digit set",
      "intuition": "The natural number `0` belongs to `A` because its base-3 expansion contains no digit outside `{0,1}`.",
      "source_statement": "lemma zero_in_A : 0 ∈ A := by\n…",
      "source_statement_sha256": "0c9fdfac90a968065e997f20cf16d34cf9a373e4de1ceae9e7d7442dcd56634c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.zero_in_B",
      "line": 94,
      "title": "Zero belongs to the base-4 digit set",
      "intuition": "The natural number `0` belongs to `B` because its base-4 expansion contains no digit outside `{0,1}`.",
      "source_statement": "lemma zero_in_B : 0 ∈ B := by\n…",
      "source_statement_sha256": "0f8680db02548837a9255a4f172d423a8d32520d2165a8b6cedb75230c925b14",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L94",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.zero_in_A_plus_B",
      "line": 97,
      "title": "The sumset contains zero",
      "intuition": "Since `0∈A` and `0∈B`, the equality `0=0+0` witnesses `0∈A+B`.",
      "source_statement": "lemma zero_in_A_plus_B : 0 ∈ A + B := by\n…",
      "source_statement_sha256": "e73ff940e09d3338245e0e7729ffa8fdfe470537f803b1339178c9d5678d8001",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L97",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.A_max_k",
      "line": 102,
      "title": "The largest k-digit base-3 number using only zero and one",
      "intuition": "If `x∈A` and `x<3^k`, then `x≤(3^k-1)/2`, the number whose first `k` base-3 digits are all one.",
      "source_statement": "lemma A_max_k (k x : ℕ) (hx : x < 3^k) (hA : x ∈ A) : x ≤ (3^k - 1) / 2 := by\n…",
      "source_statement_sha256": "0e068ff7f3824d61720ae9f0345a520a8bac0351526a1ae3b0e7e66ae8b2eca3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L102",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.B_max_m",
      "line": 107,
      "title": "The largest m-digit base-4 number using only zero and one",
      "intuition": "If `y∈B` and `y<4^m`, then `y≤(4^m-1)/3`, the number whose first `m` base-4 digits are all one.",
      "source_statement": "lemma B_max_m (m y : ℕ) (hy : y < 4^m) (hB : y ∈ B) : y ≤ (4^m - 1) / 3 := by\n…",
      "source_statement_sha256": "ec780797821c10f5bc41ce240f8315686c4584f0aa3e650151e56e6b924098fd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L107",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.A_B_gap",
      "line": 112,
      "title": "A common digit-scale gap in the sumset",
      "intuition": "If `x` is below both `3^k` and `4^m` but exceeds `(3^k-1)/2+(4^m-1)/3`, then `x∉A+B`.",
      "source_statement": "lemma A_B_gap (k m x : ℕ) (hx_gt : (3^k - 1) / 2 + (4^m - 1) / 3 < x) (hx_lt_A : x < 3^k) (hx_lt_B : x < 4^m) : x ∉ A + B := by\n…",
      "source_statement_sha256": "07bee0aa5995d25e726ad101b6c5f8460deb329234cd2ea2a3f18869cb0d8fa6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L112",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.A_decomp",
      "line": 125,
      "title": "Split a base-3 digit string into top and bottom blocks",
      "intuition": "Every `a∈A` can be written `a=a1*3^k+a0`, where both `a1` and `a0` remain in `A` and `a0<3^k`.",
      "source_statement": "lemma A_decomp (k a : ℕ) (ha : a ∈ A) : ∃ a1 a0 : ℕ, a1 ∈ A ∧ a0 ∈ A ∧ a0 < 3^k ∧ a = a1 * 3^k + a0 := by\n…",
      "source_statement_sha256": "1d3d1585d75a88ae5d2bf52f4cea9a980fd1ae0b55e1df6bfff5e5c572b75e74",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L125",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.B_decomp",
      "line": 136,
      "title": "Split a base-4 digit string into top and bottom blocks",
      "intuition": "Every `b∈B` can be written `b=b1*4^m+b0`, where both pieces remain in `B` and `b0<4^m`.",
      "source_statement": "lemma B_decomp (m b : ℕ) (hb : b ∈ B) : ∃ b1 b0 : ℕ, b1 ∈ B ∧ b0 ∈ B ∧ b0 < 4^m ∧ b = b1 * 4^m + b0 := by\n…",
      "source_statement_sha256": "86e5d8726cefe41fa77aaad287efc5969e416226e3d8eb539466e404df3f3caa",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L136",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.log_ratio_irrational",
      "line": 152,
      "title": "The logarithms of four and three have irrational ratio",
      "intuition": "`log 4 / log 3` is irrational; otherwise a rational relation would force a nontrivial equality between a power of three and a power of four.",
      "source_statement": "lemma log_ratio_irrational : Irrational (Real.log 4 / Real.log 3) := by use(·.elim fun and x =>(eq_div_iff (by norm_num)).ne.2 (and.num_div_den▸ne_of_eq_of_ne (by rw [Rat.cast_div,div_mul_eq_mul_div]) ((div_eq_iff (by norm_num)).ne.2 fun and=>?_)) x)\n…",
      "source_statement_sha256": "f65c2a37858a1707abc4f5b0ee05090418ab1a95cd144b4bee26efdf6cb0ba6b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L152",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.exists_small_pos_lin_comb_help",
      "line": 157,
      "title": "A positive irrational has arbitrarily small positive integer linear combinations",
      "intuition": "For positive irrational `α` and every `δ>0`, there are positive naturals `m,k` with `0<m*α-k<δ`.",
      "source_statement": "lemma exists_small_pos_lin_comb_help (α : ℝ) (hα : Irrational α) (hα_pos : 0 < α) (δ : ℝ) (hδ : 0 < δ) :\n  ∃ m k : ℕ, 0 < m ∧ 0 < k ∧ 0 < (m : ℝ) * α - (k : ℝ) ∧ (m : ℝ) * α - (k : ℝ) < δ := by\n…",
      "source_statement_sha256": "4967ef55ab8efb1506e297cb771914f92ac18e5a1a2e74d88f5b13dddc835acf",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L157",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.exists_small_pos_lin_comb",
      "line": 187,
      "title": "Make m log 4 minus k log 3 positive and arbitrarily small",
      "intuition": "For every `δ>0`, positive naturals `m,k` satisfy `0<m*log 4-k*log 3<δ`.",
      "source_statement": "lemma exists_small_pos_lin_comb (δ : ℝ) (hδ : 0 < δ) :\n  ∃ m k : ℕ, 0 < m ∧ 0 < k ∧ 0 < (m : ℝ) * Real.log 4 - (k : ℝ) * Real.log 3 ∧ (m : ℝ) * Real.log 4 - (k : ℝ) * Real.log 3 < δ := by\n…",
      "source_statement_sha256": "0392e55413b5676451344e5f342c14c00999d49464ed3b8b59617a3f7e682628",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L187",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.exists_small_pos_lin_comb_large_k",
      "line": 196,
      "title": "Obtain the logarithmic approximation beyond any requested scale",
      "intuition": "The exponents in the small positive logarithmic combination can also be chosen so that `3^k` exceeds any real threshold `K`.",
      "source_statement": "lemma exists_small_pos_lin_comb_large_k (δ : ℝ) (hδ : 0 < δ) (K : ℝ) :\n  ∃ m k : ℕ, 0 < m ∧ 0 < k ∧ K ≤ (3^k : ℝ) ∧ 0 < (m : ℝ) * Real.log 4 - (k : ℝ) * Real.log 3 ∧ (m : ℝ) * Real.log 4 - (k : ℝ) * Real.log 3 < δ := by\n…",
      "source_statement_sha256": "690692cd0cd40bdfafafd1c8137dc6d274eb6785b4417f44deb721437684a24f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L196",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.dirichlet_approx",
      "line": 211,
      "title": "Choose nearby powers of three and four at a large scale",
      "intuition": "For every `ε>0`, positive `k,m` can be chosen with `3^k≤4^m≤3^k*(1+ε)` and `3^k*ε≥3`.",
      "source_statement": "lemma dirichlet_approx (ε : ℝ) (hε : 0 < ε) : ∃ k m : ℕ, 0 < k ∧ 0 < m ∧ (3^k : ℝ) ≤ 4^m ∧ (4^m : ℝ) ≤ (3^k : ℝ) * (1 + ε) ∧ (3^k : ℝ) * ε ≥ 3 := by\n…",
      "source_statement_sha256": "b222e0c4e51974db8fe1d0ec63e438f928c2355744c75def0b7fa92731580047",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L211",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.hz_eq_lemma",
      "line": 225,
      "title": "Rewrite a sum using the smaller common scale",
      "intuition": "When `3^k≤4^m`, decompositions of `a` and `b` give `x=(a1+b1)*3^k+z`, with `z=a0+b0+b1*(4^m-3^k)`.",
      "source_statement": "lemma hz_eq_lemma (x a b a1 a0 b1 b0 k m : ℕ)\n  (h1 : 3^k ≤ 4^m) (h3 : x = a + b) (h4 : a = a1 * 3^k + a0) (h5 : b = b1 * 4^m + b0) :\n  x = (a1 + b1) * 3^k + (a0 + b0 + b1 * (4^m - 3^k)) := by\n…",
      "source_statement_sha256": "ec0067242f5d6840364b9924ffec503bf13094f9cea69e17f1d1eb83ee2634ae",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L225",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.scale_step",
      "line": 236,
      "title": "Reduce density by a fixed factor at a larger scale",
      "intuition": "If the count below `N` is at most `C*N`, then at a suitable scale `N'=N*3^k` the count is at most `(11/12)*C*N'`.",
      "source_statement": "lemma scale_step (N : ℕ) (hN : 0 < N) (C : ℝ) (hC : (((Finset.Ico 0 N).filter (· ∈ A + B)).card : ℝ) ≤ C * (N : ℝ)) :\n  ∃ N' > 0, (((Finset.Ico 0 N').filter (· ∈ A + B)).card : ℝ) ≤ (11/12 : ℝ) * C * (N' : ℝ) := by\n…",
      "source_statement_sha256": "ee303cc3e5ef633a26dd607d4db5d387767d09de4603d12bde16c429ffd5dc7d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L236",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.density_multi_scale",
      "line": 301,
      "title": "Iterate the contraction through arbitrarily many scales",
      "intuition": "After `d` scale steps, some positive cutoff `N` has normalized count at most `(11/12)^d`.",
      "source_statement": "lemma density_multi_scale (d : ℕ) : ∃ N > 0, (((Finset.Ico 0 N).filter (· ∈ A + B)).card : ℝ) ≤ ((11/12 : ℝ)^d) * (N : ℝ) := by\n…",
      "source_statement_sha256": "67dad94d329220bd05672987c01b0dd744f6f21ed057b51a51e50bbea03a93fa",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L301",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.limit_11_12",
      "line": 328,
      "title": "Powers of eleven twelfths fall below every positive threshold",
      "intuition": "For every `ε>0`, some natural exponent `d` satisfies `(11/12)^d≤ε`.",
      "source_statement": "lemma limit_11_12 (ε : ℝ) (hε : ε > 0) : ∃ d : ℕ, (11/12 : ℝ)^d ≤ ε := by\n…",
      "source_statement_sha256": "d61aedb40355861e5d7495fb42400938f9e032e48e77a10eb85501ea5078fffd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L328",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.density_tends_to_zero",
      "line": 331,
      "title": "Find a cutoff with density below any positive epsilon",
      "intuition": "For every `ε>0`, there is a positive `N` with `|(A+B)∩[0,N)|≤εN`.",
      "source_statement": "lemma density_tends_to_zero (ε : ℝ) (hε : ε > 0) : ∃ N > 0, (((Finset.Ico 0 N).filter (· ∈ A + B)).card : ℝ) ≤ ε * (N : ℝ) := by\n…",
      "source_statement_sha256": "4d7d39e4bb0ecf5c385567a6c891d6ab1026b2c8cc334ee561086cba8192c027",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L331",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean",
      "declaration": "Erdos125.target_theorem_0",
      "line": 342,
      "title": "Erdős problem 125: the base-3 plus base-4 digit sumset has zero lower density",
      "intuition": "Let `A` contain the nonnegative integers using only digits 0 and 1 in base 3, and `B` the analogous set in base 4. Their sumset does not have positive lower density: along suitable scales its proportion can be made arbitrarily small.",
      "source_statement": "theorem target_theorem_0\n  : answer(\n  -- EVOLVE-VALUE-START\n  False\n  -- EVOLVE-VALUE-END\n  ) ↔ 0 < (A + B).lowerDensity := by\n…",
      "source_statement_sha256": "b0fa7157541897a9764eba7316e1497bfda7899a9c3182c831c1f34b76a1a0a7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_125.variants.positive_lower_density.lean#L342",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.monoAP_guarantee_set",
      "line": 51,
      "title": "Interval lengths that force a monochromatic arithmetic progression",
      "intuition": "`monoAP_guarantee_set r k` contains exactly those natural numbers `N` for which every `r`-colouring of `{1,...,N}` contains a monochromatic arithmetic progression of length `k`.",
      "source_statement": "def monoAP_guarantee_set (r k : ℕ) : Set ℕ :=\n…",
      "source_statement_sha256": "eb3f6b2450da011a069accb67901c8243c129acdc02c87f7eb9cfcd5779413df",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L51",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.monoAPNumber",
      "line": 59,
      "title": "The least interval length that forces a monochromatic progression",
      "intuition": "`monoAPNumber r k` is the infimum of the forcing set, hence the formal van der Waerden number for `r` colours and progression length `k` once that set is known to be nonempty.",
      "source_statement": "noncomputable def monoAPNumber (r k : ℕ) : ℕ := sInf (monoAP_guarantee_set r k)",
      "source_statement_sha256": "6301ec0fe338cd06fc252a1e0cf45dec57a28036e177e17b642eb580278f516f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L59",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.W",
      "line": 66,
      "title": "The two-colour van der Waerden number `W(k)`",
      "intuition": "`W` abbreviates `monoAPNumber 2`, so `W k` is the least interval length that forces a monochromatic `k`-term arithmetic progression under every red-blue colouring.",
      "source_statement": "noncomputable abbrev W : ℕ → ℕ := monoAPNumber 2\n…",
      "source_statement_sha256": "8a09bb9e945e8905ac1117486f1af69083a6e2c533a167732b258d8e36690973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L66",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.HasMonoAP",
      "line": 103,
      "title": "An explicit monochromatic arithmetic progression inside `[1,N]`",
      "intuition": "`HasMonoAP c N k` supplies a first term `a` and positive step `d` such that all `k` terms `a+m*d` have colour `c a`, start at or above 1, and end at or before `N`.",
      "source_statement": "def HasMonoAP (c : ℕ → Fin 2) (N k : ℕ) : Prop :=\n…",
      "source_statement_sha256": "186c52f5fea47fa52d780adec316717d7072d9907980c7d712ca52260f0da65a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L103",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.ap_must_end",
      "line": 111,
      "title": "Any newly created `(k+1)`-progression must end at the new point",
      "intuition": "If the old colouring has no `(k+1)`-term progression through `N+i`, but assigning a colour at `N+i+1` creates one, that progression must use `N+i+1` as its last term.",
      "source_statement": "lemma ap_must_end (k i N : ℕ) (c : ℕ → Fin 2) (color : Fin 2)\n  (h_no_ap_k1 : ¬ HasMonoAP c (N + i) (k + 1))\n  (h_has : HasMonoAP (fun x => if x = N + i + 1 then color else c x) (N + i + 1) (k + 1)) :\n  ∃ a d, d > 0 ∧ a ≥ 1 ∧ a + k * d = N + i + 1 ∧\n         (∀ m < k, c (a + m * d) = color) := by\n…",
      "source_statement_sha256": "5ca8c540170e56e5736712e89aa394b1db65d3b0b4fe1981a7ff3bc0cedbc336",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L111",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.extend_one_step",
      "line": 161,
      "title": "At least one colour safely extends the progression-free interval",
      "intuition": "While fewer than `k` new points have been appended, a two-colouring with no old `k`-progression and no current `(k+1)`-progression can always colour the next point without creating a `(k+1)`-progression.",
      "source_statement": "lemma extend_one_step (k i N : ℕ) (c : ℕ → Fin 2)\n  (hk : k > 0)\n  (hi : i < k)\n  (h_no_ap_k : ¬ HasMonoAP c N k)\n  (h_no_ap_k1 : ¬ HasMonoAP c (N + i) (k + 1)) :\n  ∃ color : Fin 2, ¬ HasMonoAP (fun x => if x = N + i + 1 then color else c x) (N + i + 1) (k + 1) := by\n…",
      "source_statement_sha256": "485c3fffa2e34193e52a9e7a51f55e765e6746d75d470458cb7cf8af258f9b65",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L161",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.has_mono_ap_ext",
      "line": 258,
      "title": "The progression predicate depends only on colours inside `[1,N]`",
      "intuition": "If two total colourings agree at every positive integer at most `N`, then either both contain a monochromatic `k`-term progression inside that interval or neither does.",
      "source_statement": "lemma has_mono_ap_ext (c1 c2 : ℕ → Fin 2) (N k : ℕ)\n  (h_eq : ∀ x ≤ N, x ≥ 1 → c1 x = c2 x) :\n  HasMonoAP c1 N k ↔ HasMonoAP c2 N k := by\n…",
      "source_statement_sha256": "aabc3e6f2d4744fb0a9cfa938363d99275a2725a78bb07a148980dbfa0c75c45",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L258",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.extend_j_steps",
      "line": 287,
      "title": "Iterating the safe extension for up to `k` new points",
      "intuition": "From a colouring of `[1,N]` with no monochromatic `k`-term progression, the lemma constructs a colouring through `N+j`, for every `j≤k`, with no monochromatic `(k+1)`-term progression and with all original colours preserved.",
      "source_statement": "lemma extend_j_steps (k N j : ℕ) (c : ℕ → Fin 2) (hk : k > 0)\n  (hj : j ≤ k)\n  (h_no_ap_k : ¬ HasMonoAP c N k) :\n  ∃ c' : ℕ → Fin 2, (∀ x ≤ N, c' x = c x) ∧ ¬ HasMonoAP c' (N + j) (k + 1) := by\n…",
      "source_statement_sha256": "34c14229a1b0eb2f0f73220e2cf28ae83e83b8d3464244bbeba3010c172fb749",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L287",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.extend_coloring",
      "line": 335,
      "title": "Extend a finite interval colouring to every natural number",
      "intuition": "`extend_coloring N c` agrees with the subtype-valued colouring `c` on `{1,...,N}` and assigns colour `0` everywhere outside that interval.",
      "source_statement": "def extend_coloring (N : ℕ) (c : Finset.Icc 1 N → Fin 2) : ℕ → Fin 2 :=\n…",
      "source_statement_sha256": "b957bc9e2e10f2578ce854f98358a781e050f75a08335b05573a92d39e424a45",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L335",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.ap_equiv",
      "line": 338,
      "title": "Indexing a positive-step arithmetic progression by `Fin k`",
      "intuition": "For `d>0`, the map `m ↦ a+m*d` is an equivalence from the `k` indices to the set of the first `k` terms of the arithmetic progression.",
      "source_statement": "def ap_equiv (a d k : ℕ) (hd : d > 0) : Fin k ≃ { x : ℕ | ∃ m < k, x = a + m * d } where\n…",
      "source_statement_sha256": "5b86325347ef71cad1c18fd6612fedd6bf75981e0f27387565844fb6c4bf804c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L338",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.card_ap_eq_k",
      "line": 365,
      "title": "A positive-step `k`-term progression has exactly `k` elements",
      "intuition": "The underlying set `{a+m*d | m<k}` has extended-natural cardinality `k` whenever `d>0` and `k>0`.",
      "source_statement": "lemma card_ap_eq_k (a d k : ℕ) (hd : d > 0) (hk : k > 0) :\n  ENat.card ↑{ x : ℕ | ∃ m < k, x = a + m * d } = k := by\n…",
      "source_statement_sha256": "403ef62cee948eb301cfccfe202fc1508a23a602c3a7667ff26d431e0d0d25bc",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L365",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.card_ap_pos_d",
      "line": 371,
      "title": "More than one progression term forces a positive step",
      "intuition": "If `k>1` and the set `{a+m*d | m<k}` really has cardinality `k`, then `d` cannot be zero.",
      "source_statement": "lemma card_ap_pos_d (a d k : ℕ) (hk : k > 1) (h_card : ENat.card ↑{ x : ℕ | ∃ m < k, x = a + m * d } = k) :\n  d > 0 := by\n…",
      "source_statement_sha256": "a473eeb86ca855f4a42ba4a4a2836edf263176655accb24b5f67395858e77199",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L371",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.contains_mono_ap_imp",
      "line": 395,
      "title": "Convert Mathlib's set-valued progression witness to `HasMonoAP`",
      "intuition": "A monochromatic arithmetic-progression subset of `Finset.Icc 1 N` yields explicit natural-number witnesses `a,d` satisfying `HasMonoAP` for the extended colouring.",
      "source_statement": "lemma contains_mono_ap_imp (N k : ℕ) (hk : k > 0) (c : Finset.Icc 1 N → Fin 2)\n  (h : ContainsMonoAPofLength c k) :\n  HasMonoAP (extend_coloring N c) N k := by\n…",
      "source_statement_sha256": "d99eb3d08fe58c3f09e5b3e49112c81fe1312af9db250dfc31dd92233042b802",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L395",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.imp_contains_mono_ap",
      "line": 507,
      "title": "Convert `HasMonoAP` back to Mathlib's interval predicate",
      "intuition": "Explicit witnesses `a,d` for a monochromatic progression inside `[1,N]` define the required progression subset of the interval subtype.",
      "source_statement": "lemma imp_contains_mono_ap (N k : ℕ) (hk : k > 0) (c : ℕ → Fin 2)\n  (h : HasMonoAP c N k) :\n  ContainsMonoAPofLength (fun (x : Finset.Icc 1 N) => c x.1) k := by\n…",
      "source_statement_sha256": "e676dcba45b7dc0983998f5dba0519f4228f04e1f8ab5571fac14ea9603f8f30",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L507",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.not_guarantee_extend",
      "line": 566,
      "title": "A countercolouring for `k` extends to one for `k+1`",
      "intuition": "If some two-colouring of `{1,...,N}` avoids every `k`-term progression, then some two-colouring of `{1,...,N+k}` avoids every `(k+1)`-term progression.",
      "source_statement": "lemma not_guarantee_extend (k N : ℕ) (hk : k > 0) :\n    N ∉ monoAP_guarantee_set 2 k → (N + k) ∉ monoAP_guarantee_set 2 (k + 1) := by\n…",
      "source_statement_sha256": "719cf8e2a94e6596d6b6456d9961f4a8cdb512e7bd82c5bb36b9d64760b63856",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L566",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.not_in_set_of_lt_sInf",
      "line": 601,
      "title": "Nothing below a natural-number infimum belongs to the set",
      "intuition": "For any set of naturals `s`, if `n<sInf s`, then `n∉s`.",
      "source_statement": "lemma not_in_set_of_lt_sInf {s : Set ℕ} {n : ℕ} (h : n < sInf s) : n ∉ s := by\n…",
      "source_statement_sha256": "388b22532d42b5728ff773139888be6809cdb0d5d69b8a37b6b3c17cef2ad858",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L601",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.U_limit_color",
      "line": 607,
      "title": "The ultrafilter-limit colour at one integer",
      "intuition": "Given a sequence of two-colourings `C n`, `U_limit_color C U x` chooses colour `0` when the indices assigning `0` to `x` are large in `U`, and otherwise chooses colour `1`.",
      "source_statement": "noncomputable def U_limit_color (C : ℕ → ℕ → Fin 2) (U : Ultrafilter ℕ) (x : ℕ) : Fin 2 :=\n…",
      "source_statement_sha256": "f1f69c9f6cd9ca08a9ad6840a1e5411c9c79a4c7961c3a64765e9994cfa75ecd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L607",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.U_limit_color_mem",
      "line": 610,
      "title": "The chosen limit colour occurs on a `U`-large set of indices",
      "intuition": "For each position `x`, the indices `n` satisfying `C n x = U_limit_color C U x` belong to the ultrafilter.",
      "source_statement": "lemma U_limit_color_mem (C : ℕ → ℕ → Fin 2) (U : Ultrafilter ℕ) (x : ℕ) :\n  {n | C n x = U_limit_color C U x} ∈ U := by\n…",
      "source_statement_sha256": "0bc6da5b04fb1bcb29bcd707e8d711ad68f1c669d41958056bca75041bd7a5d4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L610",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.U_inter_mem",
      "line": 645,
      "title": "Finitely many progression positions agree with the limit colouring together",
      "intuition": "For fixed `a,b,k`, the indices whose colourings agree with the ultrafilter-limit colouring at every point `a*s+b` for `s<k` form a set in `U`.",
      "source_statement": "lemma U_inter_mem (C : ℕ → ℕ → Fin 2) (U : Ultrafilter ℕ) (a b k : ℕ) :\n  {n | ∀ s < k, C n (a * s + b) = U_limit_color C U (a * s + b)} ∈ U := by\n…",
      "source_statement_sha256": "122b12b43d931fedda1b1463dafc8b11bd1cd36687b0739e952c6786a20c1a0b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L645",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.U_atTop_mem_large",
      "line": 676,
      "title": "A `U`-large set contains arbitrarily large indices",
      "intuition": "When the ultrafilter lies above `atTop`, every set in it meets every tail `{n | n≥M}`.",
      "source_statement": "lemma U_atTop_mem_large (U : Ultrafilter ℕ) (h_le : ↑U ≤ (Filter.atTop : Filter ℕ))\n  (s : Set ℕ) (hs : s ∈ U) (M : ℕ) : ∃ n ∈ s, n ≥ M := by\n…",
      "source_statement_sha256": "df3efb44fa98d06bd6375cb0dcc7793a181912d5d8ba93044508dc320985e985",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L676",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.W_is_nonempty",
      "line": 685,
      "title": "Van der Waerden forcing lengths exist for every progression length",
      "intuition": "For every `k`, some finite `N` forces a monochromatic `k`-term progression under every two-colouring, so the set whose infimum defines `W k` is nonempty.",
      "source_statement": "lemma W_is_nonempty (k : ℕ) : (monoAP_guarantee_set 2 k).Nonempty := by\n…",
      "source_statement_sha256": "aa3687fd722d1124dc9d614718af22166238771a0b676bbf061a73ece8123479",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L685",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.Icc_subset_succ",
      "line": 787,
      "title": "The interval `[1,N]` sits inside `[1,N+1]`",
      "intuition": "Any natural number between `1` and `N` is also between `1` and `N+1`.",
      "source_statement": "lemma Icc_subset_succ (N x : ℕ) (hx : x ∈ Finset.Icc 1 N) : x ∈ Finset.Icc 1 (N + 1) := by\n…",
      "source_statement_sha256": "ef80d96743badebe4dcdfdf80b3fd8ca92e39157b2f7e1221e4e36bf751738c4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L787",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.guarantee_upward_closed",
      "line": 791,
      "title": "Once an interval length guarantees a progression, every larger one does",
      "intuition": "If every `r`-colouring of `{1,...,N}` contains a monochromatic `k`-term progression, then the same is true for `{1,...,N+1}`.",
      "source_statement": "lemma guarantee_upward_closed (k r N : ℕ) :\n    N ∈ monoAP_guarantee_set r k → (N + 1) ∈ monoAP_guarantee_set r k := by\n…",
      "source_statement_sha256": "0e4ab548a6275eaa8dcf79d2be028c62e256932afa7a26e9cd884de70c30bcc1",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L791",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.not_in_guarantee_lt_sInf",
      "line": 841,
      "title": "A non-guaranteeing length lies below the van der Waerden number",
      "intuition": "If `N` does not force a two-colour `k`-term progression, then `N<W k`.",
      "source_statement": "lemma not_in_guarantee_lt_sInf (k N : ℕ) (hN : N ∉ monoAP_guarantee_set 2 k) :\n    N < W k := by\n…",
      "source_statement_sha256": "432718314d3ed263c93939d03172c6a1c887c1d95839d01a702b1e75863584fc",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L841",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.W_not_guarantee",
      "line": 859,
      "title": "One less than a positive van der Waerden number is a counterexample length",
      "intuition": "When `W k≠0`, minimality gives `W k-1∉monoAP_guarantee_set 2 k`.",
      "source_statement": "lemma W_not_guarantee (k : ℕ) (hW : W k ≠ 0) : W k - 1 ∉ monoAP_guarantee_set 2 k := by\n…",
      "source_statement_sha256": "7e93e394ec276a43d9d09c73f2ffab829ee2a3d7d375d73d6e61b17982047c93",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L859",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.W_diff_bound_gt0",
      "line": 864,
      "title": "The quantitative gap bound when `W(k)` is positive",
      "intuition": "For `k>0` and `W k≠0`, the proof establishes `W(k+1)-W(k)≥k`.",
      "source_statement": "lemma W_diff_bound_gt0 (k : ℕ) (hk : k > 0) (hW : W k ≠ 0) : W (k + 1) - W k ≥ k := by\n…",
      "source_statement_sha256": "aacc1da886b8919710eb4a67da47a1467f3523255f58f8cc18ec760902d6f402",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L864",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.W_diff_ge_k",
      "line": 870,
      "title": "The gap bound for every natural `k`",
      "intuition": "For all natural numbers `k`, `W(k+1)-W(k)≥k`, including the edge cases introduced by the natural-number encoding.",
      "source_statement": "lemma W_diff_ge_k (k : ℕ) : W (k + 1) - W k ≥ k := by\n…",
      "source_statement_sha256": "39c54a8181300dbb081d7545fe8b26b13a682016f26e5ac069a490ded4a59a28",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L870",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.W_diff_ge_k_all",
      "line": 898,
      "title": "The pointwise lower bound used to prove divergence",
      "intuition": "This wrapper presents the uniform inequality `k≤W(k+1)-W(k)` in the orientation expected by `Filter.tendsto_atTop_mono`.",
      "source_statement": "lemma W_diff_ge_k_all (k : ℕ) : W (k + 1) - W k ≥ k := by\n…",
      "source_statement_sha256": "2d46c60cc00fd0ff00f44b8bc87c74dd2667a64487cd24998f78c6b821238769",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L898",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_138.variants.difference.lean",
      "declaration": "Erdos138.target_theorem_0",
      "line": 906,
      "title": "Erdős problem 138: consecutive van der Waerden numbers separate without bound",
      "intuition": "For the two-colour van der Waerden number `W(k)`, the difference `W(k+1)-W(k)` tends to infinity. The proof establishes the explicit lower bound `W(k+1) ≥ W(k)+k`, which immediately forces the differences upward.",
      "source_statement": "theorem target_theorem_0\n  : answer(\n  -- EVOLVE-VALUE-START\n  True\n  -- EVOLVE-VALUE-END\n  ) ↔ atTop.Tendsto (fun k => (W (k + 1) - W k)) atTop := by\n…",
      "source_statement_sha256": "00a0095749b0e774c95dee2bc0cef1136af186945fd05e3d2d79d5c42c3e35c4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_138.variants.difference.lean#L906",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_152.lean",
      "declaration": "Erdos152.target_theorem_0",
      "line": 505,
      "title": "Erdős problem 152: Sidon sumsets have quadratically many isolated sums",
      "intuition": "If `f(n)` is the minimum number of isolated elements in `A+A` over Sidon sets `A` of size `n`, then `f(n)` tends to infinity. In fact the formal proof derives the quantitative lower bound `(n²-100n-16)/16`.",
      "source_statement": "theorem target_theorem_0\n  : answer(\n  -- EVOLVE-VALUE-START\n  True\n  -- EVOLVE-VALUE-END\n  ) ↔ Tendsto f atTop atTop := by\n…",
      "source_statement_sha256": "463c8d940377a9234366f3646b15b393e231605b8746fcb25f917028b667f707",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_152.lean#L505",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_26.variants.tenenbaum.lean",
      "declaration": "Erdos26.target_theorem_0",
      "line": 1053,
      "title": "Erdős problem 26 variant: a thick sequence whose shifted multiples stay sparse",
      "intuition": "The universal weak-Behrend assertion is false. There is a strictly increasing sequence with divergent reciprocal sum such that every additive shift still leaves a fixed positive proportion of integers uncovered by its multiples.",
      "source_statement": "theorem target_theorem_0\n  : answer(False) ↔ ∀ᵉ (A : ℕ → ℕ), StrictMono A → IsThick A → (∀ ε > (0 : ℝ), ∃ k, IsWeaklyBehrend (A · + k) ε) := by\n…",
      "source_statement_sha256": "460dc4f47d46a85468e95fbcc71fd3f50745009c718022d5b5276baa465cfccf",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_26.variants.tenenbaum.lean#L1053",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.S_x",
      "line": 79,
      "title": "The lower observation scale in the Erdős 741 counterexample",
      "intuition": "`S_x(k)=4^(3^k)` is the rapidly growing left endpoint of the `k`th scale. Base four makes the digit split carry-free, while the double-exponential spacing prevents earlier added intervals from controlling the next density measurement.",
      "source_statement": "def S_x (k : ℕ) : ℕ := 4 ^ (3 ^ k)",
      "source_statement_sha256": "8fe3985a58b469ad308e92ca441b8f5e60e9240b6caac41c3015231513889fc6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L79",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.S_y",
      "line": 80,
      "title": "The upper endpoint: ten times the lower scale",
      "intuition": "`S_y(k)=10*S_x(k)` gives each added interval a fixed nine-scale-unit width. Those fat intervals later force many same-colour sums before the next enormous base-four scale arrives.",
      "source_statement": "def S_y (k : ℕ) : ℕ := 10 * 4 ^ (3 ^ k)",
      "source_statement_sha256": "25b263a698280bb87f4a3675eae7cfe21af074122c657363bcda39d81c6b99b4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L80",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.S_C",
      "line": 82,
      "title": "The sparse union of fat intervals that drives density oscillation",
      "intuition": "`S_C` is the union of every half-open interval `[S_x(k),S_y(k))`. Each interval is locally large, but the starting scales grow so quickly that earlier intervals become negligible at the next lower checkpoint.",
      "source_statement": "def S_C : Set ℕ := ⋃ k, Ico (S_x k) (S_y k)",
      "source_statement_sha256": "7400c5495f72285196bfc9074ed76417c4919f8f9ae8479a4098bc14a99f8e2d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.split1",
      "line": 84,
      "title": "Put each binary digit into the 0-or-1 part of a base-four number",
      "intuition": "`split1(n)` recursively keeps the parity bit of each base-four digit. Its base-four digits are therefore only 0 or 1, which prevents carries when two `B1` elements are added.",
      "source_statement": "noncomputable def split1 (n : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "542ff4cc456631c7ffc53104c22dfe018d3935a769947532bb6b903c01600adb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L84",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.split2",
      "line": 88,
      "title": "Put the complementary 0-or-2 contribution into the second base-four stream",
      "intuition": "`split2(n)` keeps `n % 4 - n % 2` at each digit, so its base-four digits are 0 or 2. Digit by digit, `split1(n)+split2(n)=n` without a carry.",
      "source_statement": "noncomputable def split2 (n : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "fddeb8f0f9254b1e7e62cd13382a26aef96d011c6b6a5ee226d66840c44a31ed",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L88",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.B1",
      "line": 92,
      "title": "The carry-free base-four set with digits 0 and 1",
      "intuition": "`B1` is the range of `split1`. Its elements use only 0-or-1 base-four digits, making `B1+B1` digitwise sparse even though `B1+B2` will cover every natural number.",
      "source_statement": "def B1 : Set ℕ := Set.range split1",
      "source_statement_sha256": "78f9ffbffcd3486b13fc73947c70738136b30a0d9f0c763279dc56465b90e259",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L92",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.B2",
      "line": 93,
      "title": "The complementary carry-free base-four set with digits 0 and 2",
      "intuition": "`B2` is the range of `split2`. It is the digitwise complement to `B1`: together the two sets represent every natural, while `B2+B2` has the same sparse-count phenomenon as `B1+B1`.",
      "source_statement": "def B2 : Set ℕ := Set.range split2",
      "source_statement_sha256": "044ce0b47862aa5809c33b12f61e7ef3f2b2acc7495e089edcc0308fa401299e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L93",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.SandorA",
      "line": 96,
      "title": "The strict-natural-density counterexample set",
      "intuition": "`SandorA` joins the carry-free covering set `B1∪B2` to the sparse union of fat intervals `S_C`. Coverage makes `SandorA+SandorA` all naturals; the intervals make every two-colouring oscillate too much for both monochromatic self-sumsets to have positive natural-density limits.",
      "source_statement": "def SandorA : Set ℕ := B ∪ S_C",
      "source_statement_sha256": "1fd46f15f64a07e3e83aa3f32610df610cfd0ef7b2d35b2f1b5d546e2922aa90",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L96",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.B1_sum_ncard",
      "line": 312,
      "title": "Below `4^d`, the first self-sumset has at most `3^d` values",
      "intuition": "A sum of two 0-or-1 base-four digit strings never has digit 3. There are only three choices at each of the first `d` positions, hence at most `3^d` possible sums below `4^d`.",
      "source_statement": "lemma B1_sum_ncard (d : ℕ) : ((B1 + B1) ∩ Iio (4^d)).ncard ≤ 3^d := by\n…",
      "source_statement_sha256": "8555f39d7a7bbdcc3936bcfb1fddaa792062037d5754415f126c6e2d7af0678c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L312",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.B2_sum_ncard",
      "line": 406,
      "title": "The second carry-free self-sumset obeys the same `3^d` bound",
      "intuition": "Because the 0-or-2 digit set is twice the 0-or-1 digit set, its self-sums below the matching power-of-four scale also have at most `3^d` possibilities.",
      "source_statement": "lemma B2_sum_ncard (d : ℕ) : ((B2 + B2) ∩ Iio (4^d)).ncard ≤ 3^d := by\n…",
      "source_statement_sha256": "e6ace5ca283420bcfb7b519f5a1e1ef0c24c86d13b8197ccebb5accad5318dce",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L406",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.B1_add_B2_eq_univ",
      "line": 435,
      "title": "The two complementary digit sets add to every natural number",
      "intuition": "For each natural `n`, the construction splits every base-four digit uniquely between `split1(n)` and `split2(n)`. Thus `n` belongs to `B1+B2`, and the cross-sumset is exactly all naturals.",
      "source_statement": "lemma B1_add_B2_eq_univ : B1 + B2 = Set.univ := by\n…",
      "source_statement_sha256": "138c3a2c4dfe1e57dde2587abb289aba9d2e6252bc38587d4faa4e4c55af62a6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L435",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.SandorA_add_SandorA_eq_univ",
      "line": 451,
      "title": "The counterexample's full self-sumset is all naturals",
      "intuition": "`SandorA` contains both `B1` and `B2`, and their cross-sumset already equals every natural number. Enlarging the summand set cannot lose those representations, so `SandorA+SandorA=Set.univ`.",
      "source_statement": "lemma SandorA_add_SandorA_eq_univ : SandorA + SandorA = Set.univ := by\n…",
      "source_statement_sha256": "4cc59702adcc9441ea291c6c5a3318265dd0c21f3a8a017b64da6b43165c8f7b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L451",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.SandorA_has_pos_density",
      "line": 471,
      "title": "The full sumset has natural density one",
      "intuition": "Since `SandorA+SandorA` is `Set.univ`, its partial-density sequence is constantly one after the zero scale. This proves `HasPosDensity` in the strict sense used here: a positive natural-density limit exists.",
      "source_statement": "lemma SandorA_has_pos_density : HasPosDensity (SandorA + SandorA) := by\n…",
      "source_statement_sha256": "cfd4d2d1d52c90657fd679d8662217aff2fbac29327096ab0127faf548bf2da3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L471",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.B_partition_alpha_bound",
      "line": 1199,
      "title": "Any convergent two-colour split pays a cross-term density deficit",
      "intuition": "If the two monochromatic self-sumsets have natural-density limits `α₁` and `α₂`, then the carry-free digit split forces `α₁+α₂ ≤ 1-α₁α₂`. In particular, if both limits are positive, their sum is uniformly below one.",
      "source_statement": "lemma B_partition_alpha_bound (A₁ A₂ : Set ℕ) (h_union : SandorA = A₁ ∪ A₂) (h_disj : Disjoint A₁ A₂)\n  (α₁ α₂ : ℝ) (h_dens1 : HasDensity (A₁ + A₁) α₁) (h_dens2 : HasDensity (A₂ + A₂) α₂) :\n  α₁ + α₂ ≤ 1 - α₁ * α₂ := by\n…",
      "source_statement_sha256": "5197b0b7ee88a3331fedb38f90eacd1dcb8bf2c9219561c448ffe5ddc05e6730",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L1199",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.sandor_cross_sums",
      "line": 1291,
      "title": "A fat interval forces almost twice its size in monochromatic sums",
      "intuition": "Partition the interval `[S_x(k),S_y(k))` between two colours. For each colour, a finite nonempty set `U` satisfies `|U+U|≥2|U|-1`; adding the two bounds yields at least `2*(S_y-S_x)-2` same-colour sums in the doubled interval.",
      "source_statement": "lemma sandor_cross_sums (A₁ A₂ : Set ℕ) (h_union : SandorA = A₁ ∪ A₂) (h_disj : Disjoint A₁ A₂) (k : ℕ) (hk : k ≥ 10) :\n  (((A₁ + A₁) ∩ Ico (2 * S_x k) (2 * S_y k)).ncard : ℝ) + (((A₂ + A₂) ∩ Ico (2 * S_x k) (2 * S_y k)).ncard : ℝ)\n    ≥ 2 * ((S_y k : ℝ) - (S_x k : ℝ)) - 2 := by\n…",
      "source_statement_sha256": "9086ac45c093bc18c583a0f6645e6b8b2468d89537fced1b118fcba5f795c194",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L1291",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.SandorA_fluctuation_bounds",
      "line": 1359,
      "title": "Positive limiting densities would still oscillate by a fixed amount",
      "intuition": "Assuming both monochromatic self-sumsets have positive natural-density limits, this lemma constructs lower checkpoints `N(k)=2*S_x(k)` and upper checkpoints `M(k)=2*S_y(k)` where their combined partial density at `M(k)` exceeds that at `N(k)` by one fixed `delta>0` forever.",
      "source_statement": "lemma SandorA_fluctuation_bounds (A₁ A₂ : Set ℕ) (h_union : SandorA = A₁ ∪ A₂) (h_disj : Disjoint A₁ A₂)\n  (α₁ α₂ : ℝ) (h_dens1 : HasDensity (A₁ + A₁) α₁) (h_dens2 : HasDensity (A₂ + A₂) α₂) (h_pos1 : α₁ > 0) (h_pos2 : α₂ > 0) :\n  ∃ (M N : ℕ → ℕ) (K0 : ℕ) (delta : ℝ), delta > 0 ∧ Filter.Tendsto M Filter.atTop Filter.atTop ∧ Filter.Tendsto N Filter.atTop Filter.atTop ∧\n  ∀ k ≥ K0, (((A₁ + A₁) ∩ Iio (M k)).ncard : ℝ) / (M k : ℝ) + (((A₂ + A₂) ∩ Iio (M k)).ncard : ℝ) / (M k : ℝ)\n     ≥ (((A₁ + A₁) ∩ Iio (N k)).ncard : ℝ) / (N k : ℝ) + (((A₂ + A₂) ∩ Iio (N k)).ncard : ℝ) / (N k : ℝ) + delta := by\n…",
      "source_statement_sha256": "4f91574aa3e11e92810e65a80c3ac6d324800a6fd9d6b4bd357f41353df74fb5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L1359",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.SandorA_no_valid_partition",
      "line": 1447,
      "title": "No disjoint two-part split can give both self-sumsets positive natural density",
      "intuition": "Suppose both component self-sumsets had positive limits. Convergence would make each combined partial density arbitrarily close to `α₁+α₂` at every sufficiently large scale, but `SandorA_fluctuation_bounds` produces two arbitrarily large scales separated by a fixed positive gap.",
      "source_statement": "lemma SandorA_no_valid_partition :\n  ∀ A₁ A₂, SandorA = A₁ ∪ A₂ → Disjoint A₁ A₂ → ¬(HasPosDensity (A₁ + A₁) ∧ HasPosDensity (A₂ + A₂)) := by\n…",
      "source_statement_sha256": "c3e34e91279be04deec2ba1ec6c2a1990d208bb0b7b841b090247bb220cf8d9d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L1447",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.exists_erdos_counterexample",
      "line": 1480,
      "title": "The explicit witness refuting the strict-natural-density statement",
      "intuition": "`SandorA` has a self-sumset of natural density one, yet every disjoint two-part split fails to give positive natural-density limits to both component self-sumsets. This is the witness needed to negate the universal strict-density version of Erdős 741(i).",
      "source_statement": "lemma exists_erdos_counterexample : ∃ A : Set ℕ, HasPosDensity (A + A) ∧ ∀ A₁ A₂, A = A₁ ∪ A₂ → Disjoint A₁ A₂ → ¬ (HasPosDensity (A₁ + A₁) ∧ HasPosDensity (A₂ + A₂)) := by\n…",
      "source_statement_sha256": "995a96f0e4b7faf0f47e6c60e4dae590df766fd494c6c942f39ea6ff7ac9c678",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L1480",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.i.lean",
      "declaration": "Erdos741.target_theorem_0",
      "line": 1486,
      "title": "Erdős problem 741(i): strict natural density is false, upper density is true",
      "intuition": "This Lean theorem proves `answer(False)` for `HasPosDensity`, which here means that a positive ordinary natural-density limit exists. The pinned companion proves a different, compatible theorem for positive upper density. The two artifacts do not disagree: density choice changes the answer. The lower-density variant is separate and remains unresolved in the cited status sources.",
      "source_statement": "theorem target_theorem_0\n  : answer(\n  -- EVOLVE-VALUE-START\n  False\n  -- EVOLVE-VALUE-END\n  ) ↔ ∀ A : Set ℕ, HasPosDensity (A + A) → ∃ A₁ A₂, A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ HasPosDensity (A₁ + A₁) ∧ HasPosDensity (A₂ + A₂) := by\n…",
      "source_statement_sha256": "df7411a8564ab1e2a36430b4f3479a163cb82758f5c7d34afa45118264e1aad7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.i.lean#L1486",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_741.parts.ii.lean",
      "declaration": "Erdos741.target_theorem_0",
      "line": 634,
      "title": "Erdős problem 741(ii): an order-two basis with no syndetic two-part split",
      "intuition": "There exists an additive basis `A ∪ {0}` of order two such that every disjoint partition `A=A₁∪A₂` leaves at least one of `A₁+A₁` and `A₂+A₂` with arbitrarily large gaps.",
      "source_statement": "theorem target_theorem_0\n  : answer(True) ↔ ∃ A : Set ℕ, IsAddBasisOfOrder (A ∪ {0}) 2 ∧ ∀ A₁ A₂, A = A₁ ∪ A₂ → Disjoint A₁ A₂ → ¬(IsSyndetic (A₁ + A₁) ∧ IsSyndetic (A₂ + A₂)) := by\n…",
      "source_statement_sha256": "2db309e23a4dc06f26ca8604be5dbe28f1a97ccf841c3824f02246075ea7fbd2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_741.parts.ii.lean#L634",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/ErdosProblems/erdos_846.lean",
      "declaration": "Erdos846.target_theorem_0",
      "line": 909,
      "title": "Erdős problem 846: robustly non-collinear sets need not split into finitely many general-position sets",
      "intuition": "The asserted implication is false. An infinite planar set can have the property that every finite subset contains a line-free subset of at least half its size, yet the whole set is not a finite union of sets with no three collinear.",
      "source_statement": "theorem target_theorem_0\n  : answer(\n  -- EVOLVE-VALUE-START\n  False\n  -- EVOLVE-VALUE-END\n  ) ↔ ∀ᵉ (A : Set ℝ²) (ε > 0), A.Infinite → NonTrilinearFor A ε → WeaklyNonTrilinear A := by\n…",
      "source_statement_sha256": "650fc8ac3b45e5627586e0a0a71c0c9c16d89021a9aca6bfab071dc990749659",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/ErdosProblems/erdos_846.lean#L909",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/OEIS/oeis_51293_conjecture_0.lean",
      "declaration": "target_theorem_0",
      "line": 517,
      "title": "OEIS A051293: a six-term asymptotic expansion for subsets with integer average",
      "intuition": "Let `a(n)` count the nonempty subsets of `{1,…,n}` whose arithmetic mean is an integer. After subtracting the displayed expansion through `541/n⁵`, the error is little-o of `2^(n+1)/n⁶`; the Lean limit states that normalized remainder tends to zero.",
      "source_statement": "theorem target_theorem_0\n  : Tendsto (fun n : ℕ => (a_real n - ((2 : ℝ) ^ (n + 1) / (n : ℝ)) * ((1 : ℝ) + 1 / (n : ℝ) + 3 / ((n : ℝ) ^ 2) + 13 / ((n : ℝ) ^ 3) + 75 / ((n : ℝ) ^ 4) + 541 / ((n : ℝ) ^ 5))) / (((2 : ℝ) ^ (n + 1)) / ((n : ℝ) ^ 6))) atTop (nhds 0) := by\n…",
      "source_statement_sha256": "2c56653171da3b1c57850f188d40866f8c05d01e3a6d243a3e72bb9274b650d8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/OEIS/oeis_51293_conjecture_0.lean#L517",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AdditiveCombinatorics/57.lean",
      "declaration": "Green57.green_57",
      "line": 1120,
      "title": "Ben Green's Problem 57: the two structured-function spaces do not always coincide",
      "intuition": "The proposed equality `Φ(G) = Φ'(G)` is false for finite abelian groups. The pinned development constructs a separating real linear functional for `G = ZMod 3`, so this single finite counterexample refutes the universal equality.",
      "source_statement": "theorem green_57 :\n  answer(False) ↔\n    ∀ (G : Type) [AddCommGroup G] [Fintype G] [DecidableEq G],\n      Φ G = Φ' G := by\n…",
      "source_statement_sha256": "a01e7070f64b28e17a32bba330258192c44a0f23fb115a33f5099f64b88a372d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AdditiveCombinatorics/57.lean#L1120",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Optimization/LastIterateConvergence.lean",
      "declaration": "Arxiv.«1905.10899».last_iterate_convergence",
      "line": 491,
      "title": "Exact last-iterate `O(1/T)` convergence for anchored gradient descent-ascent",
      "intuition": "For a differentiable convex-concave objective whose saddle operator is monotone and Lipschitz, the schedule `α_t = 1/(K√(t+γ))` and `β_t = γ/(t+γ)` with `γ ≥ 2` gives a positive constant `C` such that `‖F(z_T)‖² ≤ C/T` for every `T ≥ 1`.",
      "source_statement": "theorem last_iterate_convergence\n  (L : (Z_space n m) → ℝ)\n  (h_diff : Differentiable ℝ L)\n\n  (h_monotone : ∀ z w : Z_space n m,\n    inner ℝ (F_operator n m L z - F_operator n m L w) (z - w) ≥ (0 : ℝ)) -- Convex-Concave\n\n  (K : NNReal) (h_K_nonneg : 0 < K) -- Lipschitz-Continuous\n  (h_lipschitz : ∀ z w : Z_space n m,\n    ‖F_operator n m L z - F_operator n m L w‖ ≤ K * ‖z - w‖)\n\n  -- Algorithm Parameter\n…",
      "source_statement_sha256": "a58fd712e9128593c0b17fdc9a7f8b832d6951570b30ab31cf6fdf9a8fbb089c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Optimization/LastIterateConvergence.lean#L491",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/Graphs/GraphConjecture2.lean",
      "declaration": "WrittenOnTheWallII.GraphConjecture2.conjecture2",
      "line": 704,
      "title": "Graffiti.pc Conjecture 2: average local independence forces many spanning-tree leaves",
      "intuition": "In every finite connected simple graph, the maximum number `Ls(G)` of leaves in a spanning tree is at least `2(l(G)-1)`, where `l(G)` is the average, over vertices, of the largest independent set inside each open neighbourhood.",
      "source_statement": "theorem conjecture2 (G : SimpleGraph α) (h : G.Connected) :\n  2 * (l G - 1) ≤ Ls G := by\n…",
      "source_statement_sha256": "1096003f7fb655e87b9592c124466161acf9fb4cbe079572f7e28d4982de1e9c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/GraphConjecture2.lean#L704",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/QuantumOptics/35.lean",
      "declaration": "OpenQuantumProblem35.ame_11_5_open",
      "line": 2147,
      "title": "An explicit absolutely maximally entangled state for 11 five-level parties",
      "intuition": "There exists a normalized pure state of 11 parties, each with local dimension 5, whose reduced density matrix on every choice of 5 parties is maximally mixed. The theorem is an existence result with a fully explicit graph-state witness; it does not claim that this is the first mathematical construction of the underlying quantum code.",
      "source_statement": "theorem ame_11_5_open :\n    answer(True) ↔ ExistsAME 11 5 := by\n…",
      "source_statement_sha256": "361bcc15b87bcf063de81f09c6d84c2c3388d2f814647e4bd2ae59b9acb5c278",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/QuantumOptics/35.lean#L2147",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/QuantumOptics/MonochromaticQuantumGraph.lean",
      "declaration": "MonochromaticQuantumGraph.eqSystem_no_solution_even_ge4_d_eq_n_explicit",
      "line": 371,
      "title": "No equal-dimension monochromatic quantum graph for even numbers of parties at least four",
      "intuition": "For every even `N ≥ 4`, no complex edge-weight family on the complete `N`-vertex graph with `N` endpoint colors satisfies the monochromatic quantum-graph equations: every constant vertex coloring should have perfect-matching amplitude 1, while every nonconstant coloring should have amplitude 0. This is the equal-dimension case `D = N`, not the still broader conjecture for every `D ≥ 3` when `N ≥ 6`.",
      "source_statement": "theorem eqSystem_no_solution_even_ge4_d_eq_n_explicit :\n    answer(True) ↔\n      ∀ N : Nat, N ≥ 4 → Even N →\n        ¬ ∃ W : WeightsN N N ℂ, EqSystemN N N W := by\n…",
      "source_statement_sha256": "ef6cf38103351f06b179fad14aa2f7ba871b6ac317155e3628e38e5016d02161",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/QuantumOptics/MonochromaticQuantumGraph.lean#L371",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "Monomial",
      "line": 39,
      "title": "A monomial as its vector of natural-number exponents",
      "intuition": "The Lean type `Monomial(r)` is the finitely supported exponent vector for a monomial in `r` variables. For the manuscript's three-variable theorem, a monomial such as `x₀^a x₁^b x₂^c` is therefore represented by the vector `(a,b,c)` in `ℕ³`.",
      "source_statement": "abbrev Monomial (r : ℕ) := Fin r →₀ ℕ",
      "source_statement_sha256": "ba028d617438bb9b13bbda4dd1bd6d6c61198f94d77f033f7b78200cb836ed0a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "totalDeg",
      "line": 42,
      "title": "Total degree is the sum of a monomial's exponents",
      "intuition": "`totalDeg(m)`—written `totalDeg m` in Lean—adds every coordinate of the exponent vector. In three variables this is `m(0)+m(1)+m(2)`, the usual total degree of the represented monomial.",
      "source_statement": "def totalDeg {r : ℕ} (m : Monomial r) : ℕ :=\n…",
      "source_statement_sha256": "07320fed24bc7ee391c851c6f4759f7f0bc86a339e7944bcb140e420c3a81ac8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "OrderIdeal",
      "line": 47,
      "title": "A finite set of monomials closed downward under divisibility",
      "intuition": "An `OrderIdeal(r)` (Lean: `OrderIdeal r`) stores a finite collection of monomials together with the guarantee that every divisor of a stored monomial is stored too. In exponent-vector form, divisibility is coordinatewise `≤`.",
      "source_statement": "structure OrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "ba122a64b08ac5cf04b4b37d3c8522deb453ddb1d4672639ef155e023ad5efa4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "divisorsFinset",
      "line": 56,
      "title": "All monomial divisors of one generator as a finite set",
      "intuition": "`divisorsFinset(g)` (Lean: `divisorsFinset g`) is the principal lower set of `g`: it contains exactly the monomials `u` whose exponent vector is coordinatewise at most that of `g`. Equivalently, these are the monomial divisors of `g`.",
      "source_statement": "noncomputable def divisorsFinset {r : ℕ} (g : Monomial r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ed048c02d80a7110d87b10e92d84789d3bc0a4c363fdab8ab54ae3df1710eb6e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L56",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "IsMaximalIn",
      "line": 82,
      "title": "Maximality inside an order ideal under monomial divisibility",
      "intuition": "`IsMaximalIn(I,m)` (Lean: `IsMaximalIn I m`) says first that `m` belongs to `I`, and then that no strictly larger monomial of `I` is divisible by `m`. Any member `n` above `m` must also lie below `m`, hence equals it in the partial order.",
      "source_statement": "def IsMaximalIn {r : ℕ} (I : OrderIdeal r) (m : Monomial r) : Prop :=\n…",
      "source_statement_sha256": "a60c8e1aeee59741870c750d748a65267eff8911c395e5aa6d957019b83c6ef7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "IsPure",
      "line": 91,
      "title": "Purity means every maximal monomial has the same total degree",
      "intuition": "`IsPure(I)` (Lean: `IsPure I`) requires any two maximal monomials of `I` to have equal total degree. It does not itself assert that a maximal monomial exists; that nonemptiness is added by `PureOrderIdeal`.",
      "source_statement": "def IsPure {r : ℕ} (I : OrderIdeal r) : Prop :=\n…",
      "source_statement_sha256": "03a865f2ea87c5e18c0d95ebd9a4a40c761c6c84812507389b001eff8a0222c5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "PureOrderIdeal",
      "line": 95,
      "title": "An order ideal packaged with purity and a maximal monomial",
      "intuition": "`PureOrderIdeal(r)` (Lean: `PureOrderIdeal r`) is the structure used when later statements need a pure finite order ideal as data. It contains the ideal, a proof that its maximal monomials share one degree, and a proof that the maximal set is nonempty.",
      "source_statement": "structure PureOrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "6d350f4719325c70dec54ff5b88b3153fe545231a269e9e89eb4e1dcf9320ed4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "typePO",
      "line": 101,
      "title": "The type is the number of maximal monomials",
      "intuition": "`typePO(P)` is the cardinality of `maximalMonomials(P.ideal)`. In Lean these are written `typePO P` and `maximalMonomials P.ideal`. Thus “type 2” in the theorem means that the pure order ideal has exactly two maximal monomial generators.",
      "source_statement": "noncomputable def typePO {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "0d966ca8aba605e0d57dd15479643315e3556c2ece0a816b26838e844e8f8973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "socleDegree",
      "line": 106,
      "title": "The shared total degree of a pure ideal's maximal monomials",
      "intuition": "`socleDegree(P)` (Lean: `socleDegree P`) chooses one maximal monomial and returns its total degree. Purity proves that every other maximal monomial has the same degree, so the result is independent of which witness is chosen.",
      "source_statement": "noncomputable def socleDegree {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "5650301724fb40e83c63529839c28aa33de260f51f059acab3ea978409b21e54",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "pureOSequence",
      "line": 110,
      "title": "Degree-by-degree counts of monomials in an order ideal",
      "intuition": "`pureOSequence(I,d)` (Lean: `pureOSequence I d`) counts the monomials in `I` whose total degree is exactly `d`. Varying `d` produces the Hilbert-function sequence studied by the theorem.",
      "source_statement": "def pureOSequence {r : ℕ} (I : OrderIdeal r) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f9067ca146a946dfce29d85869385ca16043308dd3b50b841447653bbb8429b8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean",
      "declaration": "IsLogConcaveUpTo",
      "line": 122,
      "title": "Log-concavity at every internal degree through a chosen endpoint",
      "intuition": "`IsLogConcaveUpTo(e,h)` (Lean: `IsLogConcaveUpTo e h`) requires `h(i-1)·h(i+1) ≤ h(i)²` for every integer index strictly between `0` and `e`. Each middle term must therefore dominate the geometric mean of its two neighbours.",
      "source_statement": "def IsLogConcaveUpTo (e : ℕ) (h : ℕ → ℕ) : Prop :=\n…",
      "source_statement_sha256": "eaac3ce65f7cb9899a0f38b012844ec5a0a69d7f7766524de03a160408b669b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_1.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "Monomial",
      "line": 39,
      "title": "A monomial as its vector of natural-number exponents",
      "intuition": "The Lean type `Monomial(r)` is the finitely supported exponent vector for a monomial in `r` variables. For the manuscript's three-variable theorem, a monomial such as `x₀^a x₁^b x₂^c` is therefore represented by the vector `(a,b,c)` in `ℕ³`.",
      "source_statement": "abbrev Monomial (r : ℕ) := Fin r →₀ ℕ",
      "source_statement_sha256": "ba028d617438bb9b13bbda4dd1bd6d6c61198f94d77f033f7b78200cb836ed0a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "totalDeg",
      "line": 42,
      "title": "Total degree is the sum of a monomial's exponents",
      "intuition": "`totalDeg(m)`—written `totalDeg m` in Lean—adds every coordinate of the exponent vector. In three variables this is `m(0)+m(1)+m(2)`, the usual total degree of the represented monomial.",
      "source_statement": "def totalDeg {r : ℕ} (m : Monomial r) : ℕ :=\n…",
      "source_statement_sha256": "07320fed24bc7ee391c851c6f4759f7f0bc86a339e7944bcb140e420c3a81ac8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "OrderIdeal",
      "line": 47,
      "title": "A finite set of monomials closed downward under divisibility",
      "intuition": "An `OrderIdeal(r)` (Lean: `OrderIdeal r`) stores a finite collection of monomials together with the guarantee that every divisor of a stored monomial is stored too. In exponent-vector form, divisibility is coordinatewise `≤`.",
      "source_statement": "structure OrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "ba122a64b08ac5cf04b4b37d3c8522deb453ddb1d4672639ef155e023ad5efa4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "divisorsFinset",
      "line": 56,
      "title": "All monomial divisors of one generator as a finite set",
      "intuition": "`divisorsFinset(g)` (Lean: `divisorsFinset g`) is the principal lower set of `g`: it contains exactly the monomials `u` whose exponent vector is coordinatewise at most that of `g`. Equivalently, these are the monomial divisors of `g`.",
      "source_statement": "noncomputable def divisorsFinset {r : ℕ} (g : Monomial r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ed048c02d80a7110d87b10e92d84789d3bc0a4c363fdab8ab54ae3df1710eb6e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L56",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "IsMaximalIn",
      "line": 82,
      "title": "Maximality inside an order ideal under monomial divisibility",
      "intuition": "`IsMaximalIn(I,m)` (Lean: `IsMaximalIn I m`) says first that `m` belongs to `I`, and then that no strictly larger monomial of `I` is divisible by `m`. Any member `n` above `m` must also lie below `m`, hence equals it in the partial order.",
      "source_statement": "def IsMaximalIn {r : ℕ} (I : OrderIdeal r) (m : Monomial r) : Prop :=\n…",
      "source_statement_sha256": "a60c8e1aeee59741870c750d748a65267eff8911c395e5aa6d957019b83c6ef7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "IsPure",
      "line": 91,
      "title": "Purity means every maximal monomial has the same total degree",
      "intuition": "`IsPure(I)` (Lean: `IsPure I`) requires any two maximal monomials of `I` to have equal total degree. It does not itself assert that a maximal monomial exists; that nonemptiness is added by `PureOrderIdeal`.",
      "source_statement": "def IsPure {r : ℕ} (I : OrderIdeal r) : Prop :=\n…",
      "source_statement_sha256": "03a865f2ea87c5e18c0d95ebd9a4a40c761c6c84812507389b001eff8a0222c5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "PureOrderIdeal",
      "line": 95,
      "title": "An order ideal packaged with purity and a maximal monomial",
      "intuition": "`PureOrderIdeal(r)` (Lean: `PureOrderIdeal r`) is the structure used when later statements need a pure finite order ideal as data. It contains the ideal, a proof that its maximal monomials share one degree, and a proof that the maximal set is nonempty.",
      "source_statement": "structure PureOrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "6d350f4719325c70dec54ff5b88b3153fe545231a269e9e89eb4e1dcf9320ed4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "typePO",
      "line": 101,
      "title": "The type is the number of maximal monomials",
      "intuition": "`typePO(P)` is the cardinality of `maximalMonomials(P.ideal)`. In Lean these are written `typePO P` and `maximalMonomials P.ideal`. Thus “type 2” in the theorem means that the pure order ideal has exactly two maximal monomial generators.",
      "source_statement": "noncomputable def typePO {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "0d966ca8aba605e0d57dd15479643315e3556c2ece0a816b26838e844e8f8973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "socleDegree",
      "line": 106,
      "title": "The shared total degree of a pure ideal's maximal monomials",
      "intuition": "`socleDegree(P)` (Lean: `socleDegree P`) chooses one maximal monomial and returns its total degree. Purity proves that every other maximal monomial has the same degree, so the result is independent of which witness is chosen.",
      "source_statement": "noncomputable def socleDegree {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "5650301724fb40e83c63529839c28aa33de260f51f059acab3ea978409b21e54",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "pureOSequence",
      "line": 110,
      "title": "Degree-by-degree counts of monomials in an order ideal",
      "intuition": "`pureOSequence(I,d)` (Lean: `pureOSequence I d`) counts the monomials in `I` whose total degree is exactly `d`. Varying `d` produces the Hilbert-function sequence studied by the theorem.",
      "source_statement": "def pureOSequence {r : ℕ} (I : OrderIdeal r) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f9067ca146a946dfce29d85869385ca16043308dd3b50b841447653bbb8429b8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean",
      "declaration": "IsLogConcaveUpTo",
      "line": 122,
      "title": "Log-concavity at every internal degree through a chosen endpoint",
      "intuition": "`IsLogConcaveUpTo(e,h)` (Lean: `IsLogConcaveUpTo e h`) requires `h(i-1)·h(i+1) ≤ h(i)²` for every integer index strictly between `0` and `e`. Each middle term must therefore dominate the geometric mean of its two neighbours.",
      "source_statement": "def IsLogConcaveUpTo (e : ℕ) (h : ℕ → ℕ) : Prop :=\n…",
      "source_statement_sha256": "eaac3ce65f7cb9899a0f38b012844ec5a0a69d7f7766524de03a160408b669b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_2.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "Monomial",
      "line": 39,
      "title": "A monomial as its vector of natural-number exponents",
      "intuition": "The Lean type `Monomial(r)` is the finitely supported exponent vector for a monomial in `r` variables. For the manuscript's three-variable theorem, a monomial such as `x₀^a x₁^b x₂^c` is therefore represented by the vector `(a,b,c)` in `ℕ³`.",
      "source_statement": "abbrev Monomial (r : ℕ) := Fin r →₀ ℕ",
      "source_statement_sha256": "ba028d617438bb9b13bbda4dd1bd6d6c61198f94d77f033f7b78200cb836ed0a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "totalDeg",
      "line": 42,
      "title": "Total degree is the sum of a monomial's exponents",
      "intuition": "`totalDeg(m)`—written `totalDeg m` in Lean—adds every coordinate of the exponent vector. In three variables this is `m(0)+m(1)+m(2)`, the usual total degree of the represented monomial.",
      "source_statement": "def totalDeg {r : ℕ} (m : Monomial r) : ℕ :=\n…",
      "source_statement_sha256": "07320fed24bc7ee391c851c6f4759f7f0bc86a339e7944bcb140e420c3a81ac8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "OrderIdeal",
      "line": 47,
      "title": "A finite set of monomials closed downward under divisibility",
      "intuition": "An `OrderIdeal(r)` (Lean: `OrderIdeal r`) stores a finite collection of monomials together with the guarantee that every divisor of a stored monomial is stored too. In exponent-vector form, divisibility is coordinatewise `≤`.",
      "source_statement": "structure OrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "ba122a64b08ac5cf04b4b37d3c8522deb453ddb1d4672639ef155e023ad5efa4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "divisorsFinset",
      "line": 56,
      "title": "All monomial divisors of one generator as a finite set",
      "intuition": "`divisorsFinset(g)` (Lean: `divisorsFinset g`) is the principal lower set of `g`: it contains exactly the monomials `u` whose exponent vector is coordinatewise at most that of `g`. Equivalently, these are the monomial divisors of `g`.",
      "source_statement": "noncomputable def divisorsFinset {r : ℕ} (g : Monomial r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ed048c02d80a7110d87b10e92d84789d3bc0a4c363fdab8ab54ae3df1710eb6e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L56",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "IsMaximalIn",
      "line": 82,
      "title": "Maximality inside an order ideal under monomial divisibility",
      "intuition": "`IsMaximalIn(I,m)` (Lean: `IsMaximalIn I m`) says first that `m` belongs to `I`, and then that no strictly larger monomial of `I` is divisible by `m`. Any member `n` above `m` must also lie below `m`, hence equals it in the partial order.",
      "source_statement": "def IsMaximalIn {r : ℕ} (I : OrderIdeal r) (m : Monomial r) : Prop :=\n…",
      "source_statement_sha256": "a60c8e1aeee59741870c750d748a65267eff8911c395e5aa6d957019b83c6ef7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "IsPure",
      "line": 91,
      "title": "Purity means every maximal monomial has the same total degree",
      "intuition": "`IsPure(I)` (Lean: `IsPure I`) requires any two maximal monomials of `I` to have equal total degree. It does not itself assert that a maximal monomial exists; that nonemptiness is added by `PureOrderIdeal`.",
      "source_statement": "def IsPure {r : ℕ} (I : OrderIdeal r) : Prop :=\n…",
      "source_statement_sha256": "03a865f2ea87c5e18c0d95ebd9a4a40c761c6c84812507389b001eff8a0222c5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "PureOrderIdeal",
      "line": 95,
      "title": "An order ideal packaged with purity and a maximal monomial",
      "intuition": "`PureOrderIdeal(r)` (Lean: `PureOrderIdeal r`) is the structure used when later statements need a pure finite order ideal as data. It contains the ideal, a proof that its maximal monomials share one degree, and a proof that the maximal set is nonempty.",
      "source_statement": "structure PureOrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "6d350f4719325c70dec54ff5b88b3153fe545231a269e9e89eb4e1dcf9320ed4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "typePO",
      "line": 101,
      "title": "The type is the number of maximal monomials",
      "intuition": "`typePO(P)` is the cardinality of `maximalMonomials(P.ideal)`. In Lean these are written `typePO P` and `maximalMonomials P.ideal`. Thus “type 2” in the theorem means that the pure order ideal has exactly two maximal monomial generators.",
      "source_statement": "noncomputable def typePO {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "0d966ca8aba605e0d57dd15479643315e3556c2ece0a816b26838e844e8f8973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "socleDegree",
      "line": 106,
      "title": "The shared total degree of a pure ideal's maximal monomials",
      "intuition": "`socleDegree(P)` (Lean: `socleDegree P`) chooses one maximal monomial and returns its total degree. Purity proves that every other maximal monomial has the same degree, so the result is independent of which witness is chosen.",
      "source_statement": "noncomputable def socleDegree {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "5650301724fb40e83c63529839c28aa33de260f51f059acab3ea978409b21e54",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "pureOSequence",
      "line": 110,
      "title": "Degree-by-degree counts of monomials in an order ideal",
      "intuition": "`pureOSequence(I,d)` (Lean: `pureOSequence I d`) counts the monomials in `I` whose total degree is exactly `d`. Varying `d` produces the Hilbert-function sequence studied by the theorem.",
      "source_statement": "def pureOSequence {r : ℕ} (I : OrderIdeal r) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f9067ca146a946dfce29d85869385ca16043308dd3b50b841447653bbb8429b8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean",
      "declaration": "IsLogConcaveUpTo",
      "line": 122,
      "title": "Log-concavity at every internal degree through a chosen endpoint",
      "intuition": "`IsLogConcaveUpTo(e,h)` (Lean: `IsLogConcaveUpTo e h`) requires `h(i-1)·h(i+1) ≤ h(i)²` for every integer index strictly between `0` and `e`. Each middle term must therefore dominate the geometric mean of its two neighbours.",
      "source_statement": "def IsLogConcaveUpTo (e : ℕ) (h : ℕ → ℕ) : Prop :=\n…",
      "source_statement_sha256": "eaac3ce65f7cb9899a0f38b012844ec5a0a69d7f7766524de03a160408b669b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_3.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "Monomial",
      "line": 39,
      "title": "A monomial as its vector of natural-number exponents",
      "intuition": "The Lean type `Monomial(r)` is the finitely supported exponent vector for a monomial in `r` variables. For the manuscript's three-variable theorem, a monomial such as `x₀^a x₁^b x₂^c` is therefore represented by the vector `(a,b,c)` in `ℕ³`.",
      "source_statement": "abbrev Monomial (r : ℕ) := Fin r →₀ ℕ",
      "source_statement_sha256": "ba028d617438bb9b13bbda4dd1bd6d6c61198f94d77f033f7b78200cb836ed0a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "totalDeg",
      "line": 42,
      "title": "Total degree is the sum of a monomial's exponents",
      "intuition": "`totalDeg(m)`—written `totalDeg m` in Lean—adds every coordinate of the exponent vector. In three variables this is `m(0)+m(1)+m(2)`, the usual total degree of the represented monomial.",
      "source_statement": "def totalDeg {r : ℕ} (m : Monomial r) : ℕ :=\n…",
      "source_statement_sha256": "07320fed24bc7ee391c851c6f4759f7f0bc86a339e7944bcb140e420c3a81ac8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "OrderIdeal",
      "line": 47,
      "title": "A finite set of monomials closed downward under divisibility",
      "intuition": "An `OrderIdeal(r)` (Lean: `OrderIdeal r`) stores a finite collection of monomials together with the guarantee that every divisor of a stored monomial is stored too. In exponent-vector form, divisibility is coordinatewise `≤`.",
      "source_statement": "structure OrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "ba122a64b08ac5cf04b4b37d3c8522deb453ddb1d4672639ef155e023ad5efa4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "divisorsFinset",
      "line": 56,
      "title": "All monomial divisors of one generator as a finite set",
      "intuition": "`divisorsFinset(g)` (Lean: `divisorsFinset g`) is the principal lower set of `g`: it contains exactly the monomials `u` whose exponent vector is coordinatewise at most that of `g`. Equivalently, these are the monomial divisors of `g`.",
      "source_statement": "noncomputable def divisorsFinset {r : ℕ} (g : Monomial r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ed048c02d80a7110d87b10e92d84789d3bc0a4c363fdab8ab54ae3df1710eb6e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L56",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "IsMaximalIn",
      "line": 82,
      "title": "Maximality inside an order ideal under monomial divisibility",
      "intuition": "`IsMaximalIn(I,m)` (Lean: `IsMaximalIn I m`) says first that `m` belongs to `I`, and then that no strictly larger monomial of `I` is divisible by `m`. Any member `n` above `m` must also lie below `m`, hence equals it in the partial order.",
      "source_statement": "def IsMaximalIn {r : ℕ} (I : OrderIdeal r) (m : Monomial r) : Prop :=\n…",
      "source_statement_sha256": "a60c8e1aeee59741870c750d748a65267eff8911c395e5aa6d957019b83c6ef7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "IsPure",
      "line": 91,
      "title": "Purity means every maximal monomial has the same total degree",
      "intuition": "`IsPure(I)` (Lean: `IsPure I`) requires any two maximal monomials of `I` to have equal total degree. It does not itself assert that a maximal monomial exists; that nonemptiness is added by `PureOrderIdeal`.",
      "source_statement": "def IsPure {r : ℕ} (I : OrderIdeal r) : Prop :=\n…",
      "source_statement_sha256": "03a865f2ea87c5e18c0d95ebd9a4a40c761c6c84812507389b001eff8a0222c5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "PureOrderIdeal",
      "line": 95,
      "title": "An order ideal packaged with purity and a maximal monomial",
      "intuition": "`PureOrderIdeal(r)` (Lean: `PureOrderIdeal r`) is the structure used when later statements need a pure finite order ideal as data. It contains the ideal, a proof that its maximal monomials share one degree, and a proof that the maximal set is nonempty.",
      "source_statement": "structure PureOrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "6d350f4719325c70dec54ff5b88b3153fe545231a269e9e89eb4e1dcf9320ed4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "typePO",
      "line": 101,
      "title": "The type is the number of maximal monomials",
      "intuition": "`typePO(P)` is the cardinality of `maximalMonomials(P.ideal)`. In Lean these are written `typePO P` and `maximalMonomials P.ideal`. Thus “type 2” in the theorem means that the pure order ideal has exactly two maximal monomial generators.",
      "source_statement": "noncomputable def typePO {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "0d966ca8aba605e0d57dd15479643315e3556c2ece0a816b26838e844e8f8973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "socleDegree",
      "line": 106,
      "title": "The shared total degree of a pure ideal's maximal monomials",
      "intuition": "`socleDegree(P)` (Lean: `socleDegree P`) chooses one maximal monomial and returns its total degree. Purity proves that every other maximal monomial has the same degree, so the result is independent of which witness is chosen.",
      "source_statement": "noncomputable def socleDegree {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "5650301724fb40e83c63529839c28aa33de260f51f059acab3ea978409b21e54",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "pureOSequence",
      "line": 110,
      "title": "Degree-by-degree counts of monomials in an order ideal",
      "intuition": "`pureOSequence(I,d)` (Lean: `pureOSequence I d`) counts the monomials in `I` whose total degree is exactly `d`. Varying `d` produces the Hilbert-function sequence studied by the theorem.",
      "source_statement": "def pureOSequence {r : ℕ} (I : OrderIdeal r) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f9067ca146a946dfce29d85869385ca16043308dd3b50b841447653bbb8429b8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean",
      "declaration": "IsLogConcaveUpTo",
      "line": 122,
      "title": "Log-concavity at every internal degree through a chosen endpoint",
      "intuition": "`IsLogConcaveUpTo(e,h)` (Lean: `IsLogConcaveUpTo e h`) requires `h(i-1)·h(i+1) ≤ h(i)²` for every integer index strictly between `0` and `e`. Each middle term must therefore dominate the geometric mean of its two neighbours.",
      "source_statement": "def IsLogConcaveUpTo (e : ℕ) (h : ℕ → ℕ) : Prop :=\n…",
      "source_statement_sha256": "eaac3ce65f7cb9899a0f38b012844ec5a0a69d7f7766524de03a160408b669b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_4.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "Monomial",
      "line": 39,
      "title": "A monomial as its vector of natural-number exponents",
      "intuition": "The Lean type `Monomial(r)` is the finitely supported exponent vector for a monomial in `r` variables. For the manuscript's three-variable theorem, a monomial such as `x₀^a x₁^b x₂^c` is therefore represented by the vector `(a,b,c)` in `ℕ³`.",
      "source_statement": "abbrev Monomial (r : ℕ) := Fin r →₀ ℕ",
      "source_statement_sha256": "ba028d617438bb9b13bbda4dd1bd6d6c61198f94d77f033f7b78200cb836ed0a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "totalDeg",
      "line": 42,
      "title": "Total degree is the sum of a monomial's exponents",
      "intuition": "`totalDeg(m)`—written `totalDeg m` in Lean—adds every coordinate of the exponent vector. In three variables this is `m(0)+m(1)+m(2)`, the usual total degree of the represented monomial.",
      "source_statement": "def totalDeg {r : ℕ} (m : Monomial r) : ℕ :=\n…",
      "source_statement_sha256": "07320fed24bc7ee391c851c6f4759f7f0bc86a339e7944bcb140e420c3a81ac8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "OrderIdeal",
      "line": 47,
      "title": "A finite set of monomials closed downward under divisibility",
      "intuition": "An `OrderIdeal(r)` (Lean: `OrderIdeal r`) stores a finite collection of monomials together with the guarantee that every divisor of a stored monomial is stored too. In exponent-vector form, divisibility is coordinatewise `≤`.",
      "source_statement": "structure OrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "ba122a64b08ac5cf04b4b37d3c8522deb453ddb1d4672639ef155e023ad5efa4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "divisorsFinset",
      "line": 56,
      "title": "All monomial divisors of one generator as a finite set",
      "intuition": "`divisorsFinset(g)` (Lean: `divisorsFinset g`) is the principal lower set of `g`: it contains exactly the monomials `u` whose exponent vector is coordinatewise at most that of `g`. Equivalently, these are the monomial divisors of `g`.",
      "source_statement": "noncomputable def divisorsFinset {r : ℕ} (g : Monomial r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ed048c02d80a7110d87b10e92d84789d3bc0a4c363fdab8ab54ae3df1710eb6e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L56",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "IsMaximalIn",
      "line": 82,
      "title": "Maximality inside an order ideal under monomial divisibility",
      "intuition": "`IsMaximalIn(I,m)` (Lean: `IsMaximalIn I m`) says first that `m` belongs to `I`, and then that no strictly larger monomial of `I` is divisible by `m`. Any member `n` above `m` must also lie below `m`, hence equals it in the partial order.",
      "source_statement": "def IsMaximalIn {r : ℕ} (I : OrderIdeal r) (m : Monomial r) : Prop :=\n…",
      "source_statement_sha256": "a60c8e1aeee59741870c750d748a65267eff8911c395e5aa6d957019b83c6ef7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "IsPure",
      "line": 91,
      "title": "Purity means every maximal monomial has the same total degree",
      "intuition": "`IsPure(I)` (Lean: `IsPure I`) requires any two maximal monomials of `I` to have equal total degree. It does not itself assert that a maximal monomial exists; that nonemptiness is added by `PureOrderIdeal`.",
      "source_statement": "def IsPure {r : ℕ} (I : OrderIdeal r) : Prop :=\n…",
      "source_statement_sha256": "03a865f2ea87c5e18c0d95ebd9a4a40c761c6c84812507389b001eff8a0222c5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "PureOrderIdeal",
      "line": 95,
      "title": "An order ideal packaged with purity and a maximal monomial",
      "intuition": "`PureOrderIdeal(r)` (Lean: `PureOrderIdeal r`) is the structure used when later statements need a pure finite order ideal as data. It contains the ideal, a proof that its maximal monomials share one degree, and a proof that the maximal set is nonempty.",
      "source_statement": "structure PureOrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "6d350f4719325c70dec54ff5b88b3153fe545231a269e9e89eb4e1dcf9320ed4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "typePO",
      "line": 101,
      "title": "The type is the number of maximal monomials",
      "intuition": "`typePO(P)` is the cardinality of `maximalMonomials(P.ideal)`. In Lean these are written `typePO P` and `maximalMonomials P.ideal`. Thus “type 2” in the theorem means that the pure order ideal has exactly two maximal monomial generators.",
      "source_statement": "noncomputable def typePO {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "0d966ca8aba605e0d57dd15479643315e3556c2ece0a816b26838e844e8f8973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "socleDegree",
      "line": 106,
      "title": "The shared total degree of a pure ideal's maximal monomials",
      "intuition": "`socleDegree(P)` (Lean: `socleDegree P`) chooses one maximal monomial and returns its total degree. Purity proves that every other maximal monomial has the same degree, so the result is independent of which witness is chosen.",
      "source_statement": "noncomputable def socleDegree {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "5650301724fb40e83c63529839c28aa33de260f51f059acab3ea978409b21e54",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "pureOSequence",
      "line": 110,
      "title": "Degree-by-degree counts of monomials in an order ideal",
      "intuition": "`pureOSequence(I,d)` (Lean: `pureOSequence I d`) counts the monomials in `I` whose total degree is exactly `d`. Varying `d` produces the Hilbert-function sequence studied by the theorem.",
      "source_statement": "def pureOSequence {r : ℕ} (I : OrderIdeal r) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f9067ca146a946dfce29d85869385ca16043308dd3b50b841447653bbb8429b8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean",
      "declaration": "IsLogConcaveUpTo",
      "line": 122,
      "title": "Log-concavity at every internal degree through a chosen endpoint",
      "intuition": "`IsLogConcaveUpTo(e,h)` (Lean: `IsLogConcaveUpTo e h`) requires `h(i-1)·h(i+1) ≤ h(i)²` for every integer index strictly between `0` and `e`. Each middle term must therefore dominate the geometric mean of its two neighbours.",
      "source_statement": "def IsLogConcaveUpTo (e : ℕ) (h : ℕ → ℕ) : Prop :=\n…",
      "source_statement_sha256": "eaac3ce65f7cb9899a0f38b012844ec5a0a69d7f7766524de03a160408b669b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_7.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "Monomial",
      "line": 39,
      "title": "A monomial as its vector of natural-number exponents",
      "intuition": "The Lean type `Monomial(r)` is the finitely supported exponent vector for a monomial in `r` variables. For the manuscript's three-variable theorem, a monomial such as `x₀^a x₁^b x₂^c` is therefore represented by the vector `(a,b,c)` in `ℕ³`.",
      "source_statement": "abbrev Monomial (r : ℕ) := Fin r →₀ ℕ",
      "source_statement_sha256": "ba028d617438bb9b13bbda4dd1bd6d6c61198f94d77f033f7b78200cb836ed0a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "totalDeg",
      "line": 42,
      "title": "Total degree is the sum of a monomial's exponents",
      "intuition": "`totalDeg(m)`—written `totalDeg m` in Lean—adds every coordinate of the exponent vector. In three variables this is `m(0)+m(1)+m(2)`, the usual total degree of the represented monomial.",
      "source_statement": "def totalDeg {r : ℕ} (m : Monomial r) : ℕ :=\n…",
      "source_statement_sha256": "07320fed24bc7ee391c851c6f4759f7f0bc86a339e7944bcb140e420c3a81ac8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "OrderIdeal",
      "line": 47,
      "title": "A finite set of monomials closed downward under divisibility",
      "intuition": "An `OrderIdeal(r)` (Lean: `OrderIdeal r`) stores a finite collection of monomials together with the guarantee that every divisor of a stored monomial is stored too. In exponent-vector form, divisibility is coordinatewise `≤`.",
      "source_statement": "structure OrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "ba122a64b08ac5cf04b4b37d3c8522deb453ddb1d4672639ef155e023ad5efa4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "divisorsFinset",
      "line": 56,
      "title": "All monomial divisors of one generator as a finite set",
      "intuition": "`divisorsFinset(g)` (Lean: `divisorsFinset g`) is the principal lower set of `g`: it contains exactly the monomials `u` whose exponent vector is coordinatewise at most that of `g`. Equivalently, these are the monomial divisors of `g`.",
      "source_statement": "noncomputable def divisorsFinset {r : ℕ} (g : Monomial r) : Finset (Monomial r) :=\n…",
      "source_statement_sha256": "ed048c02d80a7110d87b10e92d84789d3bc0a4c363fdab8ab54ae3df1710eb6e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L56",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "IsMaximalIn",
      "line": 82,
      "title": "Maximality inside an order ideal under monomial divisibility",
      "intuition": "`IsMaximalIn(I,m)` (Lean: `IsMaximalIn I m`) says first that `m` belongs to `I`, and then that no strictly larger monomial of `I` is divisible by `m`. Any member `n` above `m` must also lie below `m`, hence equals it in the partial order.",
      "source_statement": "def IsMaximalIn {r : ℕ} (I : OrderIdeal r) (m : Monomial r) : Prop :=\n…",
      "source_statement_sha256": "a60c8e1aeee59741870c750d748a65267eff8911c395e5aa6d957019b83c6ef7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "IsPure",
      "line": 91,
      "title": "Purity means every maximal monomial has the same total degree",
      "intuition": "`IsPure(I)` (Lean: `IsPure I`) requires any two maximal monomials of `I` to have equal total degree. It does not itself assert that a maximal monomial exists; that nonemptiness is added by `PureOrderIdeal`.",
      "source_statement": "def IsPure {r : ℕ} (I : OrderIdeal r) : Prop :=\n…",
      "source_statement_sha256": "03a865f2ea87c5e18c0d95ebd9a4a40c761c6c84812507389b001eff8a0222c5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L91",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "PureOrderIdeal",
      "line": 95,
      "title": "An order ideal packaged with purity and a maximal monomial",
      "intuition": "`PureOrderIdeal(r)` (Lean: `PureOrderIdeal r`) is the structure used when later statements need a pure finite order ideal as data. It contains the ideal, a proof that its maximal monomials share one degree, and a proof that the maximal set is nonempty.",
      "source_statement": "structure PureOrderIdeal (r : ℕ) where\n…",
      "source_statement_sha256": "6d350f4719325c70dec54ff5b88b3153fe545231a269e9e89eb4e1dcf9320ed4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L95",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "typePO",
      "line": 101,
      "title": "The type is the number of maximal monomials",
      "intuition": "`typePO(P)` is the cardinality of `maximalMonomials(P.ideal)`. In Lean these are written `typePO P` and `maximalMonomials P.ideal`. Thus “type 2” in the theorem means that the pure order ideal has exactly two maximal monomial generators.",
      "source_statement": "noncomputable def typePO {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "0d966ca8aba605e0d57dd15479643315e3556c2ece0a816b26838e844e8f8973",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "socleDegree",
      "line": 106,
      "title": "The shared total degree of a pure ideal's maximal monomials",
      "intuition": "`socleDegree(P)` (Lean: `socleDegree P`) chooses one maximal monomial and returns its total degree. Purity proves that every other maximal monomial has the same degree, so the result is independent of which witness is chosen.",
      "source_statement": "noncomputable def socleDegree {r : ℕ} (P : PureOrderIdeal r) : ℕ :=\n…",
      "source_statement_sha256": "5650301724fb40e83c63529839c28aa33de260f51f059acab3ea978409b21e54",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L106",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "pureOSequence",
      "line": 110,
      "title": "Degree-by-degree counts of monomials in an order ideal",
      "intuition": "`pureOSequence(I,d)` (Lean: `pureOSequence I d`) counts the monomials in `I` whose total degree is exactly `d`. Varying `d` produces the Hilbert-function sequence studied by the theorem.",
      "source_statement": "def pureOSequence {r : ℕ} (I : OrderIdeal r) (d : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "f9067ca146a946dfce29d85869385ca16043308dd3b50b841447653bbb8429b8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L110",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "alphaproof-nexus",
      "file": "APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean",
      "declaration": "IsLogConcaveUpTo",
      "line": 122,
      "title": "Log-concavity at every internal degree through a chosen endpoint",
      "intuition": "`IsLogConcaveUpTo(e,h)` (Lean: `IsLogConcaveUpTo e h`) requires `h(i-1)·h(i+1) ≤ h(i)²` for every integer index strictly between `0` and `e`. Each middle term must therefore dominate the geometric mean of its two neighbours.",
      "source_statement": "def IsLogConcaveUpTo (e : ℕ) (h : ℕ → ℕ) : Prop :=\n…",
      "source_statement_sha256": "eaac3ce65f7cb9899a0f38b012844ec5a0a69d7f7766524de03a160408b669b3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/AlgebraicGeometry/hilbert_functions_8.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned AlphaProof Nexus results documentation",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/README.md"
        },
        {
          "label": "AlphaProof Nexus paper",
          "url": "https://doi.org/10.1126/science.aej2213"
        },
        {
          "label": "Pinned human-written natural-language proof companions",
          "url": "https://github.com/google-deepmind/alphaproof-nexus-results/tree/0647711a71183c1ea492ad60860776617ce1ea88/NaturalLanguageProofs"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/Defs.lean",
      "declaration": "ClassicalCarleson",
      "line": 298,
      "title": "Carleson's theorem (statement)",
      "intuition": "For every continuous 2*pi-periodic complex function f, the partial Fourier sums of f converge to f at almost every point.",
      "source_statement": "def ClassicalCarleson : Prop :=\n…",
      "source_statement_sha256": "dc79cabbbb56b71033b8437568009cb26a89c090c344a425d2ce2b4b5f8331b9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/Defs.lean#L298",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/Defs.lean",
      "declaration": "partialFourierSum",
      "line": 43,
      "title": "Partial Fourier sum",
      "intuition": "The N-th partial sum adds the Fourier modes of f with frequency from -N to N, using coefficients taken over one period [0, 2*pi].",
      "source_statement": "def partialFourierSum (N : ℕ) (f : ℝ → ℂ) (x : ℝ) : ℂ := ∑ n ∈ Finset.Icc (-(N : ℤ)) N,\n…",
      "source_statement_sha256": "2fedc335226744bc4320263c68627209869bf19fd33575b913c8146d0056d401",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/Defs.lean#L43",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/Defs.lean",
      "declaration": "carlesonOperator",
      "line": 228,
      "title": "The generalized Carleson operator",
      "intuition": "It takes, at each point, the largest truncated oscillatory integral of K times f against a phase, over all phases and all inner/outer truncation radii.",
      "source_statement": "def carlesonOperator [FunctionDistances ℝ X] (K : X → X → ℂ) (f : X → ℂ) (x : X) : ℝ≥0∞ :=\n…",
      "source_statement_sha256": "6134843fa501ac9c59fd401eadffeab86e03c3ea259ae46986435d2a0114a049",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/Defs.lean#L228",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/Defs.lean",
      "declaration": "KernelProofData",
      "line": 261,
      "title": "Standing assumptions for the kernel chapters",
      "intuition": "This class packages everything the early chapters need: a doubling measure, a parameter a with 4 at most a, compatible function distances, and a one-sided Calderon-Zygmund kernel K.",
      "source_statement": "class KernelProofData {X : Type*} (a : outParam ℕ) (K : outParam (X → X → ℂ))\n    [PseudoMetricSpace X] where\n…",
      "source_statement_sha256": "2a1d2be799733f3502576b40ef92ca5a62c3d0309d6ba13a0747d7908549abbc",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/Defs.lean#L261",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/Defs.lean",
      "declaration": "MetricSpaceCarleson",
      "line": 304,
      "title": "Metric space Carleson theorem (statement)",
      "intuition": "In a doubling metric measure space with a Calderon-Zygmund kernel whose truncated nontangential operator is L2-bounded, the Carleson operator obeys a restricted weak-type estimate integrated over G against F.",
      "source_statement": "def MetricSpaceCarleson : Prop :=\n…",
      "source_statement_sha256": "0fe4cd9b6c4d03001f26cca7aec424f6ac988fc0e3a939b731aac5d3239b0779",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/Defs.lean#L304",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/MetricCarleson/Main.lean",
      "declaration": "metric_carleson",
      "line": 203,
      "title": "Metric Carleson theorem (Theorem 1.1.1)",
      "intuition": "The sup over phases is reduced to a countable dense set of phases and rational radii, then to finitely many phases, where the linearized theorem applies, and a monotone limit passes the bound to the supremum.",
      "source_statement": "theorem metric_carleson [IsCancellative X (defaultτ a)]\n    (hq : q ∈ Ioc 1 2) (hqq' : q.HolderConjugate q') (mF : MeasurableSet F) (mG : MeasurableSet G)\n    (mf : Measurable f) (nf : (‖f ·‖) ≤ F.indicator 1)\n    (hT : HasBoundedStrongType (nontangentialOperator K · ·) 2 2 volume volume (C_Ts a)) :\n    ∫⁻ x in G, carlesonOperator K f x ≤\n    C1_0_2 a q * volume G ^ (q' : ℝ)⁻¹ * volume F ^ (q : ℝ)⁻¹ := by\n…",
      "source_statement_sha256": "e16f7064a5cbb12e13acc59069fbc3d471e6cff075ae80ec7620c924bcfda2fe",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/MetricCarleson/Main.lean#L203",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/MetricCarleson/Linearized.lean",
      "declaration": "linearized_metric_carleson",
      "line": 109,
      "title": "Linearized metric Carleson theorem (Theorem 1.1.2)",
      "intuition": "For a fixed phase choice Q, truncate the radii in an increasing family, bound each truncation, and pass to the limit by monotone convergence.",
      "source_statement": "theorem linearized_metric_carleson [IsCancellative X (defaultτ a)]\n    (hq : q ∈ Ioc 1 2) (hqq' : q.HolderConjugate q') (mF : MeasurableSet F) (mG : MeasurableSet G)\n    (mf : Measurable f) (nf : (‖f ·‖) ≤ F.indicator 1)\n    (hT : ∀ θ : Θ X, HasBoundedStrongType (linearizedNontangentialOperator Q θ K · ·)\n      2 2 volume volume (C_Ts a)) :\n    ∫⁻ x in G, linearizedCarlesonOperator Q K f x ≤\n    C1_0_2 a q * volume G ^ (q' : ℝ)⁻¹ * volume F ^ (q : ℝ)⁻¹ := by\n…",
      "source_statement_sha256": "bdbd33632dcf1339c6977f6d5f5fa3dfd05dfdf3a1dac5fe51aeeb9725f04cc6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/MetricCarleson/Linearized.lean#L109",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/Classical/ClassicalCarleson.lean",
      "declaration": "classical_carleson",
      "line": 131,
      "title": "Classical Carleson theorem",
      "intuition": "Convergence a.e. on one period, together with periodicity, gives convergence a.e. on the whole real line.",
      "source_statement": "theorem classical_carleson {f : ℝ → ℂ} (cont_f : Continuous f) (periodic_f : f.Periodic (2 * π)) :\n    ∀ᵐ x, Tendsto (S_ · f x) atTop (𝓝 (f x)) := by\n…",
      "source_statement_sha256": "0998b6ec506eee25ba4dc26b85bb25277e6171ae048ecfc91cfcf610a76606f0",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/Classical/ClassicalCarleson.lean#L131",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/Classical/CarlesonHunt.lean",
      "declaration": "classical_carleson_from_carleson_hunt_real",
      "line": 162,
      "title": "Classical Carleson from the Carleson-Hunt theorem",
      "intuition": "A continuous periodic function is bounded and hence lies in every L^p, so the Carleson-Hunt a.e. convergence result for L^p functions applies.",
      "source_statement": "theorem classical_carleson_from_carleson_hunt_real : ClassicalCarleson := by\n…",
      "source_statement_sha256": "48c943d9fb34838808831a86bac6a6bdbd83e9e50f74e824ca91718f86eba2fa",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/Classical/CarlesonHunt.lean#L162",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "carleson",
      "file": "Carleson/TwoSidedCarleson/RestrictedWeakType.lean",
      "declaration": "two_sided_metric_carleson_hasStrongType",
      "line": 126,
      "title": "Strong-type bound for the Carleson operator (Theorem 10.0.1)",
      "intuition": "Lorentz-type bounds at two exponents bracketing q are combined by real interpolation to give a strong (q,q) bound.",
      "source_statement": "theorem two_sided_metric_carleson_hasStrongType\n  [Countable (Θ X)] (ha : 4 ≤ a) (hq : q ∈ Ioo 1 2)\n    (hT : ∀ r > 0, HasBoundedStrongType (czOperator K r) 2 2 volume volume (C_Ts a)) :\n      HasStrongType (carlesonOperator K) q q volume volume\n        (C_carleson_hasStrongType a q) := by\n…",
      "source_statement_sha256": "ca8d6a392363ef56599284faca57393b4284127c51117959a90f6df3c7f23023",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/Carleson/TwoSidedCarleson/RestrictedWeakType.lean#L126",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/README.md"
        },
        {
          "label": "Pinned Carleson blueprint",
          "url": "https://github.com/fpvandoorn/carleson/blob/8972acdfd953c63bb2501ace20d61ced7c0697ec/blueprint/src/chapter/main.tex"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo1959P1.lean",
      "declaration": "Imo1959P1.imo1959_p1",
      "line": 50,
      "title": "IMO 1959 Problem 1: an irreducible fraction",
      "intuition": "The fraction (21n+4)/(14n+3) is in lowest terms for every natural number n.",
      "source_statement": "problem imo1959_p1 : ∀ n : ℕ, Nat.Coprime (21 * n + 4) (14 * n + 3) :=\n…",
      "source_statement_sha256": "4d2faf052058a88737d6ef23900f0422d5933c0bad32e3f1c2ddd8a33cd696a3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo1959P1.lean#L50",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo1962P1.lean",
      "declaration": "Imo1962P1.imo1962_p1",
      "line": 140,
      "title": "IMO 1962 Problem 1: moving the last digit 6 to the front",
      "intuition": "The smallest natural number ending in 6 whose value becomes four times larger when that 6 is moved to the front is 153846.",
      "source_statement": "problem imo1962_p1 : IsLeast {n | ProblemPredicate n} solution :=\n…",
      "source_statement_sha256": "9e5662f98bc43928f1e88f2d3c14e1696b47416a3e3b89f103b0461d84df1f4c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo1962P1.lean#L140",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo1964P1.lean",
      "declaration": "Imo1964P1.imo1964_p1a",
      "line": 32,
      "title": "IMO 1964 Problem 1(a): when 7 divides 2^n - 1",
      "intuition": "2^n is congruent to 1 modulo 7 exactly when n is a multiple of 3.",
      "source_statement": "problem imo1964_p1a (n : ℕ) : n ∈ solution_set ↔ 2^n ≡ 1 [MOD 7] := by\n…",
      "source_statement_sha256": "809d9bf23dda2c4739ab8d2596c0bd9226c2794b36467fb94a71b40f48c5c948",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo1964P1.lean#L32",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo1988P6.lean",
      "declaration": "Imo1988P6.imo1988_p6",
      "line": 201,
      "title": "IMO 1988 Problem 6: Vieta jumping",
      "intuition": "If ab+1 divides a^2+b^2 for natural numbers a and b, then the quotient is a perfect square.",
      "source_statement": "problem imo1988_p6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :\n    ∃ d, d ^ 2 = (a ^ 2 + b ^ 2) / (a * b + 1) := by\n…",
      "source_statement_sha256": "58b028d5a18bb4218f267057ee1a0cc8d8143e4d8729f5281793a4e56052d73e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo1988P6.lean#L201",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo2006P3.lean",
      "declaration": "Imo2006P3.imo2006_p3",
      "line": 135,
      "title": "IMO 2006 Problem 3: the least constant M",
      "intuition": "The least M with |ab(a^2-b^2) + bc(b^2-c^2) + ca(c^2-a^2)| at most M(a^2+b^2+c^2)^2 for all reals is 9 sqrt(2) / 32.",
      "source_statement": "problem imo2006_p3 :\n    IsLeast\n      { M | (∀ a b c : ℝ,\n              |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤\n              M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2) } solution := by\n…",
      "source_statement_sha256": "54f9b5b436404c4f051f75cb9d79b65e48e5d8a8b16f1d7ed5494274bd257dff",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo2006P3.lean#L135",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo2019P1.lean",
      "declaration": "Imo2019P1.imo2019_p1",
      "line": 41,
      "title": "IMO 2019 Problem 1: a functional equation on the integers",
      "intuition": "The integer functions with f(2a) + 2f(b) = f(f(a+b)) are exactly the zero function and f(z) = 2z + c.",
      "source_statement": "problem imo2019_p1 (f : ℤ → ℤ) :\n    (∀ a b, f (2 * a) + 2 * (f b) = f (f (a + b))) ↔ f ∈ solution_set := by\n…",
      "source_statement_sha256": "566208495a80fb5cff103230b3dfe537e3784431ccfc62b76c94288fbd2058d1",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo2019P1.lean#L41",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo2024P2.lean",
      "declaration": "Imo2024P2.imo2024_p2",
      "line": 197,
      "title": "IMO 2024 Problem 2: eventually constant gcd",
      "intuition": "The only pair of positive integers (a, b) for which gcd(a^n + b, b^n + a) is eventually constant is (1, 1).",
      "source_statement": "problem imo2024_p2 (a b : ℕ+) :\n    (a, b) ∈ solutionSet ↔\n    ∃ g N : ℕ+,\n      ∀ n : ℕ, N ≤ n → Nat.gcd (a^n + b) (b^n + a) = g := by\n…",
      "source_statement_sha256": "e2432878b38001ddd4a3576ae72a220d2f72d27977785adefb9984186dbb2326",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo2024P2.lean#L197",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "compfiles",
      "file": "Compfiles/Imo2021P3.lean",
      "declaration": "Imo2021P3.imo2021_p3",
      "line": 40,
      "title": "IMO 2021 Problem 3: three concurrent lines (unsolved here)",
      "intuition": "A geometry statement about a triangle with a special interior point D; the formalization is stated but its proof is the single word sorry.",
      "source_statement": "problem imo2021_p3 {A B C D E F X O₁ O₂ : P}\n    (affineIndependent_ABC : AffineIndependent ℝ ![A, B, C])\n    (acuteAngled_ABC : (⟨_, affineIndependent_ABC⟩ : Triangle ℝ P).AcuteAngled)\n    (AC_lt_AB : dist A C < dist A B)\n    (D_mem_interior_ABC : D ∈ (⟨_, affineIndependent_ABC⟩ : Triangle ℝ P).interior)\n    (angle_DAB_eq_angle_CAD : ∠ D A B = ∠ C A D) (wbtw_A_E_C : Wbtw ℝ A E C)\n    (angle_ADE_eq_angle_BCD : ∠ A D E = ∠ B C D) (wbtw_A_F_B : Wbtw ℝ A F B)\n    (angle_FDA_eq_angle_DBC : ∠ F D A = ∠ D B C) (X_mem_AC : X ∈ line[ℝ, A, C])\n    (CX_eq_BX : dist C X = dist B X)\n    (affineIndependent_ADC : AffineIndependent ℝ ![A, D, C])\n    (O₁_eq_circumcenter_ADC :\n      O₁ = (⟨_, affineIndependent_ADC⟩ : Triangle ℝ P).circumcenter)\n…",
      "source_statement_sha256": "50a10ffb7a0ce2d0569ec1b70488a205392ffacda449aebd58c24cf9c0b06c74",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles/Imo2021P3.lean#L40",
      "corpus_references": [
        {
          "label": "Pinned compfiles documentation",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/README.md"
        },
        {
          "label": "Pinned compfiles corpus index",
          "url": "https://github.com/dwrensha/compfiles/blob/47b19bbfe5f38dc49379a6b53008d2909c4d71c5/Compfiles.lean"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/Proof.lean",
      "declaration": "flt",
      "line": 111,
      "title": "Fermat's Last Theorem, conditional on unfinished work",
      "intuition": "This is Mathlib's FermatLastTheorem, obtained by chaining the reductions B4 to B3 to B2 to B1; it is not yet a complete proof, because it rests on the unproved theorem B4_proof.",
      "source_statement": "theorem flt : FermatLastTheorem :=\n…",
      "source_statement_sha256": "f759b4965fd08ca8c3243091f7e46e969b8cbda89ad891f25fe7c4b9c2201eac",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/Proof.lean#L111",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/Proof.lean",
      "declaration": "FLT.Bosses.B4_proof",
      "line": 98,
      "title": "The remaining spine gap: the p-torsion is reducible (unproved)",
      "intuition": "This is the deep core of the modularity-lifting argument, stated as B4 (the p-torsion of the Frey curve is a reducible Galois representation), and in this version of the repository its proof is just `sorry`.",
      "source_statement": "theorem B4_proof : B4 :=\n…",
      "source_statement_sha256": "5c2efc5fa87fbeee56474448e12e02c8a58c439280333c11476b518406e291da",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/Proof.lean#L98",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/Proof.lean",
      "declaration": "FLT.Bosses.B4_implies_B3",
      "line": 83,
      "title": "Reducible p-torsion rules out a Frey package",
      "intuition": "If the p-torsion of every Frey curve must be reducible, but Mazur's theorem says it is irreducible, then no Frey package can exist.",
      "source_statement": "theorem B4_implies_B3 : B4 → B3 := by\n…",
      "source_statement_sha256": "919d0d4dd0942db3604bfa108008eced6551471290faa0574ff42abc31c9545f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/Proof.lean#L83",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/Proof.lean",
      "declaration": "FLT.Bosses.B3_implies_B2",
      "line": 80,
      "title": "No Frey package implies FLT for primes at least 5",
      "intuition": "A counterexample to FLT at a prime p at least 5 can be normalised into a Frey package, so if none exist there is no counterexample.",
      "source_statement": "theorem B3_implies_B2 : B3 → B2 :=\n…",
      "source_statement_sha256": "1604f87c0c48f9b49e6f4acd8eb9011feaf498cb445c7a897c4aea069ecb3937",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/Proof.lean#L80",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/Basic/Lemmas.lean",
      "declaration": "FermatLastTheorem.of_p_ge_5",
      "line": 38,
      "title": "It suffices to prove FLT for primes at least 5",
      "intuition": "Any exponent n at least 3 is divisible by 3, by 4, or by a prime at least 5, and FLT for a divisor implies FLT for the multiple.",
      "source_statement": "lemma FermatLastTheorem.of_p_ge_5 (H : ∀ p ≥ 5, p.Prime → FermatLastTheoremFor p) :\n    FermatLastTheorem := by\n…",
      "source_statement_sha256": "5b9da06bbcd42c7871b2f125e3c3592b669af00b75b3c7a69bd5edf99e2b0e61",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/Basic/Lemmas.lean#L38",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/FreyCurve/FreyPackage.lean",
      "declaration": "FreyPackage",
      "line": 75,
      "title": "Frey package",
      "intuition": "A Frey package bundles a hypothetical counterexample a^p + b^p = c^p, with a prime p at least 5, into normalised data (a coprime to b, a congruent to 3 mod 4, b even) so that the associated elliptic curve is well behaved.",
      "source_statement": "structure FreyPackage where\n…",
      "source_statement_sha256": "8bb15a4ee435ada66928f596c17737dfd3f0447832b0226e9eb19ae276db5c06",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/FreyCurve/FreyPackage.lean#L75",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/FreyCurve/FreyPackage.lean",
      "declaration": "FreyPackage.fermatLastTheoremFor_p_ge_5",
      "line": 217,
      "title": "Empty type of Frey packages gives FLT for p at least 5",
      "intuition": "By contraposition: a counterexample for some prime p at least 5 would produce a Frey package.",
      "source_statement": "lemma fermatLastTheoremFor_p_ge_5 (h : IsEmpty FreyPackage) :\n    ∀ p ≥ 5, p.Prime → FermatLastTheoremFor p := by\n…",
      "source_statement_sha256": "c0d8e8b8edd670348b4216afa56850f89c243bf685b3e46c9d4b258dc41a3f61",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/FreyCurve/FreyPackage.lean#L217",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "flt",
      "file": "FLT/FreyCurve/Mazur.lean",
      "declaration": "FreyPackage.mazur",
      "line": 30,
      "title": "Mazur step: the p-torsion of the Frey curve is irreducible (assumed)",
      "intuition": "The statement is that the Galois representation on the p-torsion of the Frey curve is irreducible, a consequence of Mazur's work as used in Serre's 1987 Duke paper.",
      "source_statement": "theorem FreyPackage.mazur (P : FreyPackage) :\n    let E := P.freyCurve\n…",
      "source_statement_sha256": "edca63f462d9e80e11a81725f6a3741f857f58eb44667c545d551bc0a5207145",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/FLT/FreyCurve/Mazur.lean#L30",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/README.md"
        },
        {
          "label": "Pinned FLT blueprint introduction",
          "url": "https://github.com/ImperialCollegeLondon/FLT/blob/f6c6a2fe9796d6d89cd2a0d24e8a453729d881ff/blueprint/src/chapter/ch01introduction.tex"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/ErdosProblems/1.lean",
      "declaration": "Erdos1.IsSumDistinctSet",
      "line": 38,
      "title": "Sum-distinct sets",
      "intuition": "A finite set of naturals up to N is sum-distinct when no two different subsets add up to the same total.",
      "source_statement": "abbrev IsSumDistinctSet (A : Finset ℕ) (N : ℕ) : Prop :=\n…",
      "source_statement_sha256": "3d34682e7415e7eaeb89261c9c14c95c1ff65a4cd469ba15597daea20a7a85ab",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/ErdosProblems/1.lean#L38",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/ErdosProblems/1.lean",
      "declaration": "Erdos1.erdos_1",
      "line": 53,
      "title": "Erdos Problem 1: sum-distinct sets need N of order 2^n",
      "intuition": "The statement says there is no constant C with N at least C * 2^n for every sum-distinct set of size n inside 1..N, and the file marks this as solved.",
      "source_statement": "theorem erdos_1 : ¬ ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N),\n    N ≠ 0 → C * 2 ^ A.card < N := by\n…",
      "source_statement_sha256": "177bc7c0cf19eac5b4a341700b45d90b500689f9afa8c3302f38666d07b33d24",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/ErdosProblems/1.lean#L53",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/ErdosProblems/3.lean",
      "declaration": "Erdos3.erdos_3",
      "line": 47,
      "title": "Erdos Problem 3: divergent reciprocal sums and arithmetic progressions",
      "intuition": "If the reciprocals of a set of naturals sum to infinity, must the set contain arithmetic progressions of every length? The file marks this as open.",
      "source_statement": "theorem erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ,\n    (¬ Summable fun a : A ↦ 1 / (a : ℝ)) →\n    ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k := by\n…",
      "source_statement_sha256": "9963abda19751d9e6423f8a24e9eda1358e5a04ef2db55b92ff95079775d46a7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/ErdosProblems/3.lean#L47",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/Wikipedia/CollatzConjecture.lean",
      "declaration": "CollatzConjecture.collatzStep",
      "line": 43,
      "title": "One step of the Collatz map",
      "intuition": "Halve an even number, and triple an odd number then add one.",
      "source_statement": "def collatzStep (n : ℕ) : ℕ :=\n…",
      "source_statement_sha256": "0b04f992345f78c5515ae3b1e0bd46f37571ef87dec9011eb1df1a2116c02b2b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/Wikipedia/CollatzConjecture.lean#L43",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/Wikipedia/CollatzConjecture.lean",
      "declaration": "CollatzConjecture.collatz_conjecture",
      "line": 53,
      "title": "The Collatz conjecture",
      "intuition": "Starting from any positive integer and repeating the Collatz step, you eventually reach 1; the file marks this as open.",
      "source_statement": "theorem collatz_conjecture (n : ℕ) (hn : n > 0) : ∃ m, collatzStep^[m] n = 1 := by\n…",
      "source_statement_sha256": "6c761e69e4cd2c833c3b7ed5214039610f9772883a947b3f88e8ec5b5a8f45e5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/Wikipedia/CollatzConjecture.lean#L53",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/Wikipedia/TwinPrimes.lean",
      "declaration": "TwinPrimes.twin_primes",
      "line": 37,
      "title": "The twin prime question",
      "intuition": "Are there infinitely many primes p for which p + 2 is also prime? The file marks this as open.",
      "source_statement": "theorem twin_primes :\n    answer(sorry) ↔ {p : ℕ | Prime p ∧ Prime (p + 2)}.Infinite := by\n…",
      "source_statement_sha256": "5fbd4aa312269329e84eec1984ac3ca7631106365fcb6c90259ea8550e55e705",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/Wikipedia/TwinPrimes.lean#L37",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/Wikipedia/GoldbachConjecture.lean",
      "declaration": "GoldbachConjecture.goldbach",
      "line": 36,
      "title": "Goldbach's conjecture",
      "intuition": "Can every even number greater than 2 be written as a sum of two primes? The file marks this as open.",
      "source_statement": "theorem goldbach :\n    answer(sorry) ↔ ∀ n : ℕ, 2 < n → Even n → ∃ p q, Prime p ∧ Prime q ∧ n = p + q := by\n…",
      "source_statement_sha256": "c0e6524b743df3e387d07b913f06285b438947deb2e62dc9ba1f69c6066a4364",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/Wikipedia/GoldbachConjecture.lean#L36",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/Wikipedia/GoldbachConjecture.lean",
      "declaration": "TernaryGoldbachConjecture.ternaryGoldbach",
      "line": 52,
      "title": "Ternary Goldbach",
      "intuition": "Every odd number greater than 5 is a sum of three primes (primes may repeat); the file marks this as solved.",
      "source_statement": "theorem ternaryGoldbach (n : ℕ) (hn : 5 < n) (hn_odd : Odd n) :\n    ∃ p q r, Nat.Prime p ∧ Nat.Prime q ∧ Nat.Prime r ∧ n = p + q + r := by\n…",
      "source_statement_sha256": "5a6bef5ce62b9ea5d93cb06aede8838994ee890ee6d7c4a3c0d9b58b0d39a850",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/Wikipedia/GoldbachConjecture.lean#L52",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/Millennium/RiemannHypothesis.lean",
      "declaration": "RiemannHypothesis.riemannHypothesis",
      "line": 61,
      "title": "The Riemann Hypothesis",
      "intuition": "Every non-trivial zero of the Riemann zeta function has real part one half; the file marks this as open.",
      "source_statement": "theorem riemannHypothesis : RiemannHypothesis := by\n…",
      "source_statement_sha256": "6d3295a0af344672852e22b1a0bf2e8810619cc3f21fcb1e3e5d1c49a44edec9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/Millennium/RiemannHypothesis.lean#L61",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "formal-conjectures",
      "file": "FormalConjectures/Millennium/PvsNP.lean",
      "declaration": "ComplexityTheory.P_ne_NP",
      "line": 45,
      "title": "P is not equal to NP",
      "intuition": "The complexity classes P and NP differ; the file marks this as open.",
      "source_statement": "theorem P_ne_NP : P ≠ NP := by sorry\n…",
      "source_statement_sha256": "56078e80e1bb8480cc7025adf96c5ccc6f8b4445b4eab0e6a6f26775d494e3d1",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/FormalConjectures/Millennium/PvsNP.lean#L45",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/1646ca16afd6cc7a693d3bdc9f066c4d3cc01a89/README.md"
        },
        {
          "label": "Formal Conjectures paper, arXiv v1",
          "url": "https://arxiv.org/abs/2605.13171v1"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean",
      "declaration": "MyRing.add_zero",
      "line": 39,
      "title": "a + 0 = a in any ring, from commutativity and zero_add",
      "intuition": "Swap the two summands, then use the left-identity law that is already available.",
      "source_statement": "theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add]",
      "source_statement_sha256": "d08018dbed3d3fd4eb53cb659961b9cd80916ea477673c66c9e19b8470fde072",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean",
      "declaration": "MyRing.mul_zero",
      "line": 64,
      "title": "a * 0 = 0 via cancellation",
      "intuition": "Show a*0 + a*0 equals a*0 + 0, then cancel one a*0 from the left.",
      "source_statement": "theorem mul_zero (a : R) : a * 0 = 0 := by\n…",
      "source_statement_sha256": "622cae3d1f4914ff506e5a5fe880f562254604acf37f89ade5ebda6b61f1c7eb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean#L64",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean",
      "declaration": "MyRing.add_left_cancel",
      "line": 58,
      "title": "Exercise: left cancellation of addition (sorry)",
      "intuition": "If a + b = a + c then b = c; here the proof is left as a sorry for the reader.",
      "source_statement": "theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by\n…",
      "source_statement_sha256": "c0ca3c13b3b67eaf20011fab45581190538f40199873bc1457c0c9b346dec448",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C02_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean#L58",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean",
      "declaration": "C03S01.my_lemma",
      "line": 10,
      "title": "Exercise: a universally quantified statement (sorry)",
      "intuition": "A statement of the form for-all-x-y-epsilon-then-implications is a function that takes numbers and proofs.",
      "source_statement": "theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε :=\n…",
      "source_statement_sha256": "1d593fbf60aaec64c625d5123c6ebd1bb8a801ba56a640888fcbcbfd8852407e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean#L10",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean",
      "declaration": "C03S01.my_lemma4",
      "line": 6,
      "title": "Solution: |x*y| < epsilon by a calc chain",
      "intuition": "Bound |x*y| by |x|*|y|, then by |x|*epsilon, then by 1*epsilon, which is epsilon.",
      "source_statement": "theorem my_lemma4 :\n    ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε := by\n…",
      "source_statement_sha256": "d47fa636a6f216ffb91790c47dc40d24fb97f9201c88fc4ba9aec8e874692e2c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean#L6",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C03_Logic/S06_Sequences_and_Convergence.lean",
      "declaration": "C03S06.ConvergesTo",
      "line": 6,
      "title": "Convergence of a real sequence",
      "intuition": "A sequence converges to a if, for every positive epsilon, all terms past some index N stay within epsilon of a.",
      "source_statement": "def ConvergesTo (s : ℕ → ℝ) (a : ℝ) :=\n…",
      "source_statement_sha256": "36685a6f228ea31d2398c65be4b0422bb6a815618e56034efac16981f842c1c0",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C03_Logic/S06_Sequences_and_Convergence.lean#L6",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C03_Logic/S06_Sequences_and_Convergence.lean",
      "declaration": "C03S06.convergesTo_const",
      "line": 22,
      "title": "A constant sequence converges to its constant",
      "intuition": "Every term equals a, so the distance to a is 0, which is below any positive epsilon.",
      "source_statement": "theorem convergesTo_const (a : ℝ) : ConvergesTo (fun _x : ℕ ↦ a) a := by\n…",
      "source_statement_sha256": "90305379f8c8c7a6b87f93a383f8fa3215cec19fa8a5eb30ef67f6593f3e48d8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C03_Logic/S06_Sequences_and_Convergence.lean#L22",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean",
      "declaration": "C03S06.convergesTo_add",
      "line": 16,
      "title": "The limit of a sum is the sum of the limits",
      "intuition": "Give each sequence half of the tolerance epsilon and wait until both are within their half.",
      "source_statement": "theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ}\n      (cs : ConvergesTo s a) (ct : ConvergesTo t b) :\n    ConvergesTo (fun n ↦ s n + t n) (a + b) := by\n…",
      "source_statement_sha256": "026ba29697617809f3bc81c924546a0c46a304d4a9e6485f9956a012d7b7d315",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean#L16",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "mil",
      "file": "MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean",
      "declaration": "Cantor",
      "line": 237,
      "title": "Cantor: no function from α to its power set is surjective",
      "intuition": "Build the set of elements not in their own image; it cannot be anyone's image.",
      "source_statement": "theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by\n…",
      "source_statement_sha256": "8d1f4eddcb421b9b4511a349ed0056e3dfea1e52d85147a818343a1a10655c7b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean#L237",
      "corpus_references": [
        {
          "label": "Pinned Mathematics in Lean documentation",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/README.md"
        },
        {
          "label": "Pinned Mathematics in Lean textbook",
          "url": "https://github.com/leanprover-community/mathematics_in_lean/blob/dd6d752fedb14082f557913c2dccb2d4851e5173/mathematics_in_lean.pdf"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/QuasiRiemannHypothesis.lean",
      "declaration": "OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re",
      "line": 5,
      "title": "Challenge statement: Zero-free half-plane for the Riemann zeta function",
      "intuition": "The Riemann zeta function has no zeros anywhere to the right of the vertical line with real part 7/8.",
      "source_statement": "theorem riemannZeta_ne_zero_of_seven_eighths_lt_re\n    {s : ℂ} (hs : (7 / 8 : ℝ) < s.re) : riemannZeta s ≠ 0 := by\n…",
      "source_statement_sha256": "5aac0f908699da4372fc3a3143fa85a841fc6959eec713e44ac0026ee5b43d3b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/QuasiRiemannHypothesis.lean#L5",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/DirichletSevenEighths.lean",
      "declaration": "OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re",
      "line": 8,
      "title": "Challenge statement: Zero-free half-plane for Dirichlet L-functions",
      "intuition": "Every Dirichlet L-function, for every modulus and character, is nonzero to the right of real part 7/8, apart from the pole of the trivial character at 1.",
      "source_statement": "theorem LFunction_ne_zero_of_seven_eighths_lt_re\n    {q : ℕ} [NeZero q] (χ : DirichletCharacter ℂ q) {s : ℂ}\n    (hs : (7 / 8 : ℝ) < s.re) (hpole : ¬ (χ = 1 ∧ s = 1)) :\n    _root_.DirichletCharacter.LFunction χ s ≠ 0 := by\n…",
      "source_statement_sha256": "57d9a2468f4bd66022d3b613037327d5c2709ec998cd8a2d7c86935540c456f5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/DirichletSevenEighths.lean#L8",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/SiegelZeros.lean",
      "declaration": "OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gap",
      "line": 8,
      "title": "Challenge statement: Uniform gap for real zeros of L-functions",
      "intuition": "There is one positive constant c such that any real zero beta of a primitive real Dirichlet L-function of conductor q satisfies 1 - beta at least c / log q.",
      "source_statement": "theorem exists_absolute_real_zero_gap :\n    ∃ c : ℝ, 0 < c ∧\n      ∀ (q : ℕ) [NeZero q], 3 ≤ q →\n      ∀ χ : DirichletCharacter ℂ q,\n        χ.IsPrimitive → χ ≠ 1 → (∀ a : ZMod q, (χ a).im = 0) →\n        ∀ β : ℝ, 0 < β → β < 1 → χ.LFunction (β : ℂ) = 0 →\n          c ≤ (1 - β) * Real.log (q : ℝ) := by\n…",
      "source_statement_sha256": "dcbaa9e741fc5976956cc480d753833084146d91b39631371d8b7fd5f128effe",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/SiegelZeros.lean#L8",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/HadwigerCounterexample.lean",
      "declaration": "OAI.HadwigerCounterexample.not_hadwiger_conjecture",
      "line": 49,
      "title": "Challenge statement: Hadwiger's conjecture fails",
      "intuition": "The statement denies that every finite graph's chromatic number is at most its largest clique-minor size.",
      "source_statement": "theorem not_hadwiger_conjecture :\n    ¬ (∀ (m : ℕ) (G : SimpleGraph (Fin m)),\n      G.chromaticNumber ≤ ↑(hadwigerNumber G)) := by\n…",
      "source_statement_sha256": "8e2e71bbb58ae3f0542f2cde9e750dd16d2eb2edb5ddffd50eef89f2dc3427ce",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/HadwigerCounterexample.lean#L49",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/HadwigerCounterexample.lean",
      "declaration": "OAI.HadwigerCounterexample.exists_counterexamples",
      "line": 38,
      "title": "Challenge statement: Counterexamples of arbitrarily large order",
      "intuition": "For every bound M there is a graph on m >= max(M, 5) vertices with independence number at most 2 whose chromatic number is above its Hadwiger number.",
      "source_statement": "theorem exists_counterexamples (M : ℕ) : ∃ m : ℕ, max M 5 ≤ m ∧\n    ∃ G : SimpleGraph (Fin m), G.indepNum ≤ 2 ∧\n      (connectedMatchingNumber G : ℚ) < (m : ℚ) / 100 ∧\n      (hadwigerNumber G : ℚ) < 26 * (m : ℚ) / 75 + 2 / 3 ∧\n      26 * (m : ℚ) / 75 + 2 / 3 < (m : ℚ) / 2 ∧\n      (m : ℚ) / 2 ≤ (G.chromaticNumber.toNat : ℚ) ∧\n      hadwigerNumber G < G.chromaticNumber.toNat ∧\n      G.chromaticNumber ≠ ⊤ ∧\n      G.chromaticNumber = ↑(G.chromaticNumber.toNat) := by\n…",
      "source_statement_sha256": "dcde3f70f881bbee69071e016e2a9847de6a3ff4a14567b897ad9bb53fbc6664",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/HadwigerCounterexample.lean#L38",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/KaplanskyDirectFiniteness.lean",
      "declaration": "OAI.KaplanskyCounterexample.MainClaim",
      "line": 7,
      "title": "Challenge statement: Failure of direct finiteness in characteristic 2",
      "intuition": "A group algebra over a finite field of characteristic 2, of a finitely generated group, can contain elements a, b with a*b = 1 but b*a different from 1.",
      "source_statement": "def MainClaim : Prop :=\n…",
      "source_statement_sha256": "dec2192afc2b3c550a1e98e60c4bb12a1be8490b53d0d4ed3d367bc3fd443f02",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/KaplanskyDirectFiniteness.lean#L7",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/KaplanskyDirectFiniteness.lean",
      "declaration": "OAI.KaplanskyCounterexample.main_theorem",
      "line": 12,
      "title": "Challenge statement: Main theorem of the Kaplansky counterexample",
      "intuition": "States that MainClaim holds; the challenge file leaves the proof as sorry.",
      "source_statement": "theorem main_theorem : MainClaim := by\n…",
      "source_statement_sha256": "ab0b2bb6caeb8f0b74c07f03ba8a7e68ce35d49baa6cef4e8800cddb2f163e2d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/KaplanskyDirectFiniteness.lean#L12",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/MahlerConjecture.lean",
      "declaration": "OAI.SymmetricMahler.symmetric_mahler",
      "line": 15,
      "title": "Challenge statement: Symmetric Mahler volume bound",
      "intuition": "For a centrally symmetric convex body K in n dimensions, volume(K) times volume(its polar) is at least 4^n / n!.",
      "source_statement": "theorem symmetric_mahler {n : ℕ} (hn : 1 ≤ n)\n    {K : Set (Fin n → ℝ)} (hK : IsCompact K) (hconv : Convex ℝ K)\n    (hsym : ∀ x ∈ K, -x ∈ K) (hint : (interior K).Nonempty) :\n    (4:ℝ)^n/(Nat.factorial n:ℝ) ≤\n      (volume K).toReal*(volume (coordinatePolar K)).toReal := by\n…",
      "source_statement_sha256": "c4558cf5edef880cb4ffe582de28d6846b67ffd8040d63dce82cd8b7c94c54b0",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/MahlerConjecture.lean#L15",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AffineBernstein.lean",
      "declaration": "OAI.AffineBernstein.affine_bernstein",
      "line": 73,
      "title": "Challenge statement: Affine Bernstein theorem in dimensions 3 to 9",
      "intuition": "A complete, smooth, uniformly convex solution of the affine maximal surface equation on a convex domain must be a paraboloid.",
      "source_statement": "theorem affine_bernstein\n    (n : ℕ) (hn₃ : 3 ≤ n) (hn₉ : n ≤ 9)\n    (Ω : Set (Space n)) (hΩne : Ω.Nonempty) (hΩopen : IsOpen Ω)\n    (hΩconvex : Convex ℝ Ω) (u : Space n → ℝ)\n    (hu : ContDiffOn ℝ ∞ u Ω)\n    (hpos : ∀ x ∈ Ω, (hessian u x).PosDef)\n    (hmax : AffineMaximalOn Ω u)\n    (hcomplete : EuclideanGraphComplete Ω u) :\n    Ω = univ ∧\n      (∃ (A : Matrix (Fin n) (Fin n) ℝ) (b : Space n) (c : ℝ),\n        A.IsSymm ∧ A.PosDef ∧\n        ∀ x, u x = (1 / 2 : ℝ) * (∑ i : Fin n, ∑ j : Fin n, x i * A i j * x j)\n…",
      "source_statement_sha256": "c6b34eaabfc344e800207612579177e457050cd00a9e404814465d241adcf842",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AffineBernstein.lean#L73",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/MatrixMultiplication.lean",
      "declaration": "OAI.MatrixMultiplication.complex_omega_le_nine_quarters",
      "line": 118,
      "title": "Complex matrix multiplication has exponent at most 9/4",
      "intuition": "This is an asymptotic arithmetic-complexity statement: for every positive slack `ε`, sufficiently large complex `n×n` matrices can be multiplied using `O(n^(9/4+ε))` additions, subtractions, and multiplications in a division-free program. It does not claim that the construction is fastest at practical matrix sizes or in bit complexity.",
      "source_statement": "theorem complex_omega_le_nine_quarters :\n    Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 := by\n…",
      "source_statement_sha256": "cca022ff8b129ec26d9c428b6b431a27846196a0ec4e23fd641de4f2c92e4108",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/MatrixMultiplication.lean#L118",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/MatrixMultiplication.lean",
      "declaration": "OAI.MatrixMultiplication.complex_alpha_gt_93_div_200",
      "line": 122,
      "title": "The complex dual matrix-multiplication exponent is greater than 0.465",
      "intuition": "The dual exponent `α` is the supremum of aspect ratios `k` for which multiplying an `n×n^k` matrix by an `n^k×n` matrix can be done with exponent 2, up to arbitrary positive slack. This theorem proves `α > 93/200 = 0.465`; it is a strict asymptotic bound, not a benchmark at a fixed dimension.",
      "source_statement": "theorem complex_alpha_gt_93_div_200 :\n    (93 : ℝ) / 200 < Arithmetic.complexAlpha := by\n…",
      "source_statement_sha256": "9e1c2fd174b3cda82e306714c4f0561f9639ca5d647a374f3d3d9c37bc4d3e3c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/MatrixMultiplication.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/MatrixMultiplication.lean",
      "declaration": "OAI.MatrixMultiplication.complex_rectangular_omega_lt_523_div_250",
      "line": 126,
      "title": "Rectangular complex matrix multiplication at aspect ratio 0.709 has exponent below 2.092",
      "intuition": "For multiplication of an `n×n^0.709` matrix by an `n^0.709×n` matrix over `ℂ`, the arithmetic exponent is strictly less than `523/250 = 2.092`. The strict inequality means some uniform positive exponent improvement exists; it does not specify a practical crossover size.",
      "source_statement": "theorem complex_rectangular_omega_lt_523_div_250 :\n    Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 := by\n…",
      "source_statement_sha256": "7c868739de80b214d984d84e3946e6d20bf159b3eebbbd896a47ab6f77ffa926",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/MatrixMultiplication.lean#L126",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/Catalan.lean",
      "declaration": "OAI.InternalCatalan.catalan_irrational",
      "line": 7,
      "title": "Challenge statement: Catalan's constant is irrational",
      "intuition": "Catalan's constant is the alternating reciprocal-square sum G = Σ(j ≥ 0) (-1)^j/(2j+1)^2, equivalently the Dirichlet beta value β(2). This declaration says that no rational number equals that real series.",
      "source_statement": "theorem catalan_irrational :\n    Irrational (∑' j : ℕ, (-1 : ℝ) ^ j / ((2 * j + 1 : ℕ) : ℝ) ^ 2) := by\n…",
      "source_statement_sha256": "e6686b24ead62dfb91ec520e642d07287b9a3f81137dd341fac3104e56111453",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/Catalan.lean#L7",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/PiExponent.lean",
      "declaration": "OAI.PiExponent.main",
      "line": 7,
      "title": "Challenge statement: The irrationality exponent of π is exactly 2",
      "intuition": "Rational numbers approximate π infinitely often at essentially the Dirichlet rate q^(-2), but no exponent strictly larger than 2 succeeds infinitely often. The two conjuncts state both the eventual lower bound and the exact supremum characterization.",
      "source_statement": "theorem main :\n  (∀ ν : ℝ, 2 < ν → ∃ Q : ℤ, 2 ≤ Q ∧\n    ∀ p q : ℤ, Q ≤ q →\n      (q : ℝ) ^ (-ν) ≤ |Real.pi - (p : ℝ) / (q : ℝ)|) ∧\n  sSup {ν : ℝ | 0 < ν ∧\n    Set.Infinite {r : ℚ | 2 ≤ r.den ∧\n      0 < |Real.pi - (r : ℝ)| ∧\n      |Real.pi - (r : ℝ)| < (r.den : ℝ) ^ (-ν)}} = 2 := by\n…",
      "source_statement_sha256": "7151013510986f43f5fd48c1f39f0d1c5d34705249ac5de7cfff32ee791d47cd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/PiExponent.lean#L7",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/DyadicAvoidance.lean",
      "declaration": "OAI.Problem310.dyadic_affine_avoidance",
      "line": 12,
      "title": "Challenge statement: A large compact set avoiding every affine dyadic sequence",
      "intuition": "No matter how small a measure loss η is allowed, one can keep a compact subset of [0,1] with measure greater than 1-η while destroying every translated and nonzero-rescaled copy of 1/2, 1/4, 1/8, and so on.",
      "source_statement": "theorem dyadic_affine_avoidance :\n    ∀ η : ℝ, 0 < η → η < 1 → ∃ E : Set ℝ,\n      E ⊆ Set.Icc (0 : ℝ) 1 ∧ IsCompact E ∧\n      MeasureTheory.volume E > ENNReal.ofReal (1 - η) ∧\n      ∀ x s : ℝ, s ≠ 0 → ∃ n : ℕ, 1 ≤ n ∧\n        x + s * dyadicPoint n ∉ E := by\n…",
      "source_statement_sha256": "8cffd0a89cd7bb0d6afe115c4561fe1d2dc874e16480c11917ce7be4ce5e1cf3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/DyadicAvoidance.lean#L12",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Target.PermutationTable",
      "line": 9,
      "title": "A finite permutation table with a certified inverse",
      "intuition": "A `PermutationTable alphabet` stores both directions of a permutation of `Fin alphabet`: `images` is the forward table and `inverseImages` is its proposed inverse.",
      "source_statement": "structure PermutationTable (alphabet : Nat) where\n…",
      "source_statement_sha256": "4139d67feef6d19eeb3e527a896fbec279d5f6d9c8656598905a81c3506c7629",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L9",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Target.Instance",
      "line": 26,
      "title": "An explicit, unweighted list of Unique Games constraints",
      "intuition": "An `Instance alphabet` records a finite number of vertices and a nonempty list of permutation constraints over that fixed alphabet.",
      "source_statement": "structure Instance (alphabet : Nat) where\n…",
      "source_statement_sha256": "b49c4050fe8c9ef6cc12eaab52237e20a3f08388ed71bcd512bb2188193d5cf7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L26",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryFormula.Formula",
      "line": 58,
      "title": "Binary 3SAT as a finite list of three-literal clauses",
      "intuition": "A `Formula` is exactly a list of `Clause`s, and each clause is a length-three vector of signed literals whose variable names are natural numbers.",
      "source_statement": "structure Formula where\n…",
      "source_statement_sha256": "5bed079e56bab4439af13637be7433ce6d85c850b681ba0e0e9bdc6f508e6c06",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L58",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryEncoding.parseName",
      "line": 82,
      "title": "Decode one canonically framed natural-number name",
      "intuition": "`parseName` reads one framed list of bits, converts it to a natural number, and returns that number together with the unconsumed suffix only when the digits are canonical.",
      "source_statement": "def parseName (input : List Bool) : Option (Nat × List Bool) := do\n…",
      "source_statement_sha256": "6530cdb84750c9d1c108b1a97929fda3ac3ba43f98ed51734fa82f5514e287a8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryEncoding.parseClauses",
      "line": 100,
      "title": "Decode a terminated clause list with structural fuel",
      "intuition": "`parseClauses fuel input` parses a list in which `false` ends the list and `true` announces one three-literal clause followed by the rest.",
      "source_statement": "def parseClauses : Nat → List Bool → Option (List Clause × List Bool)\n  | 0, _ => none\n  | _ + 1, false :: rest => some ([], rest)\n  | fuel + 1, true :: input => do\n      let (clause, input) ← parseClause input\n      let (clauses, rest) ← parseClauses fuel input\n      return (clause :: clauses, rest)\n  | _ + 1, [] => none",
      "source_statement_sha256": "ad13208ea4514b33e4dc853446c9755c589f539974e44bce37f978c2a621382b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L100",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFiniteAlphabet.FiniteAlphabet",
      "line": 178,
      "title": "Every physical tape of the reduction machine has a finite alphabet",
      "intuition": "`FiniteAlphabet M` means `Finite (M.Γ k)` for every physical tape index `k` of the two-tape-style finite Turing-machine model.",
      "source_statement": "def FiniteAlphabet (M : FinTM2) : Prop := ∀ k, Finite (M.Γ k)\n…",
      "source_statement_sha256": "554575c9b30ae4c344b9e6278a6fe8f62bca58d019bbdd9a1cfa2e61b0b04b24",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L178",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Explicit.MachineOutputContract.SimpleBipartite",
      "line": 191,
      "title": "A certificate that the output graph is simple and bipartite",
      "intuition": "`SimpleBipartite game` assigns every vertex a Boolean side, orients every constraint from the `false` side to the `true` side, and forbids two occurrence indices from sharing the same ordered endpoints.",
      "source_statement": "structure SimpleBipartite {q : Nat} (game : Instance q) where\n…",
      "source_statement_sha256": "7bf4bc69173ce8c0a5bde70def9611b4fd818ba728efe1f934c9880608a1f423",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L191",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Explicit.MachineOutputContract.BinaryGapReduction",
      "line": 203,
      "title": "The complete machine-checkable contract for the gap reduction",
      "intuition": "A `BinaryGapReduction ε δ` packages one alphabet, binary-vector-space coordinate system, constructor, machine computation, and the two gap guarantees before any encoded formula is supplied.",
      "source_statement": "structure BinaryGapReduction (ε δ : ℝ) where\n…",
      "source_statement_sha256": "074cd111292f5dad23e50e2bdda586fa7003edbb991103367d9f5d721b7c8edb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L203",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Target.Constraint",
      "line": 15,
      "title": "One oriented permutation constraint",
      "intuition": "A `Constraint vertices alphabet` selects a source vertex, a target vertex, and the permutation table that tells which target label matches each source label.",
      "source_statement": "structure Constraint (vertices alphabet : Nat) where\n…",
      "source_statement_sha256": "84eeb6e303f456aadf0538253259e760c437bbec52cf13963429a58825da8d1a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L15",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Target.Constraint.satisfied",
      "line": 20,
      "title": "Test whether one labeling satisfies one constraint",
      "intuition": "`constraint.satisfied labeling` is the Boolean decision that the target vertex's label equals the forward image of the source vertex's label.",
      "source_statement": "def Constraint.satisfied {n q : Nat} (constraint : Constraint n q)\n    (labeling : Fin n → Fin q) : Bool :=\n…",
      "source_statement_sha256": "b7764cde677b9f7d60070b4e5a3b6014b5c2cbcf7f5b9d19922d4501e122e767",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L20",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Target.countSatisfied",
      "line": 31,
      "title": "Count satisfied constraint occurrences",
      "intuition": "`countSatisfied labeling constraints` recursively counts how many entries in the constraint list evaluate to `true` under one labeling.",
      "source_statement": "def countSatisfied {n q : Nat} (labeling : Fin n → Fin q) :\n    List (Constraint n q) → Nat\n  | [] => 0\n  | constraint :: rest =>\n      (if constraint.satisfied labeling then 1 else 0) + countSatisfied labeling rest\n\nend UniqueGamesTheorem.Foundations.Target\n\nnamespace UniqueGamesTheorem.BinaryFormula\n\nopen UniqueGamesTheorem.Foundations",
      "source_statement_sha256": "2d68eddd45e85951d14452d3203f0d2d40f6e91ebac291fb29a05af6843e1a72",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L31",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryFormula.Literal",
      "line": 43,
      "title": "A signed Boolean variable occurrence",
      "intuition": "A `Literal` pairs an arbitrary natural-number variable name with a Boolean sign.",
      "source_statement": "structure Literal where\n…",
      "source_statement_sha256": "f4bb95893d35365905fd64960c657fa1f8538e3024c117f0161ddeb4128b8be7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L43",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryFormula.Literal.eval",
      "line": 48,
      "title": "Evaluate a signed literal under a Boolean assignment",
      "intuition": "Given `assignment : Nat → Bool`, a positive literal returns the assigned bit and a negative literal returns its Boolean complement.",
      "source_statement": "def Literal.eval (literal : Literal) (assignment : Nat → Bool) : Bool :=\n…",
      "source_statement_sha256": "291d5f22039ddbe74e914b5c4aec023feed860f1e6ac4816ee7eae5d23812a57",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L48",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryFormula.Clause",
      "line": 51,
      "title": "Exactly three literal slots",
      "intuition": "`Clause` abbreviates `Vector Literal 3`, so its arity is fixed by the type rather than checked at runtime.",
      "source_statement": "abbrev Clause := Vector Literal 3",
      "source_statement_sha256": "523b4a1399327664a4feb39cb9901219039dc0a16441b4cc1ecbeef1fa4d12d0",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L51",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryFormula.Clause.eval",
      "line": 53,
      "title": "Evaluate a 3SAT clause by disjunction",
      "intuition": "A clause is true when at least one of its three signed literals evaluates to true under the assignment.",
      "source_statement": "def Clause.eval (clause : Clause) (assignment : Nat → Bool) : Bool :=\n…",
      "source_statement_sha256": "578149cfa40061d5358bc8c2df156a035b1cda8a8d94e728652ef7734eda2e09",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L53",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryFormula.Formula.Satisfiable",
      "line": 61,
      "title": "One assignment satisfies every listed clause",
      "intuition": "`F.Satisfiable` means there exists a total Boolean assignment making each clause occurring in `F.clauses` evaluate to `true`.",
      "source_statement": "def Formula.Satisfiable (F : Formula) : Prop :=\n…",
      "source_statement_sha256": "f734a21b769bb40d6e8c0f72ae26cc65949b56773b7672c539efb63bc7c8835c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L61",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryEncoding.bitsValue",
      "line": 70,
      "title": "Interpret a least-significant-bit-first Boolean list",
      "intuition": "`bitsValue` converts Lean's canonical Boolean digit list back to a natural number.",
      "source_statement": "def bitsValue : List Bool → Nat\n  | [] => 0\n  | b :: bits => Nat.bit b (bitsValue bits)",
      "source_statement_sha256": "d9bc4f5060f956876101de271ceaa75f9b0ec18a0b7b728d5d01414899d68a52",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L70",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryEncoding.parseFrame",
      "line": 74,
      "title": "Parse one self-delimiting frame of bits",
      "intuition": "`parseFrame` reads zero or more data bits, each introduced by `true`, until a `false` terminator, then returns the digits and untouched suffix.",
      "source_statement": "def parseFrame : List Bool → Option (List Bool × List Bool)\n  | false :: rest => some ([], rest)\n  | true :: b :: rest => do\n      let (bits, trailing) ← parseFrame rest\n      return (b :: bits, trailing)\n  | _ => none\n\n/-- Equality with the canonical digits rejects alternate padded encodings. -/",
      "source_statement_sha256": "7e75f5aa6a6a419fb7fbfdc4dcfed9107231fc0515c4b1a49aa7da142fd82e0f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L74",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryEncoding.parseLiteral",
      "line": 87,
      "title": "Parse a sign bit followed by one canonical name",
      "intuition": "`parseLiteral` consumes the first bit as polarity, delegates the remaining input to `parseName`, and returns one `Literal` plus the suffix.",
      "source_statement": "def parseLiteral : List Bool → Option (Literal × List Bool)\n  | sign :: input => do\n      let (name, rest) ← parseName input\n      return (⟨name, sign⟩, rest)\n  | [] => none",
      "source_statement_sha256": "278bdf14ac319ee2f7b747ce8f151ce4bf667b5f8f32674882254342db3fb69f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L87",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryEncoding.parseClause",
      "line": 93,
      "title": "Parse three literals in sequence",
      "intuition": "`parseClause` runs `parseLiteral` three times and packages the results as the length-three vector `#v[a,b,c]`.",
      "source_statement": "def parseClause (input : List Bool) : Option (Clause × List Bool) := do\n…",
      "source_statement_sha256": "503422c4f0c251fb298f99939705c40e39d80350d2cde119f3ab5d6e5b856483",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L93",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryEncoding.decodeFormula",
      "line": 109,
      "title": "Accept exactly one complete canonical formula encoding",
      "intuition": "`decodeFormula input` parses the clause list with enough fuel and succeeds only when no trailing bits remain.",
      "source_statement": "def decodeFormula (input : List Bool) : Option Formula := do\n…",
      "source_statement_sha256": "f3fc257a1e7a147ed1b79047cd94a0f7f5c98ce26af43711aa7987d1d54e14d5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L109",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.BinaryLanguage.language",
      "line": 119,
      "title": "The encoded language of satisfiable binary 3SAT formulas",
      "intuition": "`language input` holds exactly when the entire bit list decodes to some formula and that formula has a satisfying Boolean assignment.",
      "source_statement": "def language (input : List Bool) : Prop :=\n…",
      "source_statement_sha256": "8702ce2b4a8525f3fb2749145ffe7bdc33e9b60e8a93cce627530753fadcef55",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L119",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Integration.BinaryLinear.F2",
      "line": 126,
      "title": "The two-element coefficient field",
      "intuition": "`F2` abbreviates `ZMod 2`, the field whose two elements encode binary addition and translation.",
      "source_statement": "abbrev F2 := ZMod 2",
      "source_statement_sha256": "5d0d64b95007569376cb7191d8351782d45121f8eeb0f4ca6803d0b62183d6da",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L126",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Integration.BinaryLinear.Vector",
      "line": 128,
      "title": "A finite binary coordinate vector",
      "intuition": "`Vector n` abbreviates the function type `Fin n → F2`, so a vector is one modulo-two coordinate for each of `n` positions.",
      "source_statement": "abbrev Vector (n : Nat) := Fin n → F2\n…",
      "source_statement_sha256": "1707eefd6789d24f762160096ca392147abbe0a4bc9f88c56f43a899a1eabde8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L128",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Integration.TranslationTarget.IsTranslationInstance",
      "line": 137,
      "title": "Every permutation constraint is addition by one binary shift",
      "intuition": "After identifying labels with a binary vector space, `IsTranslationInstance coordinates g` requires each listed permutation to act as `x ↦ x+shift` for some shift depending on that constraint.",
      "source_statement": "def IsTranslationInstance {q s : Nat} (coordinates : Fin q ≃ BinaryLinear.Vector s)\n    (g : Instance q) : Prop :=\n…",
      "source_statement_sha256": "7111590e0c1fa9372f32bd8ad0403d71a747e68a6f89f441796eeabc10689aa5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L137",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Complexity.encodeWord",
      "line": 147,
      "title": "Encode one natural number in unary with a zero terminator",
      "intuition": "`encodeWord n` is `n` copies of `true` followed by one `false`.",
      "source_statement": "def encodeWord (n : Nat) : List Bool := List.replicate n true ++ [false]",
      "source_statement_sha256": "14c100dae5f9c52f2255831045266f43e0a14e9b0e1d9ad5d12a56aa66f8755c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L147",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Complexity.encodeWords",
      "line": 149,
      "title": "Concatenate a list of unary natural-number words",
      "intuition": "`encodeWords` maps each natural to `encodeWord` and joins the framed words in list order.",
      "source_statement": "def encodeWords : List Nat → List Bool\n  | [] => []\n  | n :: ns => encodeWord n ++ encodeWords ns\n\nend UniqueGamesTheorem.Foundations.Complexity\n\nnamespace UniqueGamesTheorem.Foundations.Complexity\n\nopen Target",
      "source_statement_sha256": "9f64bda4e527d93157c08cb8cb592b52ccd8f982f33092246fa9c2896683026d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L149",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Complexity.tableWords",
      "line": 159,
      "title": "Serialize the forward half of a permutation table",
      "intuition": "`tableWords table` lists the natural values of `table.images` in label order.",
      "source_statement": "def tableWords {q : Nat} (table : PermutationTable q) : List Nat :=\n…",
      "source_statement_sha256": "a40c0a14c6c2811ca4287efb965c2ec78a76e81b681819c75e21632d1a8899d4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L159",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Complexity.constraintWords",
      "line": 162,
      "title": "Serialize one constraint as endpoints plus a permutation table",
      "intuition": "`constraintWords` emits the source index, target index, and every forward permutation image as natural numbers.",
      "source_statement": "def constraintWords {n q : Nat} (constraint : Constraint n q) : List Nat :=\n…",
      "source_statement_sha256": "a4b1cff6b6bda75dc5ec85a512e84c4807feb97882645b3770255f2ae3938bcd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L162",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Complexity.gameWords",
      "line": 165,
      "title": "Flatten an explicit game into natural-number fields",
      "intuition": "`gameWords game` places a three-word header before the serialization of every constraint occurrence.",
      "source_statement": "def gameWords {q : Nat} (game : Instance q) : List Nat :=\n…",
      "source_statement_sha256": "1bdf115ec33b174aa37471d129b46b7ca79a963d3fe8f42b75ee81a5c0996eac",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L165",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.Foundations.Complexity.gameBits",
      "line": 168,
      "title": "The comparator's concrete output bit string",
      "intuition": "`gameBits game` applies the unary `encodeWords` representation to the complete natural-number field list from `gameWords`.",
      "source_statement": "def gameBits {q : Nat} (game : Instance q) : List Bool := encodeWords (gameWords game)\n…",
      "source_statement_sha256": "29177b012c2c2a8d3e465d6cd161ae8a172527534924517f5aeeba9c6466c144",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L168",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/UniqueGamesTheorem.lean",
      "declaration": "OAI.UniqueGamesTheorem.theorem11",
      "line": 231,
      "title": "Challenge statement: A polynomial-time binary-3SAT reduction proving the Unique Games gap",
      "intuition": "For any fixed completeness error ε and soundness bound δ between 0 and 1/2, there exists a concrete polynomial-time reduction from encoded binary 3SAT to a nonempty simple bipartite Unique Games instance. Satisfiable inputs admit a labeling of value at least 1-ε; unsatisfiable inputs give every labeling value at most δ.",
      "source_statement": "theorem theorem11 (ε δ : ℝ)\n    (hε : 0 < ε) (hεhalf : ε < 1 / 2)\n    (hδ : 0 < δ) (hδhalf : δ < 1 / 2) :\n    Nonempty (Explicit.MachineOutputContract.BinaryGapReduction ε δ) := by\n…",
      "source_statement_sha256": "4dca30fea3315d253edb808d5402e5527bf1ce668d83b95a2bd78d37b1ea3daf",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniqueGamesTheorem.lean#L231",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/QuinticLienard.lean",
      "declaration": "OAI.QuinticLienard.main",
      "line": 39,
      "title": "Challenge statement: Exactly two is the sharp limit-cycle bound for quintic Liénard systems",
      "intuition": "For the planar system x' = y-F(x), y' = -x, every real polynomial F of degree at most five has at most two geometrically distinct isolated periodic-orbit images, and some such F has exactly two.",
      "source_statement": "theorem main :\n    (∀ F : Polynomial ℝ, F.degree ≤ 5 → {C : Set Plane | IsLimitCycle F C}.encard ≤ 2) ∧\n    (∃ F : Polynomial ℝ, F.degree ≤ 5 ∧ {C : Set Plane | IsLimitCycle F C}.encard = 2) := by\n…",
      "source_statement_sha256": "5a9a07b944d771e17f304dd8fab93b6428bad7cf78a468b2c18b497328176931",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/QuinticLienard.lean#L39",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/YauCounterexample.lean",
      "declaration": "OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample",
      "line": 65,
      "title": "Challenge statement: A fixed smooth metric with unbounded normalized nodal four-volume",
      "intuition": "On S⁴×S¹, one fixed smooth Riemannian metric supports nonzero real Laplace eigenfunctions with eigenvalues tending to infinity while four-dimensional nodal measure divided by √λ also tends to infinity. This refutes a uniform O(√λ) upper bound for that metric.",
      "source_statement": "theorem yau_nodal_set_upper_bound_counterexample : MainTarget := by\n…",
      "source_statement_sha256": "0fe28cfcaeccf7e31f401e180a119c4e80c3a65f45f5bcb89ed3ee0205956efc",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/YauCounterexample.lean#L65",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/SeymourSecondNeighborhood.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.exists_goodVertex",
      "line": 34,
      "title": "Challenge statement: Seymour's second-neighborhood conjecture",
      "intuition": "Every nonempty finite oriented graph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood. Here \"second\" means directed distance exactly two: the vertex itself and vertices already reached in one step are excluded.",
      "source_statement": "theorem exists_goodVertex (r : V → V → Prop) (hr : IsOriented r) :\n    ∃ v, GoodVertex r v := by\n…",
      "source_statement_sha256": "1330cb81fa72804024c7254efe766ad292b77db4306bc99070f36fa2609454b6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/SeymourSecondNeighborhood.lean#L34",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/Model.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.IsOriented",
      "line": 10,
      "title": "The loopless, asymmetric relation underlying an oriented graph",
      "intuition": "`IsOriented r` packages the two relation-level conditions needed throughout the development: no vertex points to itself, and an arc `u → v` rules out the opposite arc `v → u`. The Lean model uses a relation rather than a graph structure, so finiteness and decidable equality are supplied separately where needed.",
      "source_statement": "structure IsOriented (r : V → V → Prop) : Prop where\n…",
      "source_statement_sha256": "8635fec259c5734a9d3f7ad4b15b6b4f6a4e9eb42e3096aa0f9e78ceb83c8d49",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/Model.lean#L10",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/Model.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.firstNeighbors",
      "line": 18,
      "title": "The first out-neighborhood of a vertex",
      "intuition": "`firstNeighbors r v` is the finite set of vertices `u` with an outgoing arc `r v u`. It is the manuscript's `N₁⁺(v)`, counting each reached vertex once.",
      "source_statement": "def firstNeighbors (r : V → V → Prop) (v : V) : Finset V := by\n…",
      "source_statement_sha256": "f1e984d36630e7947afc9e9b7c5367bbd64e550a7e8b35b9ae04e7ba087d7322",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/Model.lean#L18",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/Model.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.secondNeighbors",
      "line": 22,
      "title": "Vertices at directed distance exactly two",
      "intuition": "`secondNeighbors r v` contains a vertex `w` when some `u` gives a two-step path `v → u → w`, while explicitly excluding `v` itself and every first out-neighbor of `v`. This is exact distance two, not merely reachability within two steps.",
      "source_statement": "def secondNeighbors (r : V → V → Prop) (v : V) : Finset V := by\n…",
      "source_statement_sha256": "e59998cc4cbe28e69bbbfa3783625540363373180f5e47eeee50e501828cb0e6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/Model.lean#L22",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/Model.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.GoodVertex",
      "line": 27,
      "title": "A vertex with at least as many second as first out-neighbors",
      "intuition": "`GoodVertex r v` is precisely the conjectured cardinal inequality `|N₁⁺(v)| ≤ |N₂⁺(v)|`. It does not require either neighborhood to be nonempty, so every sink is automatically good.",
      "source_statement": "def GoodVertex (r : V → V → Prop) (v : V) : Prop :=\n…",
      "source_statement_sha256": "9e28e610dd99e0418dbba260235381419e0aa78a540fae74d9d72b288fc6f454",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/Model.lean#L27",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/ReductionCore.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.Counterexample",
      "line": 10,
      "title": "Every vertex has a strict first-neighborhood surplus",
      "intuition": "`Counterexample r` says that every vertex fails `GoodVertex`: its exact second out-neighborhood is strictly smaller than its first. This pointwise strict inequality is the negation normal form used by the minimal-counterexample argument.",
      "source_statement": "def Counterexample (r : V → V → Prop) : Prop :=\n…",
      "source_statement_sha256": "e0684bbf24f3ad7c40db11fcefb4606b51815ed3432106071f342d29c42ad8da",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/ReductionCore.lean#L10",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/ReductionCore.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.PositiveIndegree",
      "line": 29,
      "title": "Every vertex has an incoming neighbor",
      "intuition": "`PositiveIndegree r` requires an incoming witness `u → v` for each vertex `v`. In the reduction this property follows from strong connectivity of a minimal counterexample and survives the transitive-tournament blowup.",
      "source_statement": "def PositiveIndegree (r : V → V → Prop) : Prop :=\n…",
      "source_statement_sha256": "288cd1fb4ecd7951a7de59c5ffdb268e4feb932960e558178fa107f8a0dad1c2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/ReductionCore.lean#L29",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/ReductionCore.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.StrictSubsetGrowth",
      "line": 32,
      "title": "Every nonempty proper subset has a strict one-step image surplus",
      "intuition": "`StrictSubsetGrowth r` is the amplified inequality `|S|+|F²(S)| < 2|F(S)|` for every nonempty proper finite subset. It compares ordinary relation images, which may overlap `S`, rather than exact-distance neighborhoods.",
      "source_statement": "def StrictSubsetGrowth (r : V → V → Prop) : Prop :=\n…",
      "source_statement_sha256": "3db283b55569d604cd50f69154969235a39d91aa5838b68172f8766ee6852c6b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/ReductionCore.lean#L32",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/MinimalCounterexample.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.exists_minimal_counterexample",
      "line": 132,
      "title": "Select a smallest counterexample and then minimize its arcs",
      "intuition": "From any finite oriented counterexample, this theorem reindexes a counterexample onto `Fin n`, minimizes `n`, then minimizes the number of arcs. The chosen relation is oriented, remains a counterexample, is order- and arc-minimal, has nonempty boundary for every nontrivial subset, and has positive indegree.",
      "source_statement": "theorem exists_minimal_counterexample [Nonempty V] {r : V → V → Prop}\n    (hr : IsOriented r) (hbad : Counterexample r) :\n    ∃ n : ℕ, 0 < n ∧ ∃ s : Fin n → Fin n → Prop,\n      IsOriented s ∧ Counterexample s ∧ ArcMinimal s ∧\n        NonemptyBoundary s ∧ PositiveIndegree s := by\n…",
      "source_statement_sha256": "ddfc3d8c0673010120cd3f952a683ebe6806e8cfdd3b62752d2c2d1a82bfccd7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/MinimalCounterexample.lean#L132",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/SubsetDeficit.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.subsetDeficit_of_minimal",
      "line": 44,
      "title": "Minimal counterexamples satisfy the strengthened subset deficit",
      "intuition": "In an oriented counterexample minimal by arcs and equipped with a nonempty boundary, every nonempty proper set `S` has fewer newly reached second-image vertices than first-image vertices lying outside `S`. This strengthens the earlier subset perspective by using `|F(S)\\S|` on the right.",
      "source_statement": "theorem subsetDeficit_of_minimal {r : V → V → Prop}\n    (hr : IsOriented r) (hcounter : Counterexample r)\n    (hminimal : ArcMinimal r) (hboundary : NonemptyBoundary r) :\n    SubsetDeficit r := by\n…",
      "source_statement_sha256": "a28e90451b2a2664448fafc0e275cdc129096f19d5eb5a7ce7e35891c19c62ac",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/SubsetDeficit.lean#L44",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/TransitiveBlowup.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.blowup_strictSubsetGrowth",
      "line": 136,
      "title": "A large transitive-tournament blowup amplifies subset deficit into strict growth",
      "intuition": "Replace each base vertex by a transitive tournament with block size `m` larger than the number of base vertices. If the base graph has positive indegree and the strengthened subset deficit, the blowup satisfies `|U|+|F²(U)| < 2|F(U)|` for every nonempty proper subset `U`.",
      "source_statement": "theorem blowup_strictSubsetGrowth {r : V → V → Prop} {m : ℕ}\n    (_hr : IsOriented r) (hi : PositiveIndegree r) (hd : SubsetDeficit r)\n    (hm : Fintype.card V < m) : StrictSubsetGrowth (blowup r m) := by\n…",
      "source_statement_sha256": "f2d99ac0246bdcbfe8a8db8f5187c7bc98bccf606ea9af7600c0a14d5b720f58",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/TransitiveBlowup.lean#L136",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/Reduction.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.counterexample_reduction",
      "line": 11,
      "title": "Reduce any counterexample to a positive-indegree strict-growth graph",
      "intuition": "A hypothetical oriented counterexample yields a finite nonempty oriented graph `g` with positive indegree and `StrictSubsetGrowth g`. The theorem composes the minimal-counterexample selection, strengthened subset deficit, and a transitive-tournament blowup with block size `n+1`.",
      "source_statement": "theorem counterexample_reduction {r : V → V → Prop}\n    (hr : IsOriented r) (hcounter : Counterexample r) :\n    ∃ n : ℕ, 0 < n ∧\n      ∃ g : (Fin n × Fin (n + 1)) → (Fin n × Fin (n + 1)) → Prop,\n        IsOriented g ∧ PositiveIndegree g ∧ StrictSubsetGrowth g := by\n…",
      "source_statement_sha256": "fdcc38f8d16dfb2167a8ac18d00f39e7577281b3fb22b0fd4796c047ea09e3a8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/Reduction.lean#L11",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/Pruning.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.Pruning.pruning",
      "line": 10,
      "title": "Prune two pair families to compatibility with a controlled deletion cost",
      "intuition": "For any finite binary relation and two families of ordered pairs, the pruning theorem retains every member that was conflict-free initially and removes enough other members to eliminate all conflicts. The total deletions are charged to original conflict corners plus conflict corners left uncovered afterward.",
      "source_statement": "theorem pruning (r : X → X → Prop) : PruningStatement r := by\n…",
      "source_statement_sha256": "fba694cb9b1c64aa987592cdfa675fb77f41048cde06d64a04dd32c5fdde8d31",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/Pruning.lean#L10",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/ExtremalContradiction.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.Extremal.no_strictSubsetGrowth_of_pruning",
      "line": 29,
      "title": "Pruning contradicts the existence of a positive-indegree strict-growth graph",
      "intuition": "No nonempty finite oriented graph can simultaneously have positive indegree, satisfy the pruning statement, and obey `StrictSubsetGrowth`. An extremal compatible pair of ordered-pair families can always be replaced by a strictly better one, contradicting maximality.",
      "source_statement": "theorem no_strictSubsetGrowth_of_pruning {r : V → V → Prop}\n    (hr : IsOriented r) (hpos : PositiveIndegree r)\n    (hprune : PruningStatement r) : ¬ StrictSubsetGrowth r := by\n…",
      "source_statement_sha256": "b0ab18133be5769abe029ce96bd756b703de66b4498aedbd3b199bd24c385d6b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/ExtremalContradiction.lean#L29",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "OAI/Combinatorics/SecondNeighborhood/Main.lean",
      "declaration": "OAI.SeymourSecondNeighborhood.exists_goodVertex",
      "line": 11,
      "title": "Proof-bearing solution of Seymour's second-neighborhood conjecture",
      "intuition": "Every nonempty finite oriented graph has a good vertex. Unlike the comparator declaration with the same fully qualified name, this declaration imports the reduction, pruning theorem, and extremal contradiction and contains the checked proof term that composes them.",
      "source_statement": "theorem exists_goodVertex (r : V → V → Prop) (hr : IsOriented r) :\n    ∃ v, GoodVertex r v := by\n…",
      "source_statement_sha256": "1330cb81fa72804024c7254efe766ad292b77db4306bc99070f36fa2609454b6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/OAI/Combinatorics/SecondNeighborhood/Main.lean#L11",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/CannonGeometricAction.lean",
      "declaration": "OAI.CannonRelease.cannon",
      "line": 89,
      "title": "Challenge statement: Cannon's conjecture",
      "intuition": "A hyperbolic group whose boundary is a topological two-sphere admits an isometric action on hyperbolic three-space that is proper and cocompact and has finite kernel.",
      "source_statement": "theorem cannon (G : Type u) [Group G] [TopologicalSpace G] [DiscreteTopology G]\n    (D : CayleyData G) (hthin : D.ThinTriangles)\n    (hsphere : Nonempty (Boundary D ≃ₜ SphereTwo)) :\n    ∃ ρ : G →* H3Isom, ProperAction ρ ∧ CocompactAction ρ ∧\n      (ρ.ker : Set G).Finite := by\n…",
      "source_statement_sha256": "151f48ac8d254d3482b88a227a8b125c89789d57208028166d4cc968657e74cf",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/CannonGeometricAction.lean#L89",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/KoebeCircleDomains.lean",
      "declaration": "OAI.Problem047.koebe_circle_domain",
      "line": 62,
      "title": "Challenge statement: Koebe's unrestricted circle-domain uniformization",
      "intuition": "Every nonempty connected open subset of the Riemann sphere is conformally equivalent to a circle domain, with no countability restriction on its complementary components.",
      "source_statement": "theorem koebe_circle_domain :\n    ∀ U : Set Sphere, IsOpen U → IsConnected U →\n      ∃ (V : Set Sphere) (f : Sphere → Sphere),\n        IsCircleDomain V ∧ IsConformalEquivalence f U V := by\n…",
      "source_statement_sha256": "4df9bda0640379d9daa7dc453f2572808efa979a31d35de3804474f634741a4f",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/KoebeCircleDomains.lean#L62",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/KoebeCircleDomains.lean",
      "declaration": "OAI.Problem047.removability_implies_rigidity",
      "line": 68,
      "title": "Challenge statement: Conformal removability implies circle-domain rigidity",
      "intuition": "If the boundary of a circle domain is conformally removable, then every conformal equivalence from it to another circle domain is the restriction of a Möbius transformation, regardless of how many complementary components there are.",
      "source_statement": "theorem removability_implies_rigidity :\n    ∀ (U V : Set Sphere) (f : Sphere → Sphere),\n      IsCircleDomain U → IsCircleDomain V →\n      IsConformallyRemovable (frontier U) → IsConformalEquivalence f U V →\n      ∃ M : Mobius, ∀ p ∈ U, f p = M • p := by\n…",
      "source_statement_sha256": "9181c97d498292705e520878ee8c965c4f2fd010b34b47533037203cf933de2e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/KoebeCircleDomains.lean#L68",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/ThreeTorus.lean",
      "declaration": "OAI.ThreeTorus.simple_lebesgue_spectrum",
      "line": 82,
      "title": "Challenge statement: A smooth probability-preserving system with simple Lebesgue spectrum",
      "intuition": "There is a smooth volume-preserving diffeomorphism of the three-torus whose Koopman operator has simple Lebesgue spectrum on the entire complex mean-zero L² space. One real-valued vector generates a bilateral orthonormal basis under all integer iterates.",
      "source_statement": "theorem simple_lebesgue_spectrum : MainConclusion := by\n…",
      "source_statement_sha256": "046ecbcdcd6203bc8e0dda56d48d78891fbfa906e8d28ae6a7926b779ccda849",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/ThreeTorus.lean#L82",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/Rokhlin.lean",
      "declaration": "OAI.Rokhlin.mixing_all_finite_orders",
      "line": 30,
      "title": "Challenge statement: Ordinary mixing implies mixing of every finite order",
      "intuition": "Every invertible probability-preserving transformation that is mixing for pairs of measurable sets is mixing for every finite number of sets when all successive time gaps diverge.",
      "source_statement": "theorem mixing_all_finite_orders (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (T : Ω ≃ᵐ Ω) (h_pres : MeasurePreserving T μ μ) (h_mix : IsMixing μ T) :\n    ∀ k : ℕ, 3 ≤ k → MixingOfOrder μ T k := by\n…",
      "source_statement_sha256": "1cbc53851130f53560a74d1d32f7806b6a781492d14d404c123e334b3156fe50",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/Rokhlin.lean#L30",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/BorsukNine.lean",
      "declaration": "OAI.BorsukNine.main_theorem",
      "line": 25,
      "title": "Challenge statement: Borsuk's covering assertion fails in dimension nine",
      "intuition": "The compact set of rank-one orthogonal projectors onto lines in R⁴ lies in a nine-dimensional affine Euclidean space, has diameter √2, and cannot be covered by ten subsets of smaller diameter.",
      "source_statement": "theorem main_theorem :\n    IsCompact projectorSet ∧\n    projectorSet ⊆ traceOneSymmetric ∧\n    Metric.diam projectorSet = Real.sqrt 2 ∧\n    ¬ HasTenSmallCover := by\n…",
      "source_statement_sha256": "6af5180ccb7d877aeb69e1b579ee92a897750b8003b74dff33ed1f00d0eb6cca",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/BorsukNine.lean#L25",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/TingleySphereIsometry.lean",
      "declaration": "OAI.Tingley.tingley_sphere_isometry_full",
      "line": 22,
      "title": "Challenge statement: Every surjective Banach-sphere isometry extends uniquely and linearly",
      "intuition": "A surjective isometry between the unit spheres of arbitrary nonzero real Banach spaces extends uniquely to a surjective real-linear isometric equivalence of the whole spaces.",
      "source_statement": "theorem tingley_sphere_isometry_full\n    {X : Type u} {Y : Type v}\n    [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]\n    [NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]\n    [Nontrivial X] [Nontrivial Y]\n    (f : UnitSphere X → UnitSphere Y) (hf : Isometry f) (hs : Function.Surjective f) :\n    ∃ T : X ≃ₗᵢ[ℝ] Y,\n      (∀ u : UnitSphere X, T (u : X) = (f u : Y)) ∧\n      T 0 = 0 ∧\n      (∀ x : X, ∀ hx : x ≠ 0, T x = ‖x‖ • (f (normalize x hx) : Y)) ∧\n      Function.Surjective T ∧\n      (∀ L : X →ₗ[ℝ] Y, (∀ u : UnitSphere X, L (u : X) = (f u : Y)) →\n…",
      "source_statement_sha256": "f2f02df42e6b0c001c34dd36316cace5f398dde8d62022613d857b48271ec337",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/TingleySphereIsometry.lean#L22",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AbhyankarSathaye.lean",
      "declaration": "OAI.AbhyankarSathaye.exists_noncoordinate_polynomial",
      "line": 11,
      "title": "Challenge statement: Affine-space zero fibre without an ambient coordinate",
      "intuition": "In every complex ambient dimension at least four, there is a polynomial whose zero fibre is affine space of one lower dimension even though no polynomial change of coordinates turns that polynomial into a coordinate variable.",
      "source_statement": "theorem exists_noncoordinate_polynomial (n : ℕ) (hn : 4 ≤ n) :\n    ∃ F : MvPolynomial (Fin n) ℂ,\n      Nonempty ((MvPolynomial (Fin n) ℂ ⧸ Ideal.span {F}) ≃ₐ[ℂ]\n        MvPolynomial (Fin (n - 1)) ℂ) ∧\n      ¬ ∃ (equiv : MvPolynomial (Fin n) ℂ ≃ₐ[ℂ] MvPolynomial (Fin n) ℂ)\n        (index : Fin n), equiv (MvPolynomial.X index) = F := by\n…",
      "source_statement_sha256": "53b2d25075f0f4d1e7e6c9fc93c750b4d646b2750788da653338161a9a7303fb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AbhyankarSathaye.lean#L11",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/CirculantHadamard.lean",
      "declaration": "OAI.CirculantHadamard.exists_iff_order_one_or_four",
      "line": 24,
      "title": "Challenge statement: Real circulant Hadamard matrices have order one or four",
      "intuition": "A positive-order real matrix whose entries are signs, whose rows are cyclic shifts, and whose rows are mutually orthogonal exists exactly in orders one and four.",
      "source_statement": "theorem exists_iff_order_one_or_four (n : ℕ) (hn : 0 < n) :\n    ExistsRealCirculantHadamard n ↔ n = 1 ∨ n = 4 := by\n…",
      "source_statement_sha256": "42b0de2a49ee177d69923212e297afbd01f8ffcaa1d8dc9b8c05225d66d7de31",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/CirculantHadamard.lean#L24",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/EilenbergGanea.lean",
      "declaration": "OAI.EilenbergGanea.source_main",
      "line": 113,
      "title": "Challenge statement: A finitely generated Eilenberg-Ganea counterexample",
      "intuition": "One explicit height-kernel group is finitely generated and residually finite, has integral cohomological dimension two, admits a three-dimensional classifying space, and admits no two-dimensional classifying space.",
      "source_statement": "theorem source_main :\n    Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧\n    cohomologicalDimension SourceGroup = 2 ∧\n    HasClassifyingSpace.{0} SourceGroup 3 ∧\n    ¬ HasClassifyingSpace.{u} SourceGroup 2 := by\n…",
      "source_statement_sha256": "ed83200d5d56323d7656ffe954620b3a966f6808a85d84fac61f73b90b80bc96",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/EilenbergGanea.lean#L113",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/ShortEgyptianFractions.lean",
      "declaration": "OAI.ShortEgyptian.main",
      "line": 17,
      "title": "Challenge statement: The optimal uniform length of Egyptian-fraction expansions",
      "intuition": "Every proper positive rational number has an expansion into distinct increasing unit fractions, and the worst minimum number of terms among fractions with denominator `b` grows on the order of `log(log b)`.",
      "source_statement": "theorem main :\n    (∀ a b : ℕ, 1 ≤ a → a < b → ∃ ns : List ℕ, IsExpansion a b ns) ∧\n    ∃ c₁ c₂ : ℝ, 0 < c₁ ∧ 0 < c₂ ∧ ∃ b₀ : ℕ, 2 ≤ b₀ ∧\n      ∀ b : ℕ, b₀ ≤ b →\n        c₁ * Real.log (Real.log (b : ℝ)) ≤ (maxMinLength b : ℝ) ∧\n        (maxMinLength b : ℝ) ≤ c₂ * Real.log (Real.log (b : ℝ)) := by\n…",
      "source_statement_sha256": "071803734707ab38d179f62a511f4dc9593ee48d33b158e34fada4b3f07f99d6",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/ShortEgyptianFractions.lean#L17",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/CompleteCrouzeix.lean",
      "declaration": "OAI.CompleteCrouzeix.main",
      "line": 33,
      "title": "Challenge statement: The complete Crouzeix inequality has sharp constant two",
      "intuition": "Evaluate a matrix-valued polynomial on a complex matrix by replacing each scalar power $z^k$ with the matrix power $A^k$. Its operator norm is at most twice the largest coefficient-matrix norm attained while $z$ ranges over the numerical range of $A$, and no smaller constant works uniformly.",
      "source_statement": "theorem main : UniversalBound 2 ∧ ∀ C : ℝ, UniversalBound C → 2 ≤ C := by\n…",
      "source_statement_sha256": "8ee1c4e8c06b2a56188b294f588c3bcc124b469ef8920d32fa238162b6739098",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/CompleteCrouzeix.lean#L33",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/EuclideanFiveColor.lean",
      "declaration": "OAI.EuclideanFiveColor.no_proper_five_coloring",
      "line": 10,
      "title": "Challenge statement: Five colors cannot properly color the Euclidean plane",
      "intuition": "However five labels are assigned to all points of the plane, some two points exactly one unit apart receive the same label. The coloring is an arbitrary function: measurability, continuity, and region-shaped color classes are not assumed.",
      "source_statement": "theorem no_proper_five_coloring : ¬ ∃ coloring : ℂ → Fin 5, ProperColoring 5 coloring := by\n…",
      "source_statement_sha256": "9f9314ec489c042eb971e70981074b8d4bd7cef085e3d9eb7eee9248c23e0595",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/EuclideanFiveColor.lean#L10",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/GeneralMahler.lean",
      "declaration": "OAI.GeneralMahler.general_mahler",
      "line": 24,
      "title": "Challenge statement: The general Mahler bound, with equality exactly for simplices",
      "intuition": "Among all compact convex bodies with interior in $\\mathbb R^n$, the smallest possible product of the body's volume and the volume of an optimally centered polar is $(n+1)^{n+1}/(n!)^2$. Exactly the simplices attain it.",
      "source_statement": "theorem general_mahler {n : ℕ} (hn : 1 ≤ n)\n    (K : Set (EuclideanSpace ℝ (Fin n)))\n    (hcompact : IsCompact K) (hconvex : Convex ℝ K)\n    (hinterior : (interior K).Nonempty) :\n    ((n : ℝ) + 1) ^ (n + 1) / (Nat.factorial n : ℝ) ^ 2 ≤ volumeProduct K ∧\n      (volumeProduct K = ((n : ℝ) + 1) ^ (n + 1) / (Nat.factorial n : ℝ) ^ 2 ↔\n        ∃ vertices : Fin (n + 1) → EuclideanSpace ℝ (Fin n),\n          AffineIndependent ℝ vertices ∧ K = convexHull ℝ (range vertices)) := by\n…",
      "source_statement_sha256": "ccf35763e49c6125ae9fdc9e927b359ccc2bcb40ae03a057550a7362b9d77010",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/GeneralMahler.lean#L24",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/ThompsonNonamenability.lean",
      "declaration": "OAI.ThompsonNonamenability.thompson_F_nonamenable_composition",
      "line": 78,
      "title": "Challenge statement: Thompson's group F has no invariant mean",
      "intuition": "Thompson's group $F$ is modeled by increasing, dyadic piecewise-linear homeomorphisms of the unit interval. The theorem supplies the composition group law and rules out every positive normalized left-invariant mean on its bounded real-valued functions; that is the asserted nonamenability.",
      "source_statement": "theorem thompson_F_nonamenable_composition :\n    ∃ group : Group F, letI := group;\n…",
      "source_statement_sha256": "dc4a0e48ee26d1faf2735b79f2f00ae0519e493d2f43d387c21b8cb09a79c0bd",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/ThompsonNonamenability.lean#L78",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.StrongThinTree.MultiGraph",
      "line": 9,
      "title": "A finite loopless multigraph with individually indexed edge copies",
      "intuition": "`MultiGraph n m` has vertices `Fin n` and edge copies `Fin m`. Each edge copy has a left and right endpoint, and `loopless` rules out an edge whose two endpoints coincide.",
      "source_statement": "structure MultiGraph (n m : ℕ) where\n…",
      "source_statement_sha256": "b8b2702e50b34c2315befd108054fea3ab97bb18d3e1f76185c732ee49078559",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L9",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.StrongThinTree.MultiGraph.cut",
      "line": 16,
      "title": "The selected edges crossing a vertex cut",
      "intuition": "`G.cut T S` keeps exactly those edge copies in `T` with one endpoint in the vertex set `S` and the other endpoint outside it.",
      "source_statement": "def cut {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m))\n    (S : Finset (Fin n)) : Finset (Fin m) :=\n…",
      "source_statement_sha256": "7ade1864d39b9283a88a1017919069db7b26e76ff2f452628f23ba7518ae92f9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L16",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.StrongThinTree.MultiGraph.Connected",
      "line": 21,
      "title": "Connectivity of a chosen edge set",
      "intuition": "`G.Connected T` says every nonempty proper set of vertices has at least one edge of `T` crossing to its complement.",
      "source_statement": "def Connected {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop :=\n…",
      "source_statement_sha256": "a4f1ffa20b40d1b2a5601bceb7b3d407f7b80f6d875a3f32d104785dcf88a251",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L21",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.StrongThinTree.MultiGraph.SpanningTree",
      "line": 24,
      "title": "A minimally connected spanning edge set",
      "intuition": "`G.SpanningTree T` means `T` connects all vertices and deleting any one edge of `T` destroys that connectivity.",
      "source_statement": "def SpanningTree {n m : ℕ} (G : MultiGraph n m) (T : Finset (Fin m)) : Prop :=\n…",
      "source_statement_sha256": "2a28fbf7211ee02ba164ae3047754db0a80f544dc38c1def4087580d29e547e5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L24",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.StrongThinTree.MultiGraph.EdgeConnected",
      "line": 27,
      "title": "At least $k$ edge copies cross every nontrivial cut",
      "intuition": "`G.EdgeConnected k` says each nonempty proper vertex set has a whole-graph cut containing at least `k` edge copies.",
      "source_statement": "def EdgeConnected {n m : ℕ} (G : MultiGraph n m) (k : ℕ) : Prop :=\n…",
      "source_statement_sha256": "17f9cb0644f148b53fe718c4a8fd65ee6bd8b5033fc02e82fe9b2490cde1c755",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L27",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.frame",
      "line": 36,
      "title": "A self-delimiting Boolean-list frame",
      "intuition": "`frame` encodes each payload bit as `false` followed by that bit, then terminates the framed list with `true`.",
      "source_statement": "def frame : List Bool → List Bool\n  | [] => [true]\n  | b :: bs => false :: b :: frame bs",
      "source_statement_sha256": "527c80d732b42254a2eb7cecf61baaf6f8445400e8e98db95541b48223e8cab3",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L36",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.encodeNat",
      "line": 40,
      "title": "A framed bit encoding of a natural number",
      "intuition": "`encodeNat n` applies the file's self-delimiting `frame` to Mathlib's computability-oriented Boolean encoding of `n`.",
      "source_statement": "def encodeNat (n : ℕ) : List Bool := frame (Computability.encodeNat n)",
      "source_statement_sha256": "7b44dbe9466be9fd0a2173311b8ed7594cd1e6bf81c0536cfe4dd85aa964ab0d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L40",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.StackAction",
      "line": 42,
      "title": "The three permitted updates to one Boolean stack",
      "intuition": "A `StackAction` either leaves a stack alone, removes its head, or pushes one specified Boolean bit.",
      "source_statement": "inductive StackAction where\n…",
      "source_statement_sha256": "47a653e6a695c02bf6cd4a3233f78551703de8a9a694af1b26e07e67b4b862fe",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L42",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.StackAction.apply",
      "line": 48,
      "title": "Execute one action on one Boolean stack",
      "intuition": "`stay` returns the stack unchanged, `pop` takes its tail, and `push b` prepends `b`.",
      "source_statement": "def StackAction.apply : StackAction → List Bool → List Bool\n  | .stay, xs => xs\n  | .pop, xs => xs.tail\n  | .push b, xs => b :: xs",
      "source_statement_sha256": "fef312abc591338228014098f1363aad32c4988eb27cbd5552dcbd71f6c493ff",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L48",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.Instruction",
      "line": 53,
      "title": "One transition of a finite multi-stack machine",
      "intuition": "An `Instruction k q` contains an optional next state among `q` states and one stack action for each of `k` stacks.",
      "source_statement": "structure Instruction (k q : ℕ) where\n…",
      "source_statement_sha256": "d2e0ad7ed01e6c37a083516d5e5f3e5068c99c28cd7f74f54aaa06ed81a2d7a8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L53",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.Instruction.halt",
      "line": 57,
      "title": "The default halting instruction",
      "intuition": "`Instruction.halt k q` sets the next state to `none` and leaves every stack unchanged.",
      "source_statement": "def Instruction.halt (k q : ℕ) : Instruction k q := ⟨none, fun _ => .stay⟩",
      "source_statement_sha256": "e13f96b2da016ef83d59a52246138c6f10c7149a43beb2b2de4b4b30011cd534",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L57",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.Machine",
      "line": 59,
      "title": "A finite Boolean-stack machine with designated input and output stacks",
      "intuition": "`Machine` records finite numbers of stacks and states, an initial state, designated input and output stacks, and a finite instruction table.",
      "source_statement": "structure Machine where\n…",
      "source_statement_sha256": "51974a70e8786659389ffa462afe4f7320f575737deba6db6c52dfce1d5c808b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L59",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.Config",
      "line": 67,
      "title": "A machine state together with all current stacks",
      "intuition": "`Config k q` stores either a current state among `q` states or `none` for halting, plus one Boolean list for each of `k` stacks.",
      "source_statement": "structure Config (k q : ℕ) where\n…",
      "source_statement_sha256": "e5e38062f1c9bd898efab2f4e42684fcdee5003d5fbbdcd64ec8eb18c397b1bb",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L67",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.topCode",
      "line": 71,
      "title": "Encode a stack's visible head as one of three cases",
      "intuition": "`topCode` returns `0` for an empty stack, `1` for a leading `false`, and `2` for a leading `true`.",
      "source_statement": "def topCode : List Bool → ℕ\n  | [] => 0\n  | false :: _ => 1\n  | true :: _ => 2",
      "source_statement_sha256": "e2caef25c0f22a787695572355f23e0797e021611b10cc118b0844f3e8a16a21",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L71",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.headsCode",
      "line": 76,
      "title": "Combine all visible stack heads into one base-$3$ number",
      "intuition": "`headsCode k s` recursively packs the empty/false/true status of each of the `k` stack heads into a natural number.",
      "source_statement": "def headsCode : (k : ℕ) → (Fin k → List Bool) → ℕ\n  | 0, _ => 0\n  | k + 1, s => topCode (s 0) + 3 * headsCode k (fun j => s j.succ)",
      "source_statement_sha256": "bba2daec776791ee293a2b9a116af65061dfd0090701dc3aac53ffbcd171178a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L76",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.Machine.step",
      "line": 80,
      "title": "Perform one deterministic machine transition",
      "intuition": "A halted configuration stays fixed. Otherwise the current state and all visible stack heads select one instruction, whose next state and stack actions produce the next configuration.",
      "source_statement": "def Machine.step (M : Machine) (c : Config M.stackCount M.states) :\n    Config M.stackCount M.states :=\n…",
      "source_statement_sha256": "4a4275ea156582a61d069fc969ce7e84cb24aa4bb062c4b1f4a74ca3b8a5ad18",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L80",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.Machine.init",
      "line": 90,
      "title": "Initialize the machine from an encoded input",
      "intuition": "`M.init input` starts in `M.initial`, places the entire Boolean input on `M.inputStack`, and makes every other stack empty.",
      "source_statement": "def Machine.init (M : Machine) (input : List Bool) : Config M.stackCount M.states :=\n…",
      "source_statement_sha256": "7c02fc47ca8a1b0d3780c21868175b410de04b4e55304cf024b0f1740236f73a",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L90",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.CurrentKS.Machine.run",
      "line": 93,
      "title": "Run a machine for exactly the supplied fuel",
      "intuition": "`M.run input fuel` initializes `M` and iterates its one-step transition exactly `fuel` times.",
      "source_statement": "def Machine.run (M : Machine) (input : List Bool) (fuel : ℕ) :\n    Config M.stackCount M.states := (M.step^[fuel]) (M.init input)\n…",
      "source_statement_sha256": "8b0a6425c0abcdfff8013a85af9c64ca6c62c45a118a80e158910dd60f678043",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L93",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.ExplicitInput",
      "line": 101,
      "title": "A thin-tree instance with every edge copy listed explicitly",
      "intuition": "`ExplicitInput` packages the number of vertices `n`, number of edge copies `m`, connectivity parameter `k`, and an actual `MultiGraph n m`.",
      "source_statement": "structure ExplicitInput where\n…",
      "source_statement_sha256": "8ac20a21e92693c4295a0e55960cdcd125d64de4c3c69c979af91b9bf5096674",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L101",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.BinaryInput",
      "line": 107,
      "title": "A multiplicity-compressed multigraph input",
      "intuition": "`BinaryInput` lists `p` distinct ordered endpoint pairs and assigns a natural-number multiplicity to each pair instead of enumerating every parallel edge copy.",
      "source_statement": "structure BinaryInput where\n…",
      "source_statement_sha256": "e1df59be7d4aca3e5d95a90b1db991ac96fc6487177cbe3ffd842f39593949f2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L107",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.BinaryInput.Edge",
      "line": 117,
      "title": "One concrete edge copy in a compressed binary input",
      "intuition": "`x.Edge` is a dependent pair: an endpoint-pair index `i`, followed by a copy index below `x.multiplicity i`.",
      "source_statement": "abbrev BinaryInput.Edge (x : BinaryInput) := Σ i : Fin x.p, Fin (x.multiplicity i)",
      "source_statement_sha256": "f20df3b959b7c1931c8a4db675ae75d2fe1fe27c77485d4221ac282a464d7d6c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L117",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.BinaryInput.edgeEquiv",
      "line": 119,
      "title": "Number all expanded binary-input edge copies",
      "intuition": "`edgeEquiv` chooses an equivalence between the finite type `x.Edge` and consecutive indices `Fin (Fintype.card x.Edge)`.",
      "source_statement": "noncomputable def BinaryInput.edgeEquiv (x : BinaryInput) :\n    x.Edge ≃ Fin (Fintype.card x.Edge) := Fintype.equivFin x.Edge",
      "source_statement_sha256": "44ae4d3952fe20ef844f4b70d32ff98848340aaaff654e774e242379aa5546ce",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L119",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.BinaryInput.graph",
      "line": 122,
      "title": "Expand compressed multiplicities into a loopless multigraph",
      "intuition": "`x.graph` turns every inhabitant of `x.Edge` into its own indexed edge copy and assigns it the endpoints of its pair index.",
      "source_statement": "noncomputable def BinaryInput.graph (x : BinaryInput) :\n    MultiGraph x.n (Fintype.card x.Edge) where\n…",
      "source_statement_sha256": "2e3d842c06b1022d6e7add11955715c421e22b4660c4ff7aa27c783180fc88f5",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L122",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.Input",
      "line": 128,
      "title": "Either an explicit or multiplicity-compressed graph instance",
      "intuition": "`Input` is a tagged union whose `explicit` branch contains every edge copy and whose `binary` branch contains endpoint pairs with multiplicities.",
      "source_statement": "inductive Input where\n…",
      "source_statement_sha256": "c0f30b83268d64f629f47817084529ee926e8be1e5aeddc34aa326cf57cac025",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L128",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.ExplicitInput.encode",
      "line": 132,
      "title": "Serialize an explicitly listed multigraph",
      "intuition": "The encoding concatenates framed natural numbers for `n`, `m`, and `k`, then for every edge index concatenates its two endpoint indices.",
      "source_statement": "def ExplicitInput.encode (x : ExplicitInput) : List Bool :=\n…",
      "source_statement_sha256": "c002ed96bf197a9f4e094ffd5697424f6c5abda45801e9dfa7892b1b3e7a3c94",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L132",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.BinaryInput.encode",
      "line": 138,
      "title": "Serialize a multiplicity-compressed multigraph",
      "intuition": "The encoding writes `n`, the number `p` of endpoint pairs, and `k`, followed by each pair's endpoints and multiplicity.",
      "source_statement": "def BinaryInput.encode (x : BinaryInput) : List Bool :=\n…",
      "source_statement_sha256": "7fa12be42bab9bf95f9c5a966ceb0e4a740c9aaa79ced60ca70f8ec47212a90c",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L138",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.Input.encode",
      "line": 144,
      "title": "Tag and serialize either input representation",
      "intuition": "`Input.encode` prefixes `false` for an explicit input and `true` for a binary-multiplicity input, then appends that branch's encoding.",
      "source_statement": "def Input.encode : Input → List Bool\n  | .explicit x => false :: x.encode\n  | .binary x => true :: x.encode",
      "source_statement_sha256": "d1fb61c02d35c84638dcea523f97e5efd4f63095375c8c0f8b0186e6d4a8cca4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L144",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.Input.length",
      "line": 148,
      "title": "The bit length used by the polynomial runtime bound",
      "intuition": "`x.length` is exactly the length of `x.encode`, including the representation tag and all framing bits.",
      "source_statement": "def Input.length (x : Input) : ℕ := x.encode.length",
      "source_statement_sha256": "8ff69a2fb6c4990ba73bbd616439b75e1afd0c233ded9d8489ae32696ba9c0d1",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L148",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.Input.n",
      "line": 150,
      "title": "Read the vertex count from either input branch",
      "intuition": "`Input.n` returns the stored number of vertices whether the graph was represented explicitly or by binary multiplicities.",
      "source_statement": "def Input.n : Input → ℕ\n  | .explicit x => x.n\n  | .binary x => x.n",
      "source_statement_sha256": "093c62944e907c515abc516f6b0fa64d1928e7f670cb2847f71e1c09c006393b",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L150",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.Valid",
      "line": 154,
      "title": "The admissible $k$-edge-connected inputs",
      "intuition": "An input is `Valid` when it has at least one vertex, has positive `k`, and its explicit or expanded multigraph is $k$-edge-connected.",
      "source_statement": "def Valid : Input → Prop\n  | .explicit x => 1 ≤ x.n ∧ 1 ≤ x.k ∧ x.graph.EdgeConnected x.k\n  | .binary x => 1 ≤ x.n ∧ 1 ≤ x.k ∧ x.graph.EdgeConnected x.k",
      "source_statement_sha256": "1e67de896467f5beef95f8d596c5ab8f6769c2d5bbcca9d62e91486712d37fe4",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L154",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.ThinTree",
      "line": 158,
      "title": "A spanning tree uniformly thin across every cut",
      "intuition": "`ThinTree C k G T` requires `T` to be a spanning tree and to use at most a $C/k$ fraction of the whole graph's edges across every nontrivial vertex cut.",
      "source_statement": "def ThinTree {n m : ℕ} (C : ℝ) (k : ℕ) (G : MultiGraph n m)\n    (T : Finset (Fin m)) : Prop :=\n…",
      "source_statement_sha256": "da772759bee70af2275c1f58b94cd1ba1e3a9ea75313042faa3971941e37ca64",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L158",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.ExplicitInput.encodeEdges",
      "line": 163,
      "title": "Serialize a list of explicit edge indices",
      "intuition": "`encodeEdges` concatenates the framed natural-number value of every explicit edge index in the proposed output list.",
      "source_statement": "def ExplicitInput.encodeEdges (x : ExplicitInput) (edges : List (Fin x.m)) : List Bool :=\n…",
      "source_statement_sha256": "74330f5d5105beab678521b8f5c3479f91e60ad68ff38dc0a078147420327679",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L163",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.BinaryInput.encodeEdges",
      "line": 166,
      "title": "Serialize selected copies from a compressed multigraph",
      "intuition": "Each output edge is encoded by its endpoint-pair index and its copy index within that pair's multiplicity.",
      "source_statement": "def BinaryInput.encodeEdges (x : BinaryInput) (edges : List x.Edge) : List Bool :=\n…",
      "source_statement_sha256": "4bde569705b16a90b8c2ec0ecaaabd1cae9985e0fb60c6cea4868a3b69f09864",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L166",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.BinaryInput.selected",
      "line": 169,
      "title": "Convert decoded binary-input edge copies to multigraph indices",
      "intuition": "`selected` removes repetitions from a decoded edge list and maps each dependent edge copy through `edgeEquiv` into the `Fin` indices used by `x.graph`.",
      "source_statement": "noncomputable def BinaryInput.selected (x : BinaryInput) (edges : List x.Edge) :\n    Finset (Fin (Fintype.card x.Edge)) := edges.toFinset.map x.edgeEquiv.toEmbedding",
      "source_statement_sha256": "950f969780713b991bf59e8ff39645658665fed3604606aa6de5bbd27ccf13a7",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L169",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.TreeOutput",
      "line": 172,
      "title": "The output bits encode a duplicate-free thin spanning tree",
      "intuition": "`TreeOutput C x bits` means there exists a duplicate-free list of valid edge copies whose branch-specific serialization is exactly `bits` and whose selected edge set is `ThinTree C`.",
      "source_statement": "def TreeOutput (C : ℝ) : Input → List Bool → Prop\n  | .explicit x, bits => ∃ edges : List (Fin x.m), edges.Nodup ∧\n      bits = x.encodeEdges edges ∧ ThinTree C x.k x.graph edges.toFinset\n  | .binary x, bits => ∃ edges : List x.Edge, edges.Nodup ∧\n      bits = x.encodeEdges edges ∧ ThinTree C x.k x.graph (x.selected edges)",
      "source_statement_sha256": "2d6c1be494a3b2345fefc362da2baadf2f7955487a38f372c02dc484b4a6c340",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L172",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.GoodOutput",
      "line": 178,
      "title": "A valid tree output with a canonical one-vertex case",
      "intuition": "`GoodOutput C x bits` requires a `TreeOutput` and additionally requires `bits = []` whenever the input has exactly one vertex.",
      "source_statement": "def GoodOutput (C : ℝ) (x : Input) (bits : List Bool) : Prop :=\n…",
      "source_statement_sha256": "75d2af6f1b45c52805f7682df6c55b4ba9d3af66e772480b0666a2938d016253",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L178",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.AlgorithmicStrongThinTrees",
      "line": 181,
      "title": "The complete deterministic polynomial-time strong thin-tree proposition",
      "intuition": "There exist one positive real constant `C`, one fixed finite stack machine `M`, and natural numbers `a` and `degree` with `a > 0` such that every valid input is solved within `a * (length + 1) ^ degree` steps.",
      "source_statement": "def AlgorithmicStrongThinTrees : Prop :=\n…",
      "source_statement_sha256": "98d60da3855573480e689f575a2bcaa65eaaeff3c9224e57a162267235c8e4f9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L181",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AlgorithmicThinTrees.lean",
      "declaration": "OAI.AlgorithmicThinTrees.algorithmic_strong_thin_trees",
      "line": 191,
      "title": "Challenge statement: A deterministic polynomial-time strong thin-tree construction",
      "intuition": "There is one universal constant $C>0$ and one deterministic finite stack machine that, from either of the statement's two encodings of any finite loopless $k$-edge-connected multigraph, halts in polynomially many steps and outputs a spanning tree whose share of every nontrivial cut is at most $C/k$ times the graph's share of that cut.",
      "source_statement": "theorem algorithmic_strong_thin_trees : AlgorithmicStrongThinTrees := by\n…",
      "source_statement_sha256": "0be3e9e75083361bffb71af8466b21e9492a53016f41878158f53dea83a93d03",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AlgorithmicThinTrees.lean#L191",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/BarnetteHamiltonian.lean",
      "declaration": "OAI.Barnette.main",
      "line": 48,
      "title": "Challenge statement: Barnette's Hamiltonian-cycle conjecture",
      "intuition": "Every finite simple planar graph that is cubic, bipartite, and remains connected after deletion of any two vertices has one cycle visiting every vertex exactly once.",
      "source_statement": "theorem main : MainStatement.{u} := by\n…",
      "source_statement_sha256": "cf0472ff3dc63f886a00b008bf6a26412e8301ce1b159a0de0893ae80b4bff84",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/BarnetteHamiltonian.lean#L48",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/ArtinCAT0.lean",
      "declaration": "OAI.ArtinCAT0.main",
      "line": 65,
      "title": "Challenge statement: An explicit Artin group with no geometric $\\mathrm{CAT}(0)$ action",
      "intuition": "An explicitly presented Artin group on 116 generators cannot act geometrically on any nonempty proper $\\mathrm{CAT}(0)$ metric space, in any dimension.",
      "source_statement": "theorem main :\n    (∀ s t : Letters, explicitMatrix s t = explicitMatrix t s) ∧\n    (∀ s : Letters, explicitMatrix s s = 1) ∧\n    (∀ s t : Letters, s ≠ t →\n      explicitMatrix s t = 2 ∨ explicitMatrix s t = 3 ∨ explicitMatrix s t = ⊤) ∧\n    (∀ (X : Type u) [MetricSpace X] [ProperSpace X] [Nonempty X]\n      [MulAction (ArtinGroup explicitMatrix) X],\n      ¬ GeometricAction (ArtinGroup explicitMatrix) X) := by\n…",
      "source_statement_sha256": "7b4236d76576f5108e00b3bd734e94ed67e49001cd7f459371c83c9cc9773445",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/ArtinCAT0.lean#L65",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "oai",
      "file": "ComparatorChallenges/AmplitudeDamping.lean",
      "declaration": "OAI.GAD.main",
      "line": 113,
      "title": "Challenge statement: Exact unassisted classical capacity of generalized amplitude damping",
      "intuition": "For every damping strength and thermal occupation in $[0,1]$, the unrestricted $n$-use Holevo information is exactly $n$ times its one-use value, an explicit one-variable maximum attains that value, and the operational unassisted classical capacity equals it.",
      "source_statement": "theorem main (γ ν : ℝ) (hγ : γ ∈ Set.Icc (0 : ℝ) 1)\n    (hν : ν ∈ Set.Icc (0 : ℝ) 1) :\n    (∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) * holevo γ ν 1) ∧\n    (∃ p : ℝ, Maximizes γ ν p ∧\n      ∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) / Real.log 2 * objective γ ν p) ∧\n    capacity γ ν = holevo γ ν 1 ∧\n    (∀ p : ℝ, Maximizes γ ν p → ∀ n : ℕ, 1 ≤ n →\n      (∀ s : Basis n, IsState (phaseState p s)) ∧\n      holevo γ ν n = phaseValue γ ν p n) := by\n…",
      "source_statement_sha256": "7396f0bdd16e46e96bf3d2cd231a083f807918fa3760e3c5fc2ce7e2f0f7328d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/AmplitudeDamping.lean#L113",
      "corpus_references": [
        {
          "label": "Pinned openai/math documentation",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/README.md"
        },
        {
          "label": "Pinned formalization catalogue",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/formalization.yaml"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/Main.lean",
      "declaration": "PFR_conjecture",
      "line": 261,
      "title": "Polynomial Freiman-Ruzsa conjecture (characteristic 2)",
      "intuition": "A finite set in an F2-vector space with small doubling is covered by about K^12 translates of a subgroup no bigger than the set itself.",
      "source_statement": "theorem PFR_conjecture {G : Type*} [AddCommGroup G] [Module (ZMod 2) G]\n    {A : Set G} {K : ℝ} (hA₀ : A.Nonempty) (Afin : A.Finite)\n    (hA : (A + A).ncard ≤ K * A.ncard) :\n    ∃ (H : Submodule (ZMod 2) G) (c : Set G), c.Finite ∧ (H : Set G).Finite ∧\n      Nat.card c < 2 * K ^ 12 ∧ (H : Set G).ncard ≤ A.ncard ∧ A ⊆ c + H := by\n…",
      "source_statement_sha256": "72233e3e00d40861ead7f9c5ceab4ce732e6cfaf4646030be6dc92b25510293e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/Main.lean#L261",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/Main.lean",
      "declaration": "PFR_conjecture_aux",
      "line": 153,
      "title": "Covering lemma behind PFR",
      "intuition": "A set of doubling K is covered by few cosets of a subgroup whose size is within a power of K of the set's size.",
      "source_statement": "lemma PFR_conjecture_aux (hA₀ : A.Nonempty) (hA : (A + A).ncard ≤ K * A.ncard) :\n    ∃ (H : Submodule (ZMod 2) G) (c : Set G),\n    Nat.card c ≤ K ^ (13/2 : ℝ) * A.ncard ^ (1/2 : ℝ) * (H : Set G).ncard ^ (-1/2 : ℝ) ∧\n      (H : Set G).ncard ≤ K ^ 11 * A.ncard ∧ A.ncard ≤ K ^ 11 * (H : Set G).ncard ∧ A ⊆ c + H := by\n…",
      "source_statement_sha256": "63281c8cecf12ff2b421cce72b1b3a66e164615d8995dab4dd96fcde2af36233",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/Main.lean#L153",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/Main.lean",
      "declaration": "rdist_le_of_isUniform_of_card_sub_le",
      "line": 115,
      "title": "Small doubling gives small self Ruzsa distance",
      "intuition": "A uniform random variable on a set with |A - A| at most K|A| is within log K of itself in Ruzsa distance.",
      "source_statement": "theorem rdist_le_of_isUniform_of_card_sub_le [A_fin : Finite A] [MeasurableSpace G]\n    [MeasurableSingletonClass G]\n    (hA₀ : A.Nonempty) (hA : (A - A).ncard ≤ K * A.ncard)\n    {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)] {U₀ : Ω → G}\n    (U₀unif : IsUniform A U₀) (U₀meas : Measurable U₀) : d[U₀ # U₀] ≤ log K := by\n…",
      "source_statement_sha256": "5690547072d102f1581be2adc094769d30591a7336f0e38648b093d9bc4abcc2",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/Main.lean#L115",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/EntropyPFR.lean",
      "declaration": "entropic_PFR_conjecture",
      "line": 54,
      "title": "Entropic PFR (constants 11 and 6)",
      "intuition": "Any two random variables in an F2-vector space are within a constant multiple of their Ruzsa distance of a uniform distribution on some subgroup.",
      "source_statement": "theorem entropic_PFR_conjecture (hpη : p.η = 1 / 9) :\n    ∃ H : Submodule (ZMod 2) G, ∃ Ω : Type uG, ∃ mΩ : MeasureSpace Ω, ∃ U : Ω → G,\n    IsProbabilityMeasure (ℙ : Measure Ω) ∧ Measurable U ∧\n    IsUniform H U ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 11 * d[p.X₀₁ # p.X₀₂] ∧\n      d[p.X₀₁ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] ∧\n      d[p.X₀₂ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] := by\n…",
      "source_statement_sha256": "3eac56cc576df0aa4872aaffe415b8c087265b296310594dd09cea74b4a5be37",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/EntropyPFR.lean#L54",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/ImprovedPFR.lean",
      "declaration": "entropic_PFR_conjecture_improv",
      "line": 813,
      "title": "Entropic PFR, improved constants (10 and 11/2)",
      "intuition": "The same entropic statement with eta = 1/8, giving sum of distances at most 10 d and each distance at most 11/2 d.",
      "source_statement": "theorem entropic_PFR_conjecture_improv (hpη : p.η = 1 / 8) :\n    ∃ (H : Submodule (ZMod 2) G) (Ω : Type uG) (mΩ : MeasureSpace Ω) (U : Ω → G),\n    IsProbabilityMeasure (ℙ : Measure Ω) ∧ Measurable U ∧\n    IsUniform H U ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 10 * d[p.X₀₁ # p.X₀₂] ∧\n      d[p.X₀₁ # U] ≤ 11/2 * d[p.X₀₁ # p.X₀₂] ∧\n      d[p.X₀₂ # U] ≤ 11/2 * d[p.X₀₁ # p.X₀₂] := by\n…",
      "source_statement_sha256": "842d7323d6a79d77a763af7f42d9dbb80c3481e54fc21e43d0902a67979fc217",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/ImprovedPFR.lean#L813",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/ImprovedPFR.lean",
      "declaration": "PFR_conjecture_improv",
      "line": 953,
      "title": "PFR with exponent 11",
      "intuition": "The improved form of PFR: fewer than 2 * K^11 cosets of a subgroup of size at most |A| cover A.",
      "source_statement": "theorem PFR_conjecture_improv {G : Type*} [AddCommGroup G] [Module (ZMod 2) G]\n    {A : Set G} {K : ℝ} (h₀A : A.Nonempty) (Afin : A.Finite)\n    (hA : Nat.card (A + A) ≤ K * A.ncard) :\n    ∃ (H : Submodule (ZMod 2) G) (c : Set G), c.Finite ∧ (H : Set G).Finite ∧\n      Nat.card c < 2 * K ^ 11 ∧ (H : Set G).ncard ≤ A.ncard ∧ A ⊆ c + H := by\n…",
      "source_statement_sha256": "a47e5a78bc4fa0e00ed1824c85ec769e9fa8492f3e51466ee7526ab616b01373",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/ImprovedPFR.lean#L953",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/WeakPFR.lean",
      "declaration": "weak_PFR_int",
      "line": 1031,
      "title": "Weak PFR over the integers",
      "intuition": "A finite set A in a lattice with |A - A| at most K|A| has a subset of density K^(-17) that lies in an affine subspace of dimension O(log K).",
      "source_statement": "theorem weak_PFR_int\n    {G : Type*} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G]\n    {A : Set G} [A_fin : Finite A] (hnA : A.Nonempty) {K : ℝ}\n    (hA : Nat.card (A - A) ≤ K * Nat.card A) :\n    ∃ A' : Set G, A' ⊆ A ∧ Nat.card A' ≥ K ^ (-17 : ℝ) * Nat.card A ∧\n      AffineSpace.finrank ℤ A' ≤ (40 / log 2) * log K := by\n…",
      "source_statement_sha256": "744464d3ca5fd7d499d99f62b7f7f58fd5b11a19ff8404e62206ba106c1383a8",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/WeakPFR.lean#L1031",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/WeakPFR.lean",
      "declaration": "weak_PFR_int_sumset",
      "line": 1061,
      "title": "Weak PFR from a sumset hypothesis",
      "intuition": "The same weak PFR conclusion when the hypothesis is |A + A| at most K|A|, at the cost of squaring K: density K^(-34) and dimension at most (80 / log 2) log K.",
      "source_statement": "theorem weak_PFR_int_sumset\n    {G : Type*} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G]\n    {A : Set G} (hA : A.Finite) (hnA : A.Nonempty) {K : ℝ}\n    (hAK : Nat.card (A + A) ≤ K * Nat.card A) :\n    ∃ A' ⊆ A, Nat.card A' ≥ K ^ (-34 : ℝ) * Nat.card A ∧\n      AffineSpace.finrank ℤ A' ≤ (80 / log 2) * log K := by\n…",
      "source_statement_sha256": "d697012270b6e3385217b1d2feed91e1e20e50d9a475f54eef0b078295192aa9",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/WeakPFR.lean#L1061",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "pfr",
      "file": "PFR/HomPFR.lean",
      "declaration": "homomorphism_pfr",
      "line": 75,
      "title": "Homomorphism PFR",
      "intuition": "A map between F2-vector spaces that is almost additive differs from a true homomorphism by a set of size at most |S|^10.",
      "source_statement": "theorem homomorphism_pfr (f : G → G') (S : Set G') (hS : ∀ x y : G, f (x+y) - (f x) - (f y) ∈ S) :\n    ∃ (φ : G →+ G') (T : Set G'), Nat.card T ≤ Nat.card S ^ 10 ∧ ∀ x : G, (f x) - (φ x) ∈ T := by\n…",
      "source_statement_sha256": "e3cd46cb517fac824c86699158025929fe1a20a7ee4c457059af709fbf551327",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/PFR/HomPFR.lean#L75",
      "corpus_references": [
        {
          "label": "Pinned project documentation",
          "url": "https://github.com/teorth/pfr/blob/ce7ac95e620afe0ae448df63477de0b29e4dcf4d/README.md"
        },
        {
          "label": "Polynomial Freiman-Ruzsa proof paper, arXiv v2",
          "url": "https://arxiv.org/abs/2311.05762v2"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/PropositionsAndProofs.lean",
      "declaration": "and_commut",
      "line": 81,
      "title": "An axiom: and is commutative, in a toy proof system",
      "intuition": "The book first models proofs as data (Proof p), and an axiom is just a constant of such a type.",
      "source_statement": "axiom and_commut (p q : Prop) : Proof (Implies (And p q) (And q p))\n\nvariable (p q : Prop)\n\n#check and_commut p q     -- message: and_commut p q : Proof (Implies (p ∧ q) (q ∧ p))\n```\n\n\n\nIn addition to axioms, however, we would also need rules to build new\nproofs from old ones. For example, in many proof systems for\npropositional logic, we have the rule of _modus ponens_:\n…",
      "source_statement_sha256": "954237144c3ddfc207baad880a3228c08272859f62cdea90907869b84014e675",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/PropositionsAndProofs.lean#L81",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/PropositionsAndProofs.lean",
      "declaration": "modus_ponens",
      "line": 103,
      "title": "Modus ponens as an inference rule on Proof",
      "intuition": "From a proof of p implies q and a proof of p, the rule produces a proof of q.",
      "source_statement": "axiom modus_ponens (p q : Prop) :\n  Proof (Implies p q) → Proof p →\n  Proof q\n```\n\nSystems of natural deduction for propositional logic also typically rely on the following rule:\n\n> Suppose that, assuming {lean}`p` as a hypothesis, we have a proof of {lean}`q`. Then we can “cancel” the hypothesis and obtain a proof of {lean}`Implies p q`.\n\nWe could render this as follows:\n\n```lean",
      "source_statement_sha256": "2bf69242ce324661769fae8474a03d149d6a5b50a4806964da2e0e6fa3f08062",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/PropositionsAndProofs.lean#L103",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/DependentTypeTheory.lean",
      "declaration": "five",
      "line": 693,
      "title": "Example that fails: a numeral at a type that merely unfolds to Nat",
      "intuition": "def Natural := Nat makes Natural a new name, but numeral literals still need an OfNat Natural 5 instance that does not exist.",
      "source_statement": "def five : Natural := 5\n…",
      "source_statement_sha256": "975c3be0d326e5c0e9521306a3463fd7039f2a0836af0a4b1c16d7e270380011",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/DependentTypeTheory.lean#L693",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/Tactics.lean",
      "declaration": "swap_pair",
      "line": 786,
      "title": "Tactics can define data: swapping a pair",
      "intuition": "The same tactic script that proves p and q implies q and p also builds a function that swaps the two components of a pair.",
      "source_statement": "def swap_pair : α × β → β × α := by\n…",
      "source_statement_sha256": "1d098058ff890e19f4554e2d56fc7fb54e1b0904733e56e7fa4493c2d149be0e",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/Tactics.lean#L786",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/Tactics.lean",
      "declaration": "swap_sum",
      "line": 791,
      "title": "Swapping the two sides of a sum type",
      "intuition": "cases splits on which side of the sum we are in, and each branch rebuilds the value on the opposite side.",
      "source_statement": "def swap_sum : Sum α β → Sum β α := by\n…",
      "source_statement_sha256": "d4a26b02e4782b8c2a9d17cb090e145856f5e7c6ce3e447e067f4fd287cee314",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/Tactics.lean#L791",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/InductiveTypes.lean",
      "declaration": "Weekday.next_previous",
      "line": 364,
      "title": "next undoes previous, by cases on the constructors",
      "intuition": "Weekday has seven constructors, so match gives seven cases, each true by computation.",
      "source_statement": "theorem next_previous (d : Weekday) : next (previous d) = d :=\n…",
      "source_statement_sha256": "9d841e6d3a308c123120d2d95fb75f726a14dea1cfe3fecee5d4ce59d95aad27",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/InductiveTypes.lean#L364",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/InductiveTypes.lean",
      "declaration": "Weekday.Weekday.add_comm",
      "line": 1105,
      "title": "First attempt at commutativity of Nat addition (contains sorry)",
      "intuition": "Induct on the second argument; one equation in the step is not yet proved and is left as sorry.",
      "source_statement": "theorem add_comm (m n : Nat) : m + n = n + m :=\n…",
      "source_statement_sha256": "69b9289201bf272ad4048d2c4908006f2788dfda6971be39c2209788abc1c182",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/InductiveTypes.lean#L1105",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    },
    {
      "corpus": "tpil",
      "file": "book/TPiL/QuantifiersEquality.lean",
      "declaration": "divides_trans",
      "line": 571,
      "title": "Divisibility is transitive, so calc can chain it",
      "intuition": "Compose the two witnesses by multiplying them, which is the witness for the combined divisibility.",
      "source_statement": "def divides_trans (h₁ : divides x y) (h₂ : divides y z) : divides x z :=\n…",
      "source_statement_sha256": "7ee4435347fb8b5149a7e72ffdfe2208406a0af7da47ad5e1e0090db4a18de9d",
      "review_state": "source-linked-unreviewed",
      "exact_source_url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL/QuantifiersEquality.lean#L571",
      "corpus_references": [
        {
          "label": "Pinned Theorem Proving in Lean documentation",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/README.md"
        },
        {
          "label": "Pinned Theorem Proving in Lean 4 book index",
          "url": "https://github.com/leanprover/theorem_proving_in_lean4/blob/4e28129fdd58037f8f5857548d5e99fe4fb0cc57/book/TPiL.lean"
        }
      ]
    }
  ],
  "limitations": "Generated wording is checked against pinned source-controlled text. Curated wording is source-linked and target-checked but remains AI-generated-unreviewed until an identified human reviewer signs it off.",
  "totals": {
    "annotations": 15374,
    "generated_source_matched": 14844,
    "curated_source_linked_unreviewed": 530
  }
}