slug: oeis-a373312-unit-reversion-square-parity bibkey: hanna2024a373312 doi: null url: https://oeis.org/A373312 triage: theorem motivation_gids:

• D5/S1/Recurrence/Invariants/UnitReversionSquareParity

Square parity of the unit-reversion series A373312

Problem

OEIS A373312, Paul D. Hanna, Jun 25 2024. The following NAME and COMMENT are copied verbatim from the claim in Library/Arith/hanna2024a373312.md.

NAME:

Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^2 ).

COMMENT:

Conjecture: a(n) == 1 (mod 2) iff n = 2^k - 1 for k >= 1.

The formalization constructs the integer solution normalized by A(0)=0 and a(1)=1 and proves the parity assertion for every positive index.

Motivation

This is a first-tier recent OEIS conjecture. The KPI is open problems resolved, and the target is the unbounded parity assertion.

Gap

The supplied search report found no proof in the searched scope: the search seat read the OEIS entry and revision history on 2026-09-08 and searched arXiv, MathOverflow, and GitHub by identifier. This Stage-B seat has no network access and did not independently repeat that search. The missing argument is the passage from the normalized integer generating equation to the exact mod-two coefficient support.

Route

Write A = X*B. Stabilized iterates of a contracting equation for B construct the integer series without coefficient division. generating_equation proves the equation with the unit-denominator innerSeries, and generating_unique proves uniqueness by degree induction. Over ZMod 2, Frobenius, Mathlib compositional-inverse cancellation (subst_substInvOfIsUnit_left), and clearing the unit denominator give F = X + X*F(X^2) in mod_two_fixed. The private lacunary_coeff identifies its support with the Mersenne indices 2^k - 1, k >= 1, by strong induction, giving hanna_conjecture. The module has only Mathlib imports, no D5 import, and generality G.

Falsifier

A positive index n for which the normalized integer coefficient a(n) is odd but n is not 2^k - 1 for any k >= 1, or is even at such an index, would contradict the assertion. The orchestrator’s exact check is supporting evidence only; a finite check does not establish the universal assertion.

Evidence

• Lean module: D5/S1/Recurrence/Invariants/UnitReversionSquareParity.lean.
• Main public theorem: hanna_conjecture.
• Companions: generating_equation, generating_unique, mod_two_fixed.
• Axioms: std3 (propext, Classical.choice, Quot.sound), exactly as reported for all four public theorems by the implementation seat.
• The orchestrator reported make lean, make lean-report, and make emit verification before the Stage-B edits; this seat did not rerun those gates.

Triage

theorem. The formal proof closes the universal parity assertion for the normalized integer solution recorded by OEIS.

ASSUMED-UNVERIFIED

The quotations, attribution, and date were supplied by the orchestrator and recorded in the Library note. The OEIS revision history was read by the search seat, not by this seat. The literature search scope was the OEIS entry and revision history read on 2026-09-08 plus identifier searches on arXiv, MathOverflow, and GitHub; it was not an exhaustive search of MathSciNet, zbMATH, or all third-party Lean libraries. This seat has no network access. First-publication priority and the source-to-Lean identification are not kernel-checked facts. The implementation seat’s axiom report and the orchestrator’s gate and exact-check results are attributed evidence, not independent Stage-B measurements.