The integer source and dyadic parity of A397592

Paul D. Hanna's OEIS A397592 (July 10, 2026), revision 18, has NAME:

G.f. A(x) such that the sum of the first n coefficients in A(x/n)^n equals 2^(n-1) for n >= 1.

Its sole conjecture is:

Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).

The FORMULA states A(x)=Sum_{n>=0} a(n)*x^n; the original offset is 0 and a(0)=1. For A in Z[[X]], A_Q is its coefficientwise image in Q[[X]], namely A.map (Int.castRingHom Rat). The exact source and single result are:

def SourceA397592 (A : PowerSeries Int) : Prop :=
  PowerSeries.constantCoeff A = 1 ∧
    ∀ m : Nat, 0 < m →
      (Finset.range m).sum (fun j => PowerSeries.coeff j
        ((PowerSeries.rescale ((m : Rat)⁻¹)
          (A.map (Int.castRingHom Rat))) ^ m)) = (2 : Rat) ^ (m - 1)

theorem result :
  (∃! A : PowerSeries Int, SourceA397592 A) ∧
  (∀ A : PowerSeries Int, SourceA397592 A → ∀ n : Nat, 3 < n →
    (Odd (PowerSeries.coeff n A) ↔
      ∃ k : Nat, 1 < k ∧ (n = 2 ^ k - 1 ∨ n = 2 ^ k + 1)))

These are ordinary formal powers, not compositional iterates; there is no convergence or factorial normalization. The sum includes j=0,...,m-1 for every natural m>0. The separate integer-existence-and-uniqueness conjunct excludes an unproved integrality premise.

This first-tier 2026 external named conjecture is the question_answered in #8006, created 2026-09-15T03:15:36Z before probes. A397591 supplies the distinct y=0 endpoint; A397592 is the y=2 source. The same parity pattern does not identify their series.

The A397591 series F has constant term zero, offset 1, and satisfies X^(m-1)^m/(1-mX)=0 for m>1. Its endpoint does not by itself establish integer existence for A397592 or an all-degree parity comparison. A397590 attests A(X,0)=1-F(X) and A397592(n)=Sum_{k=0..n} 2^k T(n,k). These source identities do not supply a proof of general triangle integrality.

Multiply the rational NAME sum in row m=n+1 by m^n. The coefficient rescaling law and the geometric-series convolution give the integer row R_n=[X^n]A^(n+1)/(1-(n+1)X)=(2(n+1))^n. Both directions hold for every m>0, including m=1. The existing exact normalization gives R_n=(n+1)N_n for n>0. Cancellation takes place in the integers and gives N_n=2^n(n+1)^(n-1).

Let t_n=2^n(n+1)^(n-1) for n>0 and give the forcing series constant term zero. Add this forcing series to the existing coefficient-correction map advance. Its agreement property increases the number of stable coefficients by one. Iterating from 1 therefore stabilizes coefficientwise to an integer series A fixed by the forced map. The fixed-point equation gives N_n=t_n, and the complete source equivalence gives SourceA397592(A). Strong induction using normalized_change proves uniqueness among integer source solutions.

Only then map to ZMod 2. For positive n the forcing term maps to zero, so normalized_unique identifies every satisfying source modulo two with the existing homogeneous solution. The latter equals 1 minus the A397591 series. Above degree three their coefficients agree modulo two, and hanna_conjecture gives exactly the two dyadic alternatives.

The new iteration, stabilization, source equivalences and parity adapters are local to the sole theorem. A397594/y=4, arbitrary even parameters, full triangle integrality, positivity and asymptotics are outside its claim.

An integer source and an n>3 violating either direction of the parity iff would refute the parity assertion. Failure of integer existence, or two distinct integer sources, would refute the well-posedness conjunct. A finite prefix, a conditional integer-source assumption, a shifted index or a missing strict bound changes the target instead of resolving it.

The module D5/S1/Recurrence/LinearRows/DoubledLinearExponentDyadicSupport contains the source predicate SourceA397592 and the single theorem result. Its source citation is D5/L/hanna2026a397592.

The existing LinearExponentDyadicSupport endpoint has stored identities:

• hanna_conjecture: sha256:257441c32f0b491832283533f85a4a8f2894bd3bb24da35790b5c8539a682503.
• generatingSeries: sha256:5edab5bd8bbb3730bb76f331bab1d2a071ce873a70eecd471722169450fdcefe.

Those identities locate the y=0 endpoint, not an A397592 proof. The numerical appendix of Library/Words/oeis2026triage0911b.md records an N=100 integer recurrence experiment and zero parity counterexamples at n=4,...,100. Its program and data are finite research evidence, not evidence for all indices.

The admission route is open-problem-resolution under CLAUDE 3.2 for the exact external conjecture registered in #8006. The theorem has proof_shape: content: the forced integer construction and stabilization supply a new unbounded existence argument. The source predicate is a necessary definition. escape_witness: none uses the external named-problem admission basis; computational_content.kind: none applies to both declarations. The public mathematical surface consists of the necessary source predicate and one result; adapting facts belong inside its proof. The stated problem is unbounded and symbolic. It includes neither A397594 nor a full A397590 triangle result.

The source snapshot in #8006 is revision 18, Jul 11 2026 03:34:44. The live text request returned HTTP200 (4,200 bytes); the complete mirror entry is 3,990 bytes, SHA-256 ec2009bc22f7209c0403adc0c2909bffbc24016ef40c9eb8b0c131585c8e51d8. The Library note fixes the source locators; later source changes are outside that snapshot.

Bounded exact A397592 searches recorded Crossref HTTP200/0, OpenAlex HTTP200/0, MathOverflow HTTP200/empty with no further page, and a public GitHub Lean-code index count of 0. D5 and pinned Mathlib db584cd6d46c92f209a44c0f1c829460d327499d supplied the y=0 endpoint and generic series APIs, with no exact resolution in the inspected scope: dominating_theorem_search: not-found-in-searched-scope.

Only 20 of 34 broad OpenAlex records and nine MathOverflow titles were screened; not all answers were read, and broad Crossref ranking is not an absence result. Fried's arXiv:2607.24832 and Heninger–Rains–Sloane were only partially read; no verified reduction from the latter's fixed-n root integrality to this varying-row source is supplied. arXiv query HTTP429 and timeouts, Google's JavaScript shell, unread papers or portions, b-files and other unopened material give no negative evidence. Differently named, differently indexed, unindexed or new proofs remain unexcluded. Neither global priority nor exhaustive absence of prior proofs is claimed.