Signed Catalan cubic substitution modulo three

OEIS A386666 (Paul D. Hanna, Sep 07 2025) states:

G.f. A(x) satisfies: A(x)^2 = A( x^2 + 4*A(x)^3 ).

Conjecture: a(3n) (mod 3) == 2a(3n+1) (mod 3) == a(3n+2) (mod 3) == 2*A039966(n) for n > 0, where A039966(n) equals the number of partitions of n into distinct powers of 3.

The conjecture is formalized with distinctPowersOfThree n standing for A039966(n).

This is a first-tier recent OEIS conjecture. KPI: open problems resolved.

No proof was found in the OEIS entry or revision history, or by identifier searches on arXiv, MathOverflow, and GitHub, as reported by the search seat on 2026-09-08. This seat has no network access.

generatingSeries is constructed directly over ℤ by even residual corrections and compositional inversion. generating_equation proves A(0) = 0, a(1) = 1, and A² = A(X² + 4A³), with integrality requiring no extra hypothesis. generating_unique proves uniqueness among normalized solutions by the private agreement/divisibility equivalence. Over ZMod 3, signedCatalanSeries satisfies S + S² = X and the reduced cubic equation. The ternary digit-support series has the self-similarity D = (1 + X) · D(X³); Frobenius and unit cancellation give (1 + X) · D² = 1, yielding the three coefficient residues in signed_catalan_mod_three. First-difference uniqueness of normalized compositional inverses identifies A modulo 3 with S, and hanna_conjecture reads off the stated residues for p = distinctPowersOfThree, for n > 0. There is no D5 import; generality is G.

A counterexample index n > 0 at which any of the three asserted congruences fails would falsify the conjecture. The orchestrator's exact finite check is supporting evidence only.

• Lean module: D5/S1/Recurrence/Invariants/SignedCatalanCubicSubstitutionModThree.lean.
• Main theorem: hanna_conjecture.
• Companion theorems: generating_equation, generating_unique, signed_catalan_mod_three.
• Definition: def distinctPowersOfThree (n : ℕ) : ℕ := if (Nat.digits 3 n).all (· != 2) then 1 else 0.
• Axioms: propext, Classical.choice, Quot.sound (std3).

theorem.

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