Coefficients of the A396797 three-four iterate product series modulo six

OEIS A396797, Paul D. Hanna, Jun 15 2026. The following NAME and COMMENT quotes are supplied by the orchestrator and recorded in Library/Arith/hanna2026a396797.md:

NAME:

G.f. satisfies A(x) = x + A^3(x)*A^4(x).

Here A^n(x) denotes the n-th compositional iteration, not an ordinary power.

COMMENT:

Conjecture: a(n) = 1 (mod 6) for n >= 1.

范围与诚实边界

The resolution concerns the first conjecture for every positive index of the unique zero-constant integer formal power series satisfying the NAME equation. The second comment, asserting [x^n] A^k(x) ≡ k^(n−1) (mod 6) for all k, is not resolved here. Existence and uniqueness identify the constructed series with the defining equation; no coefficient congruence is assumed.

This is a first-tier recent OEIS conjecture, selected in the lane brief under the recent-small-conjecture route. KPI = open problems resolved. The target is the universal positive-index assertion, not a bounded numerical check.

The supplied search-seat report found no proof in its searched scope: the OEIS entry and revision history were read on 2026-09-08, and identifier searches were made on arXiv, MathOverflow, and GitHub. This Stage-B seat had no network access and did not repeat those searches. This is not an exhaustive literature search or a first-publication priority claim.

Stabilized successive approximations construct the integer coefficient sequence a and generatingSeries; generating_equation proves its zero constant term and defining functional equation. Degree induction gives generating_unique. The private product_contract shows that a product of two zero-constant compositional iterates gains one degree of agreement, for arbitrary iterate counts, supplying both stabilization and uniqueness.

Over ZMod 6, mod_six_fixed proves that x/(1-x) = mobius 1 is a fixed point of the three-four transformation. The frozen mobius_iterate formula reduces the iterates to mobius 3 and mobius 4; the identity (1-3X)(1-4X) = 1-X in ZMod 6 and cancellation of the unit 1-X give the formal-series equality. hanna_conjecture maps the proved integer equation to ZMod 6, applies the private uniqueness theorem over that ring, and reads off every positive-degree coefficient of mobius 1.

The generality-I module binds the frozen prerequisite D5/S1/Recurrence/Invariants/CompositionalIterateCongruence, reusing iterate, mobius, and mobius_iterate. Its freeze prerequisite statement_id is sha256:4063c4732963765b6b16b5dda51775b4d179cfc3a203c91ed15e3ebffb0467e7, read from Golden/Frozen/state/D5/S1/Recurrence/Invariants/CompositionalIterateCongruence.lean.json. The prerequisite's five-six fixed-point result does not substitute for the three-four construction and uniqueness argument.

An index n ≥ 1 whose coefficient in the unique zero-constant integer solution satisfies a(n) % 6 ≠ 1 would contradict the assertion. The orchestrator's exact check is supporting evidence only; no finite check proves the unbounded statement, and this seat did not rerun that check.

• Lean module: D5/S1/Recurrence/Invariants/ThreeFourIterateProductModSix.lean.
• Main theorem: hanna_conjecture.
• Companions: generating_equation, generating_unique, mod_six_fixed.
• Public definitions: a, generatingSeries.
• Axioms: std3, exactly propext, Classical.choice, Quot.sound for each public theorem, as reported by the implementation seat and recorded in its compiler log /tmp/threefour-iterate-mod-six.lean.log, read by Stage-B.
• The Scribe resolution claim for this slug is attached only to the Describe node of hanna_conjecture.

theorem. The formal proof closes the first OEIS conjecture for all positive indices. The second comment is outside the resolution claim's scope.

The OEIS quotes, authorship, and date were supplied by the orchestrator. The search seat, not this Stage-B seat, read the OEIS revision history on 2026-09-08. The reported literature search covered that entry and history plus identifier searches on arXiv, MathOverflow, and GitHub; it did not establish exhaustive coverage of the literature or private indexes. Stage-B had no network access. First-publication priority and the source-to-Lean identification are not kernel-checked facts. Stage-B did not rerun Lean, C# compilation, emission, freezing, or admission checks; those operations belong to the orchestrator's verification and handoff, not to the dossier's evidence of local execution.