a325046_odd_terms_at_k_times_k_plus_1.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 348 curated AI-written annotations and 373 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 12 manuscript-grounded curated result annotations were composed from pinned companion PDFs and named source records 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. Twelve supporting declarations for the Hilbert-functions result were upgraded on 2026-10-14 from mechanically condensed doc comments to declaration-specific explanations bound to exact page-one terminology in the pinned human-written companion. The other Hilbert challenge statements were deliberately not described as companion-backed results because that manuscript proves P5 specifically. Fifteen previously unannotated declarations along the Hilbert P5 proof spine were annotated on 2026-10-14 from the pinned companion's seven-page argument, covering the two-generator decomposition, finite-difference bounds, critical-case reduction, and final log-concavity step. Each quotation is bound to its exact manuscript page and Lean declaration; no human review is claimed. Twenty-four further declarations along the Hilbert P5 box-count and finite-difference spine were annotated on 2026-10-14, covering the two-dimensional slice count, three-dimensional box formula, first- and second-difference identities, the seven-threshold curvature expression, boundary handling, pointwise transport, sum linearity, and lower-or-upper region classification. Every annotation is bound to an exact quotation on pages 2 or 3 of the pinned human-written companion; Lean-specific boundary and threshold helpers are identified as implementation lemmas rather than separate manuscript claims. No human review is claimed. Twenty-eight further declarations along the Hilbert P5 argument were annotated on 2026-10-14 from pages 2 through 6 of the pinned human-written companion. They expose the shifted residual sequence, exact lower- and upper-region formulas, symmetry by complementary divisors, the curvature-one and curvature-two splits, and the final algebraic reductions. Lean-only transport and integer-arithmetic helpers are labeled as proof infrastructure rather than new manuscript claims; each quotation is page-bound in the source ledger, and no human review is claimed. Twenty high-level declarations completing the Hilbert P5 curvature-one and curvature-two case spine were annotated on 2026-10-14 from pages 3 through 7 of the pinned human-written companion. The explanations separate cases A and B, lower and upper monomial regions, shifted residual evaluation, impossible mixed regions, and the surviving upper-upper estimate; wrapper lemmas are explicitly described as proof structure. Every quotation is page-bound, and no human review is claimed. Twenty-two graph-reconstruction declarations were upgraded or added on 2026-10-14 from the pinned four-page human-written companion, covering the incidence-deletion deck, edge and degree recovery, unique isolated vertices, vertex-type reconstruction, neighboring-type descent, adjacency recovery, and final bipartite isomorphism. The result annotation also records that `G_not_regular` is present in the Lean statement but unused by its proof term and absent from the companion theorem. Exact historical quotations from Paul J. Kelly (1957) and Bondy with Hemminger (1977) are linked separately; no human review is claimed. Eighteen declarations along the Erdős 741(i) strict-natural-density counterexample were added on 2026-10-14, covering the carry-free base-four digit split, full cross-sum coverage, sparse pure self-sums, fixed-factor fat intervals, the lower-scale density deficit, upper-scale sum growth, and the final convergence contradiction. The result annotation was corrected to distinguish the negative strict-natural-density Lean theorem from the compatible affirmative upper-density companion theorem; Ulam AI's written counterexample is explicitly identified as using a different interval length with the same mechanism. The quotations are page- and hash-bound where the PDFs expose text; Paul Erdős's scanned 1994 question was visually checked and is linked separately without a machine-audit claim. No human review is claimed. Twenty-eight additional declaration-level annotations were added on 2026-10-14 for the Erdős 12(i) proof, and its IsGood definition was upgraded from a source-comment condensation. They cover the prime and Chinese-remainder moduli, the actual Lean congruence-block witness, exclusion of cross-block and same-block triples, lower and upper counting bounds, and the positive liminf. The prior result annotation was corrected because the pinned human-written companion proves the same theorem with a different base-2-to-base-3 construction; it is no longer presented as the Lean proof. The original property-P definition from Erdős and Sárközy (1970) is separately linked and page-bound; no human review is claimed. The remaining fifty declarations in the Erdős 12(i) file were annotated on 2026-10-14, completing declaration-level coverage of all eighty definitions, lemmas, and theorems in that file. These additions expose the prime-recursion invariants, divisibility transport, Chinese-remainder witness bounds, numerical density envelopes, injective block coding, and liminf support lemmas instead of leaving the proof spine as raw Lean; no human review is claimed. Eighteen supporting declarations for Erdős 125 were added on 2026-10-14, completing all nineteen declarations in the file. The annotations follow the pinned companion through digit-block decomposition, irrational logarithmic approximation, the explicit Lean contraction factor 11/12, geometric iteration, and the lower-density conclusion. The original 1996 question of Burr, Erdős, Graham, and Li and the 2026 discussion contributions of GTSoukalas, Terence Tao, and Giuseppe Melfi are linked separately; no human review is claimed. Twenty-eight supporting declarations for Erdős 138 were upgraded or added on 2026-10-14, completing all twenty-nine declarations in the file. The annotations separate the greedy difference proof from the Lean-specific representation bridges and ultrafilter compactness proof of existence, and bind quotations from the pinned companion, the AlphaProof Nexus paper, Erdős's 1981 source, van der Waerden's recollection, and the 2026 Erdős Problems discussion. No human review is claimed. The curated Erdős 12(ii), 138, and 152 annotations were further linked on 2026-10-14 to exact quotations from Felix Behrend, Bartel van der Waerden, and Paul Erdős with Pál Turán; the historical quotation sources are labeled separately from the pinned AlphaProof proof companions. The curated Hilbert-functions result, its OrderIdeal and pureOSequence definitions, and the Erdős 846 annotation were further linked on 2026-10-14 to exact quotations from Richard P. Stanley and from Moe Putterman, Mehtaab Sawhney, and Gregory Valiant. The Stanley PDF is hash- and page-bound in the source ledger; these historical sources are labeled separately from the pinned AlphaProof proof companions. Three additional target-specific annotations for Ben Green's Problem 57, exact last-iterate anchored-GDA convergence, and Graffiti.pc Conjecture 2 were composed on 2026-10-14 from the exact pinned Lean statements, quotations from the AlphaProof Nexus paper, and quotations from Green, Ryu with Yuan and Yin, Surina et al., and Wang with Dai and Tong; no human review is claimed. Two additional target-specific quantum-optics annotations for the explicit AME(11,5) construction and the even N >= 4, D = N monochromatic-quantum-graph theorem were composed on 2026-10-14 from the exact pinned Lean statements, page-bound quotations from the AlphaProof Nexus paper, Helwig et al., and Krenn, Gu, and Zeilinger, and Mario Krenn's public priority correction. The AME annotation makes no first-discovery claim; the monochromatic annotation states that the local AlphaProof wrapper contains sorry and links the separate pinned proof-bearing Formal Conjectures artifact without silently treating it as local verification. Twelve declarations that the pinned source repeats verbatim in several files were annotated in only one of the copies, so the build rejected them as differing copies. On 2026-10-16 each curated annotation was copied unchanged to every identical copy: the eleven Hilbert-functions definitions from hilbert_functions_5 to hilbert_functions_1, _2, _3, _4, _7 and _8, and Erdos12.IsGood from erdos_12.parts.i to erdos_12.parts.ii (the better-sourced annotation was kept in each case). Each copy's quotations are bound in the source ledger like the original's; no new text was written and no human review is claimed. The remaining source-derived annotations were generated mechanically from pinned 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: Google DeepMind published the AlphaProof Nexus proof artifacts. The pinned repository identifies the mathematical sources and collaborators, including OEIS, Erdős Problems, Gergely Bérczi, and Mario Krenn; those source records remain the authority for individual credit.

Project sources: Pinned AlphaProof Nexus results documentation, AlphaProof Nexus paper, Pinned human-written natural-language proof companions

14 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

FormalConjectures.Util.ProblemImports

def a A325046: G.f.: $\sum_{n \ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 - x^{n+1})^{n+1}}$ line 52 source
Source documentation
A325046: G.f.: $\sum_{n \ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 - x^{n+1})^{n+1}}$.
The term $a(N)$ is the coefficient of $x^N$ in the generating function. Expanding the terms, we get a formula for $a(N)$: $$a(N) = \sum_{n=0}^N \sum_{k=0}^n \mathbf{1}_{n + nk + (n+1)j = N} \binom{n}{k} \binom{n+j}{j}$$ where $j = \frac{N - n(k+1)}{n+1}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HSub.hSub, HMul.hMul, ite, And, LE.le, Eq, HMod.hMod, HDiv.hDiv, Nat.choose
16 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/A309132_conjecture_carmichael.lean:45 (different declaration text), APNOutputs/OEIS/a091669_conjecture_primitive_root.lean:46 (different declaration text), APNOutputs/OEIS/oeis_103311_conjecture_0.lean:47 (different declaration text), APNOutputs/OEIS/oeis_108_conjecture_2.lean:45 (different declaration text), APNOutputs/OEIS/oeis_113254_conjecture_0.lean:48 (different declaration text), APNOutputs/OEIS/oeis_175386_conjecture_0.lean:46 (different declaration text), APNOutputs/OEIS/oeis_194806_conjecture_0.lean:50 (different declaration text), APNOutputs/OEIS/oeis_227582_conjecture_0.lean:60 (different declaration text), APNOutputs/OEIS/oeis_228143_conjecture_1.lean:54 (different declaration text), APNOutputs/OEIS/oeis_243106_conjecture_0.lean:47 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_0.lean:43 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_4.lean:43 (different declaration text), APNOutputs/OEIS/oeis_267581_conjecture_0.lean:71 (different declaration text), APNOutputs/OEIS/oeis_271591_conjecture_0.lean:57 (different declaration text), APNOutputs/OEIS/oeis_2897_conjecture_0.lean:46 (different declaration text), APNOutputs/OEIS/oeis_306424_conjecture_0.lean:52 (different declaration text), APNOutputs/OEIS/oeis_307865_conjecture_0.lean:46 (different declaration text), APNOutputs/OEIS/oeis_323557_conjecture_0.lean:48 (different declaration text), APNOutputs/OEIS/oeis_341254_conjecture_0.lean:52 (different declaration text), APNOutputs/OEIS/oeis_372761_conjecture_2.lean:83 (different declaration text), APNOutputs/OEIS/oeis_62567_conjecture_0.lean:49 (different declaration text), APNOutputs/OEIS/oeis_A028859_conjecture_1.lean:43 (different declaration text), APNOutputs/OEIS/oeis_a211417_conjecture_specific.lean:44 (different declaration text), APNOutputs/OEIS/oeis_a237271_conjecture_2.lean:49 (different declaration text), APNOutputs/OEIS/oeis_a300997_finite_difference_is_one_or_two.lean:46 (different declaration text), APNOutputs/OEIS/oeis_a363102_conjecture_1.lean:56 (different declaration text), APNOutputs/OEIS/oeis_a368692_conjecture_integrality.lean:47 (different declaration text). Qualify references as a@path.
```
def a (N : ℕ) : ℕ :=
…
```
lemma sum_involution_zmod2 line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Finset, ZMod, OfNat.ofNat, Nat, Membership.mem, Eq, Finset.sum, Finset.filter
Proof uses: HAdd.hAdd, Ne, Eq.mpr, id, congrArg, Not, AddCommMonoid, Finset.sum_filter_add_sum_filter_not, Eq.refl, Finset.sum_involution, sum_involution_zmod2.match_1, Eq.rec, HMul.hMul, MulZeroClass.zero_mul, two_mul, Eq.symm, Finset.filter_subset, of_eq_true, Eq.trans, True and 11 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
def V line 120 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Prod
Proof uses: Finset.filter, And, LE.le, Prod.snd, Prod.fst, HMul.hMul, HAdd.hAdd, OfNat.ofNat, Eq, HMod.hMod, HSub.hSub, SProd.sprod, Finset.Iic
18 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/oeis_62567_conjecture_0.lean:246 (different declaration text). Qualify references as V@path.
def I_map line 124 source
no axioms what this means
Statement uses: Nat, Prod
Proof uses: Prod.fst, Prod.snd, HDiv.hDiv, HSub.hSub, HMul.hMul, HAdd.hAdd, OfNat.ofNat, Prod.mk
9 instances, coercions and auxiliary constants left out.
def T line 130 source
no axioms what this means
Statement uses: Nat, Prod
Proof uses: Prod.fst, Prod.snd, HDiv.hDiv, HSub.hSub, HMul.hMul, HAdd.hAdd, OfNat.ofNat, Nat.choose
9 instances, coercions and auxiliary constants left out.
lemma a_eq_sum_T line 136 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, a, Finset.sum, Prod, V, T
Proof uses: id, Finset.range, HAdd.hAdd, OfNat.ofNat, HSub.hSub, HMul.hMul, ite, And, LE.le, HMod.hMod, HDiv.hDiv, Nat.choose, Finset.filter, Prod.snd, Prod.fst, SProd.sprod, Finset, Finset.Iic, Eq.rec, Finset.Ico and 39 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma I_mem_V line 141 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Prod, Membership.mem, Finset, V, I_map
Proof uses: id, Eq.mpr, Finset.filter, And, LE.le, Prod.snd, Prod.fst, HMul.hMul, HAdd.hAdd, OfNat.ofNat, Eq, HMod.hMod, HSub.hSub, SProd.sprod, Finset.Iic, HDiv.hDiv, Prod.mk, Eq.trans, True, congr and 93 more
134 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma I_I_eq line 147 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Prod, Membership.mem, Finset, V, Eq, I_map
Proof uses: id, Eq.mpr, Prod.mk, HAdd.hAdd, Prod.snd, HDiv.hDiv, HSub.hSub, HMul.hMul, Prod.fst, OfNat.ofNat, congrArg, Nat.div_eq_of_eq_mul_left, Nat.zero_lt_succ, Nat.sub_eq_of_eq_add, Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt, Not.intro, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp and 82 more
117 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma T_I_eq line 152 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Prod, Membership.mem, Finset, V, Eq, T, I_map
Proof uses: id, HMul.hMul, Nat.choose, Prod.fst, Prod.snd, HAdd.hAdd, HDiv.hDiv, HSub.hSub, OfNat.ofNat, Exists, Nat.exists_eq_add_of_le, And.left, LE.le, And, HMod.hMod, And.right, Eq.mp, Eq.trans, Finset.filter, SProd.sprod and 89 more
129 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma T_even_step1 line 164 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, Eq, HMul.hMul, Nat.choose, HSub.hSub, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, HAdd.hAdd, True, congr, congrArg, two_mul, Nat.add_sub_assoc, mul_comm, Nat.choose_mul, Nat.add_sub_cancel, Nat.choose_symm, eq_self
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma T_even_step2 line 168 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, HMod.hMod, Nat.choose, HMul.hMul
Proof uses: Eq.mpr, HSub.hSub, id, congrArg, Eq.symm, Nat.mul_mod_right, Nat.choose_mul_right, LT.lt.ne', Eq.refl
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma T_even_of_lt line 171 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, Eq, HMod.hMod, HMul.hMul, Nat.choose, HSub.hSub, OfNat.ofNat
Proof uses: Eq.mpr, Even, id, congrArg, Eq.symm, propext, Nat.even_iff, HAdd.hAdd, two_mul, Nat.add_sub_assoc, le_of_lt, Finset.sum, Prod, Finset.HasAntidiagonal.antidiagonal, Prod.fst, Prod.snd, Nat.add_choose_eq, mul_comm, Finset.range, Eq.trans and 49 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma fixed_point_imp line 180 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Prod, Membership.mem, Finset, V, Eq, I_map, Prod.fst, Prod.snd, HMul.hMul, HAdd.hAdd, OfNat.ofNat
Proof uses: Eq.mpr, id, congrArg, Eq.trans, mul_add, mul_one, LT.lt, HSub.hSub, Eq.rec, HMod.hMod, Nat.mod_eq_of_lt, And.right, LE.le, And, Eq.mp, Finset.filter, SProd.sprod, Finset.Iic, Finset.filter.congr_simp, funext and 24 more
60 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem target_theorem_0 line 187 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, HMod.hMod, a, OfNat.ofNat, Exists, HMul.hMul, HAdd.hAdd
Proof uses: ZMod, Eq.rec, rfl, ZMod.natCast_mod, Finset.sum, Prod, V, T, Eq.mpr, Finset.filter, And, LE.le, Prod.snd, Prod.fst, HSub.hSub, SProd.sprod, Finset, Finset.Iic, Nat.choose, HDiv.hDiv and 77 more
104 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/A224515_conjecture_existence.lean:269 (different declaration text), APNOutputs/OEIS/A309132_conjecture_carmichael.lean:375 (different declaration text), APNOutputs/OEIS/A382590_conjecture_kth_prime_factor_is_eventually_periodic.lean:303 (different declaration text), APNOutputs/OEIS/a091669_conjecture_primitive_root.lean:275 (different declaration text), APNOutputs/OEIS/oeis_103311_conjecture_0.lean:240 (different declaration text), APNOutputs/OEIS/oeis_108_conjecture_2.lean:281 (different declaration text), APNOutputs/OEIS/oeis_113254_conjecture_0.lean:153 (different declaration text), APNOutputs/OEIS/oeis_175386_conjecture_0.lean:326 (different declaration text), APNOutputs/OEIS/oeis_194806_conjecture_0.lean:495 (different declaration text), APNOutputs/OEIS/oeis_227582_conjecture_0.lean:306 (different declaration text), APNOutputs/OEIS/oeis_228143_conjecture_1.lean:730 (different declaration text), APNOutputs/OEIS/oeis_243106_conjecture_0.lean:164 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_0.lean:223 (different declaration text), APNOutputs/OEIS/oeis_248802_conjecture_4.lean:841 (different declaration text), APNOutputs/OEIS/oeis_256012_conjecture_0.lean:116 (different declaration text), APNOutputs/OEIS/oeis_267581_conjecture_0.lean:212 (different declaration text), APNOutputs/OEIS/oeis_271591_conjecture_0.lean:518 (different declaration text), APNOutputs/OEIS/oeis_278070_conjecture_0.lean:91 (different declaration text), APNOutputs/OEIS/oeis_282779_conjecture_0.lean:141 (different declaration text), APNOutputs/OEIS/oeis_289411_conjecture_0.lean:189 (different declaration text), APNOutputs/OEIS/oeis_2897_conjecture_0.lean:427 (different declaration text), APNOutputs/OEIS/oeis_306424_conjecture_0.lean:300 (different declaration text), APNOutputs/OEIS/oeis_307865_conjecture_0.lean:212 (different declaration text), APNOutputs/OEIS/oeis_323557_conjecture_0.lean:214 (different declaration text), APNOutputs/OEIS/oeis_340737_conjecture_0.lean:456 (different declaration text), APNOutputs/OEIS/oeis_341254_conjecture_0.lean:136 (different declaration text), APNOutputs/OEIS/oeis_363347_conjecture_2.lean:848 (different declaration text), APNOutputs/OEIS/oeis_372761_conjecture_2.lean:756 (different declaration text), APNOutputs/OEIS/oeis_51293_conjecture_0.lean:517 (different declaration text), APNOutputs/OEIS/oeis_62567_conjecture_0.lean:329 (different declaration text), APNOutputs/OEIS/oeis_A028859_conjecture_1.lean:432 (different declaration text), APNOutputs/OEIS/oeis_A258667_conjecture_0.lean:446 (different declaration text), APNOutputs/OEIS/oeis_a211417_conjecture_specific.lean:264 (different declaration text), APNOutputs/OEIS/oeis_a237271_conjecture_2.lean:124 (different declaration text), APNOutputs/OEIS/oeis_a300997_finite_difference_is_one_or_two.lean:1111 (different declaration text), APNOutputs/OEIS/oeis_a363102_conjecture_1.lean:302 (different declaration text), APNOutputs/OEIS/oeis_a368692_conjecture_integrality.lean:182 (different declaration text). Qualify references as target_theorem_0@path.

Tactic trace: 50 steps
Open to load the checked before-and-after goal states.