The full alternating Laurent-row formula in A391810

OEIS A391810 formula (5), in its original positive-index range, is

$$ -\sum_{k=0}^{\lfloor(n+1)/2\rfloor^2}(-1)^k T(n,k) =2\operatorname{round}(2^n/3)=2,\mathrm{A001045}(n),\qquad n\ge1. $$

Here T(n,k) is the signed coefficient of x to the power -k in the formal expansion at zero of the literal bilateral sum Sum_{j=-oo..oo} x^j*(1-x^j)^(n+j). Integer powers with negative exponent use the Laurent-field inverse. The range includes k=0 and its upper endpoint. Rounding acts on the exact rational number, with nearest-integer ties toward positive infinity. A001045 has offset zero and recurrence J(0)=0, J(1)=1, J(n+2)=J(n+1)+2*J(n).

This exact whole formula is the externally published target registered at https://github.com/the-omega-institute/trureturing/issues/24532. The zero row sum, first moment and squared-coefficient questions are separate events and do not supply this alternating-row conclusion.

The qualified official A391810 revision 39 and A001045 revision 1350 fix the source conventions. Their export fingerprints are given in the cited library notes. The bounded qualification found no exact or explicitly stronger supplier in the selected repository delta, reached private declarations, pinned Mathlib, indexed ownership records, and bounded arXiv/Crossref surfaces. These negative observations do not certify global priority or solvedness. SeqFan, complete revision history and unrestricted literature completeness remain ASSUMED-UNVERIFIED.

The existing original-Laurent supplier proves that every nonpositive coefficient is the corresponding coefficient of the integer polynomial P_n(y)=-1+Sum_{j=1..n} y^j*(1-y^j)^(n-j), and bounds its degree by the published endpoint. Its bilateral summability and coefficient bridge retain every integer outer index, including the constant contribution at j=-(n+1).

For n>0, a finite-index shift gives P_(n+2)(-1)=P_n(-1)-2^(n+1). The recurrence for J gives J(n+2)=J(n)+2^n. Two-step induction from n=1 and n=2 yields P_n(-1)=-2*J(n). Finite polynomial evaluation and the degree bound then give the complete alternating signed row sum. Separately, two-step induction gives 3*J(n)=2^n-(-1)^n. The parity term places 2^n/3 in the exact nearest-integer interval for J(n), establishing the rounding equality. No value of the bilateral series at 1 or -1 is used.

A positive n at which either equality fails, with the original signed coefficients and full inclusive bound, falsifies formula (5). A mismatch between an original Laurent coefficient and the finite polynomial, or a nonzero nonpositive coefficient beyond the bound, falsifies the corresponding source bridge or support assertion.

D5/S3/ArithSums/HannaLaurentAlternatingJacobsthal.full_formula_five states the integer alternating/rounding equality, the integer rounding/Jacobsthal equality, the rational original-Laurent alternating/rounding equality, the coefficient bridge for every natural k, and vanishing past the bound. The rational Laurent/Jacobsthal equality follows by casting the second clause. Its exact declaration is the single terminal result, with no companion alias.

The module retains the offset-zero definition and six private consumed helpers. polynomial_at_neg_one_skip and alternating_coefficient_bridge are bind-only helpers consumed by polynomial_at_neg_one and full_formula_five, respectively. The closed-form and positive-index polynomial inductions provide content on the terminal live proof path. Admission is the preregistered exact external open-problem resolution; classical Jacobsthal identities carry no independent novelty claim.

The statement is an explicit external conjecture with a universal positive-n endpoint. The finite polynomial at n=0 equals -1 whereas J(0)=0, so this argument does not extend the row identity to zero. Formal source fidelity, registration, freezing and delivered publication are distinct obligations.

Global literature priority and solvedness outside the bounded qualification remain unverified. This dossier makes no delivery or KPI claim and does not settle any other A391810 formula.

The terminal theorem's four-slot information registration is unfinished. For the signed row-entry observation its remaining local dependence goal is -1 != -(Polynomial.coeff 1 1)+1, comparing entries zero and one in row one. The complete-family zero intervention, full-telescope equivalence and source selection have no accepted current binding evidence; no compiled Reg mirror or declared_validated status is claimed. This local proof gap does not establish a limitation of the registration producer.