`ComparatorChallenges/DepthThree.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

25 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.DepthThreeLowerBound.Cube line 9
Also declared: OAI/Computability/DepthThree/Model.lean:9 (same declaration text).
abbrev OAI.DepthThreeLowerBound.Literal line 10
Also declared: OAI/Computability/DepthThree/Model.lean:10 (same declaration text).
def OAI.DepthThreeLowerBound.Literal.eval line 16
Also declared: OAI/Computability/DepthThree/Model.lean:16 (same declaration text).
abbrev OAI.DepthThreeLowerBound.GateInput line 20
Also declared: OAI/Computability/DepthThree/Model.lean:20 (same declaration text).
abbrev OAI.DepthThreeLowerBound.RawClause line 21
Also declared: OAI/Computability/DepthThree/Model.lean:21 (same declaration text).
def OAI.DepthThreeLowerBound.GateInput.eval line 25
Also declared: OAI/Computability/DepthThree/Model.lean:25 (same declaration text).
def OAI.DepthThreeLowerBound.RawClause.eval line 33
Also declared: OAI/Computability/DepthThree/Model.lean:33 (same declaration text).
structure OAI.DepthThreeLowerBound.Circuit3 line 39
Also declared: OAI/Computability/DepthThree/Model.lean:39 (same declaration text).
def OAI.DepthThreeLowerBound.Circuit3.gateCount line 48
Also declared: OAI/Computability/DepthThree/Model.lean:48 (same declaration text).
def OAI.DepthThreeLowerBound.Circuit3.middleEval line 50
Also declared: OAI/Computability/DepthThree/Model.lean:50 (same declaration text).
def OAI.DepthThreeLowerBound.Circuit3.eval line 55
Also declared: OAI/Computability/DepthThree/Model.lean:55 (same declaration text).
def OAI.DepthThreeLowerBound.Circuit3.Computes line 59
Also declared: OAI/Computability/DepthThree/Model.lean:59 (same declaration text).
def OAI.DepthThreeLowerBound.runSteps line 63
Also declared: OAI/Computability/DepthThree/Model.lean:63 (same declaration text).
inductive OAI.DepthThreeLowerBound.HeadMove line 66
Also declared: OAI/Computability/DepthThree/Model.lean:66 (same declaration text).
def OAI.DepthThreeLowerBound.HeadMove.apply line 71
Also declared: OAI/Computability/DepthThree/Model.lean:71 (same declaration text).
structure OAI.DepthThreeLowerBound.MultiTapeCfg line 76
Also declared: OAI/Computability/DepthThree/Model.lean:76 (same declaration text).
abbrev OAI.DepthThreeLowerBound.MultiTapeCode line 80
Also declared: OAI/Computability/DepthThree/Model.lean:80 (same declaration text).
def OAI.DepthThreeLowerBound.multiTapeUpdate line 83
Also declared: OAI/Computability/DepthThree/Model.lean:83 (same declaration text).
def OAI.DepthThreeLowerBound.multiTapeStep line 87
Also declared: OAI/Computability/DepthThree/Model.lean:87 (same declaration text).
structure OAI.DepthThreeLowerBound.FiniteMultiTapeMachine line 92
Also declared: OAI/Computability/DepthThree/Model.lean:92 (same declaration text).
abbrev OAI.DepthThreeLowerBound.FiniteMultiTapeMachine.Cfg line 115
Also declared: OAI/Computability/DepthThree/Model.lean:115 (same declaration text).
abbrev OAI.DepthThreeLowerBound.FiniteMultiTapeMachine.step line 117
Also declared: OAI/Computability/DepthThree/Model.lean:117 (same declaration text).
def OAI.DepthThreeLowerBound.FiniteMultiTapeMachine.init line 120
Also declared: OAI/Computability/DepthThree/Model.lean:120 (same declaration text).
def OAI.DepthThreeLowerBound.MultiTapeHaltsIn line 127
Also declared: OAI/Computability/DepthThree/Model.lean:127 (same declaration text).
theorem OAI.DepthThreeLowerBound.exists_polynomial_time_language_depth_three_lower_bound Challenge statement: Beyond the square-root exponent for depth-three circuits sorry line 133
L4DEL explanation
The formalized result gives one polynomial-time Boolean language whose exact depth-three OR–AND–OR circuit size eventually exceeds $2^{A\sqrt n}$ for every fixed $A>0$. The language and its polynomial-time algorithm are fixed before $A$ is chosen, while circuits may vary with the input length. Gates have unrestricted finite fan-in and sharing, with free input negations and constants. The result does not assert a fixed exponent $n^{1/2+\varepsilon}$ or a lower bound for unrestricted depth.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.Computability.DepthThree.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Computability/DepthThree/Main.lean:8 (same statement text; different proof or body). Qualify references as OAI.DepthThreeLowerBound.exists_polynomial_time_language_depth_three_lower_bound@path.
```
theorem exists_polynomial_time_language_depth_three_lower_bound :
    ∃ (L : List Bool → Bool) (M : FiniteMultiTapeMachine) (C a : ℕ),
      0 < C ∧ 0 < a ∧
      (∀ w : List Bool, MultiTapeHaltsIn M w (L w) (C * (w.length + 1) ^ a)) ∧
      ∀ A : ℝ, 0 < A → ∃ N : ℕ, ∀ n : ℕ, N ≤ n →
        ∀ D : Circuit3 (Fin n), D.Computes (fun x => L (List.ofFn x)) →
          (2 : ℝ) ^ (A * Real.sqrt (n : ℝ)) < (D.gateCount : ℝ) := by
…
```
Mentions: OAI.DepthThreeLowerBound.FiniteMultiTapeMachine, OAI.DepthThreeLowerBound.MultiTapeHaltsIn, OAI.DepthThreeLowerBound.Circuit3