oeis_267581_conjecture_0.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

13 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

FormalConjectures.Util.ProblemImports

def ca_rule_167 The rule function for Rule 167 line 43 source
Source documentation
The rule function for Rule 167. Inputs must be 0 or 1.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: OfNat.ofNat, HAdd.hAdd, HMul.hMul, HMod.hMod, HDiv.hDiv, HPow.hPow
12 instances, coercions and auxiliary constants left out.
```
def ca_rule_167 (c_L c_C c_R : ℕ) : ℕ :=
…
```
def ca_state The state of the Rule 167 elementary cellular automaton at time $t$ and position $x$ line 54 source
Source documentation
The state of the Rule 167 elementary cellular automaton at time $t$ and position $x$. The initial condition is a single ON cell at $x=0$. $C(t, x)$ is structurally recursive on $t$.

standard axioms: propext what this means
Statement uses: Nat, Int
Proof uses: ca_state.match_1, Unit, OfNat.ofNat, ite, Eq, Nat.succ, ca_rule_167, HSub.hSub, HAdd.hAdd
9 instances, coercions and auxiliary constants left out.
```
def ca_state (t : ℕ) (x : ℤ) : ℕ :=
…
```
def middle_column_bit The sequence of bits forming the middle column of the CA pattern, $C_{t, 0}$ line 62 source
Source documentation
The sequence of bits forming the middle column of the CA pattern, $C_{t, 0}$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: ca_state, OfNat.ofNat, Int
1 instances, coercions and auxiliary constants left out.
```
def middle_column_bit (t : ℕ) : ℕ := ca_state t 0
```
Mentions: ca_state
def a A267581: Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell line 71 source
Source documentation
A267581: Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell. The term $a(n)$ is the decimal value of the binary number $C_{0, 0} C_{1, 0} \dots C_{n, 0}$, where $C_{i, 0}$ is the state of the center cell at time $i$. $$a(n) = \sum_{k=0}^n C_{k, 0} \cdot 2^{n-k}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, middle_column_bit, HPow.hPow, HSub.hSub
11 instances, coercions and auxiliary constants left out.

Distinct definitions of this name: APNOutputs/OEIS/A309132_conjecture_carmichael.lean:45 (same statement text; different proof or body), APNOutputs/OEIS/a091669_conjecture_primitive_root.lean:46 (same statement text; different proof or body), APNOutputs/OEIS/a325046_odd_terms_at_k_times_k_plus_1.lean:52 (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 (same statement text; different proof or body), APNOutputs/OEIS/oeis_227582_conjecture_0.lean:60 (same statement text; different proof or body), APNOutputs/OEIS/oeis_228143_conjecture_1.lean:54 (same statement text; different proof or body), 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_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 (same statement text; different proof or body), 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 (same statement text; different proof or body), APNOutputs/OEIS/oeis_372761_conjecture_2.lean:83 (same statement text; different proof or body), APNOutputs/OEIS/oeis_62567_conjecture_0.lean:49 (same statement text; different proof or body), 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 (same statement text; different proof or body), 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.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
def oeis_floor_term The floor term in the conjectured recurrence relation for A267581 line 78 source
Source documentation
The floor term in the conjectured recurrence relation for A267581. This term, $\lfloor (1/2)^{(2^{n+1} \bmod n)} \rfloor$, simplifies to 1 if $(2^{n+1} \bmod n) = 0$ (i.e., $n \mid 2^{n+1}$), and 0 otherwise. Since the recurrence is only stated for $n \ge 2$, the $n=0$ case is irrelevant to the conjecture.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HMod.hMod, HPow.hPow, HAdd.hAdd
9 instances, coercions and auxiliary constants left out.
```
def oeis_floor_term (n : ℕ) : ℕ :=
…
```
def int_choose line 117 source
no axioms what this means
Statement uses: Nat, Int
Proof uses: ite, And, LE.le, OfNat.ofNat, Nat.choose, Int.toNat
7 instances, coercions and auxiliary constants left out.
def expected_state line 120 source
no axioms what this means
Statement uses: Nat, Int
Proof uses: ite, Eq, HMod.hMod, HAdd.hAdd, OfNat.ofNat, HSub.hSub, int_choose, HDiv.hDiv
15 instances, coercions and auxiliary constants left out.
lemma int_choose_pascal_mod2 line 125 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, Eq, HMod.hMod, HAdd.hAdd, int_choose, OfNat.ofNat
Proof uses: Eq.mpr, ite, And, LE.le, Nat.choose, Int.toNat, LT.lt, id, congr, congrArg, Eq.trans, ite_congr, Eq.refl, Not, Nat.add_mod_mod, Nat.mod_add_mod, Nat.cast_add, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat and 35 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
lemma ca_rule_167_cases line 138 source
standard axioms: propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, ca_rule_167, ite, And
Proof uses: ca_rule_167_cases.match_1, ca_rule_167_cases.match_2, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma ca_state_eq_expected line 149 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, LE.le, OfNat.ofNat, Eq, ca_state, expected_state
Proof uses: Nat.recAux, Decidable.byContradiction, Not, HAdd.hAdd, LT.lt, Or, eq_or_lt_of_le, Iff.mp, Nat.succ, Nat.succ_le_succ_iff, Eq.mpr, ca_rule_167, ite, HSub.hSub, Dvd.dvd, HMod.hMod, int_choose, HDiv.hDiv, id, congrArg and 119 more
195 instances, coercions and auxiliary constants left out.

Tactic trace: 66 steps
Open to load the checked before-and-after goal states.
lemma expected_state_zero_eq line 183 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, expected_state, Int, HSub.hSub, oeis_floor_term
Proof uses: Eq.mpr, ite, Dvd.dvd, HMod.hMod, int_choose, HDiv.hDiv, HPow.hPow, HAdd.hAdd, id, congrArg, ite_congr, Eq.trans, zero_add, Not, Eq.refl, Even, And, Nat.choose, Int.toNat, Exists and 113 more
133 instances, coercions and auxiliary constants left out.

Tactic trace: 47 steps
Open to load the checked before-and-after goal states.
lemma a_rec line 205 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, a, HAdd.hAdd, HMul.hMul, HSub.hSub, middle_column_bit
Proof uses: id, Eq.trans, Finset.sum, Finset.range, HPow.hPow, Finset.sum_range_succ, a_rec.match_1, of_eq_true, True, congr, congrArg, Finset.sum_congr, Eq.refl, Finset, Membership.mem, Nat.succ, Nat.succ_sub, eq_true, Nat.pow_succ', Nat.Simproc.add_sub_add_le and 10 more
41 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 212 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, a, HSub.hSub, HAdd.hAdd, HMul.hMul, oeis_floor_term
Proof uses: Decidable.byContradiction, Not, middle_column_bit, a_rec, expected_state, Int, id, Eq.mpr, ca_state, congrArg, ca_state_eq_expected, Eq.refl, expected_state_zero_eq, Decidable, ite, HMod.hMod, HPow.hPow, if_neg, Nat.zero_le, if_pos and 1 more
27 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/a325046_odd_terms_at_k_times_k_plus_1.lean:187 (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_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: 19 steps
Open to load the checked before-and-after goal states.