The A115805 to A115803 carryless containment question

OEIS A115805 revision 14 defines the natural numbers k satisfying 13k = A048720(29,k), and its cross-reference conjectures containment in A115803. A115803 revision 13 defines the natural numbers k satisfying 11k = A048720(31,k). Both sequences include zero. The exact question is

$$ \forall k\in\mathbb N,\qquad 13k=C(29,k)\ \Longrightarrow\ 11k=C(31,k). $$

Here C is the binary carryless product of A048720 revision 62. If a_i and b_i are the binary digits of a and b, the t-th binary digit of C(a,b) is the sum of a_i*b_(t-i) over 0 <= i <= t, reduced modulo two. Thus C is coefficient convolution over GF(2). Multiplication outside C is ordinary natural-number multiplication. Every occurrence of k denotes the same natural number, with no positivity or size restriction.

The question compares solution sets for two different pairs of ordinary and carryless multipliers. Antti Karttunen authored the multiplier entries in 2006 and the operation entry in 1999. The OEIS Foundation citation year 2026 is an institutional database citation, not the question's original publication date. The proof is repository-derived and acknowledges both the externally authored question and its operation.

The qualified source scope is A115805 revision 14 and its containment cross-reference, A115803 revision 13, A048720 revision 62, and their inspected OEIS internal pages and revision histories. The source containment is labelled conjectural in that scope. The associated bounded literature qualification is identified by issue 23302. No published resolution was identified within that qualification; this does not establish worldwide priority or rule out private or unindexed proofs. A finite list of sequence terms alone cannot establish the all-natural-number implication.

The shared owner CarrylessMultiplierEquivalence defines genuine binary coefficient convolution and proves its equality to binaryProduct, the XOR recursion. Its holds predicate records an ordinary-multiplication carry and a carryless-output residue; holds_step relates reading one binary digit to the residual input.

The new relation contains 50 pairs of these states. Kernel-checked finite certificates establish initial membership, the implication at residual input zero, and closure under each Boolean input digit, giving 100 pair-digit transition obligations. Nat.evenOddRec then proves the state implication for every natural input. Applying it to the initial pair gives the exact source implication; the finite certificate is an invariant for the universal induction.

A natural number k, including zero, for which ordinary 13k equals C(29,k) but ordinary 11k differs from C(31,k) would refute the question's implication. The operation must use the full GF(2) convolution definition.

The theorem is D5/S1/Digit/Admissibility/CarrylessThirteenElevenContainment.result. Its compiled type quantifies over every natural k with exactly the stated premise and conclusion. Its axiom closure is propext, Classical.choice and Quot.sound. The 50-pair certificate and the Nat.evenOddRec induction are retained in the same Lean module. The shared semantic bridge is D5/S1/Digit/Admissibility/CarrylessMultiplierEquivalence.carryless_eq.

The source-linked Library note uses bibkey oeis2026a115803. Source locators are https://oeis.org/A115805/internal, https://oeis.org/A115803/internal and https://oeis.org/A048720/internal. OEIS source material is licensed CC-BY-SA-4.0; citation-only describes this note's usage, not that license.

The retained typed Reg mirror has audit status declared_unresolved with IE-C050/source.residual_requires_open. The unfinished audit is disclosed at https://github.com/the-omega-institute/trureturing/issues/23302#issuecomment-6480201651. This disclosure does not assert declared_validated or waive kernel, axiom, sorry or frozen-state checks.

Proved: the exact universal containment implication is the result theorem above. Its Scribe carries one Proved binding for this one external question. No converse, classification of either solution set, or other question is claimed. A finite carry characterization remains an unresolved internal candidate and supplies no additional resolution.

What the settlement shows

The source solution set A115805 is contained in A115803 for every natural input, by the result theorem. No dependent theorem or separate follow-up question relying on this conjecture was identified in the retained source scope. Questions beyond this implication remain outside the result.

First-publication priority and exhaustive absence of an earlier proof are unverified because the literature qualification is bounded. The source to Lean interpretation is documentary and independently reviewed; the kernel checks the stated convolution and implication. The Reg audit remains unfinished at the linked issue and is not reported as validated.