The fifth compositional iterate of A396798 modulo eight

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 fifth comment states:

Conjecture: [x^n] A^5(x) == [5,1,1,5] 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 exact target is forall n:Nat, 1<n -> 8 divides coeff n (I5(A)) - r(n), where r(n)=if (n-2)%4=0 or (n-2)%4=3 then 5 else 1. The four-term period begins at n=2. Multiplication is ordinary series multiplication, with no factorial normalization or convergence assumption. The source is the existing D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries.

This first-tier external named conjecture was preregistered in #7961. The fourth and eighth public results on that same source provide the high-degree coefficients needed to turn the fifth-iterate question into linear identities of formal series.

The complete revision-12 source and its four direct references A396797/A396807/A396099/A213591 do not supply this fifth-clause proof. Exact-A-number Crossref, arXiv and public Lean-code searches and all-state repository records found no fifth-clause settlement in their stated scope. The earlier #6436/#6375 queue concerns the second-iterate strong congruence. Unavailable searches provide no negative evidence. Differently named or unindexed general results are not excluded, and no global priority is claimed.

The fixed source definition and degree contraction give its zero constant coefficient, fixed-point equation and identity. The first two positive coefficients follow from the source product: a1=a2=1. The general iterate-coefficient identity then gives coeff2(I4(A))=4. Combining these low coefficients with the public fourth and eighth results gives, over ZMod8, P=I4(F)=X+4X^2 and I8(F)=X, where F is the reduction of A.

Write J=I5(F). The true source equation is F=X+PJ. Substituting P into that equation and using iteration addition at 1+4, 4+4, 5+4 and 8+1 gives J=P+XF. Eliminating F gives J*(1-X^2-4X^3)=X+5X^2. Multiplication by 1+X^2+4X^3 yields (J-X)*(1-X^4)=5X^2+X^3+X^4+5X^5.

The denominator 1-X^4 has constant coefficient 1 and is a unit even over ZMod8. With Q the geometric series supported at multiples of four, this gives the complete identity J=X+X^2*(5+X+X^2+5X^3)*Q. Its coefficients at every index n>1 are 5,1,1,5 according to the residue of n-2 modulo four. The argument keeps the lower-index bounds in convolution and does not replace the universal claim with a finite prefix. No division by 2 or 4, nonzero-factor cancellation, second-iterate strong congruence or assumed formula for A is used.

An n>1 whose fifth-iterate coefficient differs from r(n) modulo eight would refute this exact source assertion. Starting the cycle at n=1, substituting a different source, or assuming an unproved source equation would invalidate fidelity to the problem.

The public result is D5/S1/Recurrence/Residue/IterateProductFourFiveFifthModEight.result. It uses the existing source and the two separately frozen iteration results. All source adaptations, low-degree identities, linear elimination and coefficient arguments are local to result. There is no new source definition or companion theorem. Canonical reporting and freezing supply the declaration and dependency identities.

The single result settles only the fifth comment under admission_basis: open-problem-resolution. Its proof_shape: bind-only classification reflects reuse of existing results and normalization; no escape witness is claimed. The argument is unbounded and symbolic, with utility.kind: none.

The fourth comment is separately settled for n>2 and the eighth for n>1. The third comment is separately settled by IterateProductFourFiveThirdModEight.result for n>1 with period 3,1,7,5 from n=2. 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 recorded contradiction to the original first-comment period remains non-kernel evidence and is not a new formal refutation here.

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.