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slug: oeis-a396798-third-iterate-mod-eight bibkey: hanna2026a396798 doi: null url: https://oeis.org/A396798 triage: theorem motivation_gids:

  • D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight
  • D5/S1/Recurrence/Residue/IterateProductFourFiveFifthModEight
  • D5/S1/Recurrence/Residue/IterateProductFourFiveFourthModEight

The third compositional iterate of A396798 modulo eight

Problem

Paul D. Hanna’s OEIS A396798, revision 12, defines the ordinary generating series by “G.f. satisfies A(x) = x + A^4(x)*A^5(x).” Its formula field says that A^k denotes the k-th iteration. The third comment states:

Conjecture: [x^n] A^3(x) == [3,1,7,5] repeating (mod 8) for n > 1.

Let I0(A)=X and I(k+1)(A)=Ik(A) composed with A. For the unique zero-constant integer series satisfying A=X+I4(A)*I5(A), the target is forall n:Nat, 1<n -> 8 divides coeff n (I3(A)) - r(n), where r(n)=if (n-2)%4=0 then 3 else if (n-2)%4=1 then 1 else if (n-2)%4=2 then 7 else 5. The period starts at n=2. Multiplication is ordinary series multiplication, with no factorial normalization or convergence assumption. The source is D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries.

Motivation

This first-tier external named conjecture was preregistered in #7982. The public fifth and eighth results on that same source reduce the third-iterate question to a compositional inverse identity over ZMod8.

Gap

The complete revision-12 source and its four direct references A396797/A396807/A396099/A213591 do not supply the third-clause proof. Exact-A-number Crossref, arXiv and public Lean-code searches and all-state repository records found no third-clause settlement within their checked scope. The #6436/#6375 queue concerns the second-iterate strong congruence. Unavailable searches provide no negative evidence; a current arXiv request returned HTTP429 and the public code index omitted known repository results. Differently named or unindexed results are not excluded, and no global priority is claimed.

Route

Reduce the fixed integer source to F over ZMod8. The frozen coefficientwise definition supplies its zero constant coefficient and linear coefficient one. These low coefficients supplement the public eighth coefficient theorem to give I8(F)=X. The public fifth theorem gives J=I5(F)=X+X^2*(5+X+X^2+5*X^3)*Q, where Q has coefficient one precisely at multiples of four and satisfies Q*(1-X^4)=1.

Set K=X+X^2*(3+X+7*X^2+5*X^3)*Q. Clearing powers of the unit 1-X^4 and the unit 1-J^4 reduces K composed with J = X to an exact polynomial identity in characteristic eight. Only proved units are cancelled. Iteration addition gives I3(F) composed with I5(F)=I8(F)=X; compositional cancellation then identifies I3(F) with K. Extracting every coefficient above degree one gives the period 3,1,7,5 with the natural subtraction bounds retained. No second-iterate conjecture or assumed rational formula for the original source is a premise.

Falsifier

An n>1 whose third-iterate coefficient differs from r(n) modulo eight would refute this exact assertion. Starting the cycle at n=1, substituting a different source, cancelling a nonunit or replacing an infinite series identity by a finite prefix would invalidate the result.

Evidence

The public result is D5/S1/Recurrence/Residue/IterateProductFourFiveThirdModEight.result. All source adaptations, rational identities, unit proofs and coefficient arguments are local to this single result. No source definition or companion theorem is added. Canonical reporting and freezing supply its declaration and dependency identities.

Triage

The single result settles only the third comment under admission_basis: open-problem-resolution, with proof_shape: bind-only and no escape witness claimed. Its unbounded symbolic proof has utility.kind: none. The fourth, fifth and eighth comments have separate existing results. IterateProductFourFiveSeventhModEight.result separately settles the seventh comment for every n>1, with period 7,1,3,5 from n=2, using the same source and (n-2)%4. Comments 2 and 6 remain with #6436 (and the #6375 referral); the proposed repair of the first comment retains its own obligation. The original first-comment contradiction remains non-kernel evidence and is not a new formal refutation here.

ASSUMED-UNVERIFIED

The proof does not certify external-page authenticity or completeness of the bounded literature search. It makes no global-priority, nonnegativity, analytic-convergence or remaining-clause claim.