The full Laurent rows in A391810 sum to zero

Paul D. Hanna's OEIS A391810 formula (2) states:

(2) Sum_{k=0..floor((n+1)/2)^2} T(n,k) = 0 for n >= 1 (conjecture).

The defining NAME is:

Irregular triangle, read by rows, where row n lists the coefficients of nonpositive powers of x in the doubly infinite series Sum_{k=-oo..oo} x^k * (1 - x^k)^(n+k) for n >= 1.

Formula (1) identifies the truncated Laurent expansion, including the constant term, with Sum_{k=0..floor((n+1)/2)^2} T(n,k)*x^(-k). All rows with positive integer n are included. Expansion takes place at zero; negative integer exponents mean Laurent-field inverses.

This is a Tier 1 explicitly published arithmetic conjecture. Its universal endpoint is the full row sum, with the exact inclusive row support bound. The prospective question and its source qualification are registered at https://github.com/the-omega-institute/trureturing/issues/16322.

The source qualification supplied bounded searches of the current official A391810 and A391811 records, exact-ID OEIS and Crossref results, SeqFan, the full primary TeX of arXiv:2607.24832v1 and JIS Article 26.1.8, and four public Lean collections. Those searches found no settlement of the exact question. Repository searches included private D5 declarations and the pinned Mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d. The current lane search again found no exact identifier or corresponding bilateral-row result. Existing Laurent and finite-binomial infrastructure is reused directly. These bounded checks do not establish global priority.

Some search providers returned HTTP 403, rate limits, interstitial pages or bot challenges; those responses provide no negative evidence. OEIS history was inaccessible, so the version evidence is the matching official current JSON and internal page, not a complete revision-history audit.

Let y=x^(-1). The negative outer indices -1 through -n contribute y^j*(1-y^j)^(n-j) for j=1 through n. The outer index -(n+1) contributes constant -1. Every other outer index has strictly positive Laurent order and contributes nothing to a nonpositive coefficient. A uniform lower support bound and finite coefficient contribution sets certify the bilateral formal sum, rather than assuming an infinite sum exists.

The inverse-variable polynomial is therefore P_n(y)=-1+Sum_{j=1..n} y^j*(1-y^j)^(n-j). The exact coefficient bridge proves its integer coefficient at m is the coefficient at -m of the original rational Laurent series.

For even n, reflecting j to n+1-j cancels the highest monomials of the binomial expansions. Every remaining exponent j*(r+1) has j+(r+1)<=n and is bounded by floor((n+1)/2)^2. For odd n, the highest monomials themselves satisfy the same bound. Thus no nonpositive coefficient is omitted by the published row endpoint.

Evaluate this finite polynomial at one. Only j=n survives, with value one, which cancels its constant minus one. Polynomial evaluation as a sum of coefficients proves the integer row sum. The coefficient bridge transfers that sum and the full support bound to the original bilateral Laurent series. No evaluation of the original infinite expression at one is used.

A positive n with a nonzero full inclusive row sum would falsify formula (2). A nonpositive Laurent coefficient disagreeing with the integer polynomial, or a nonzero coefficient beyond the published bound, would falsify the corresponding bridge or support assertion.

The mathematical declarations are in D5/S3/ArithSums/HannaLaurentRowSums.lean, with exposition in its canonical Blueprint Scribe. result states the integer zero row sum, the corresponding rational Laurent zero row sum, and vanishing of every coefficient beyond the inclusive bound. series_coefficient identifies every nonpositive coefficient with its integer polynomial coefficient.

The expanded proof uses pinned Laurent order and formal-sum interfaces, finite binomial expansion, a reflected cancellation, polynomial degree bounds, and polynomial evaluation. Its declaration classification is conservatively bind-only; the prospective basis is open-problem-resolution. Consumed helpers organize these same arguments and do not establish a separate novelty claim.

The source is an explicit published conjecture; its integer coefficient identification and exact row endpoint are part of the mathematical claim. Independent source fidelity, admission and frozen delivery are separate from this draft's mathematical proof.

What the settlement shows

• Proved (D5/S3/ArithSums/HannaLaurentRowSums.series_coefficient, D5/S3/ArithSums/HannaLaurentRowSums.result): the decisive mechanism is the finite polynomial P_n(y)=-1+Sum_{j=1..n} y^j*(1-y^j)^(n-j). The coefficient bridge includes the constant coefficient. For positive n, evaluation at y=1 leaves only the j=n term, whose value one cancels minus one. The zero sum is thus symbolic for every positive integer n, with no parity restriction or finite-n ceiling. It does not require numerical convergence or substitution into the bilateral expression at x=1.
• Proved (D5/S3/ArithSums/HannaLaurentRowSums.row_degree, D5/S3/ArithSums/HannaLaurentRowSums.result): the degree inequality ensures that the published inclusive endpoint omits no nonpositive coefficient. The polynomial degree inequality itself holds for every natural n; the original Laurent coefficient bridge and zero-row-sum conclusion retain 0<n. No additional parity hypothesis needs to be imposed or removed.
• Computed (python3 -c 'n=0; print(((n+1)//2)**2, -1 + sum(1**j * (1-1**j)**(n-j) for j in range(1,n+1)))', output 0 -1): the finite polynomial expression at n=0 has bound zero and row sum minus one. Positivity cannot be deleted from that expression's zero-sum assertion. This is not an interpretation of a source-defined row S(0), which the published positive-n question does not provide.
• Open: row_degree proves an upper bound, not equality of the degree with the bound or optimality of that endpoint. Sharpness for each n, a smaller uniform endpoint, and extensions to other coefficient fields, expansion conventions or nonpositive row indices have not been proved here.
• Proved (D5/S3/ArithSums/HannaLaurentRowSums.series_coefficient, D5/S3/ArithSums/HannaLaurentRowSums.row_degree, D5/S3/ArithSums/HannaLaurentRowSums.result): these statements justify formula (1)'s finite nonpositive truncation and formula (2)'s zero row sum in their original positive-n domain. Formula (2) can therefore supply that exact premise wherever a source argument assumes it.
• Open: no other source conclusion depending on formula (2) has been reverified here; its remaining hypotheses and proof steps would still need checking. No other A391810 conjecture is settled, and no new claim about the related A391811 squared-sum sequence is established. Applicability of this cancellation argument to those neighboring questions remains open.

Literature completeness outside the bounded checked surfaces and global priority are unverified. No other A391810 or A391811 conjecture is settled by this result. The mirrored information audit is unfinished. The full-telescope result realization has an unfinished observation-dependence obligation -1 != -(Polynomial.coeff 1 1)+1. No compiled Reg mirror is retained. The unfinished audits of summand_order, summand_lower_bound, coefficient_finite, series_coefficient, row_degree, and result, including this proof gap and missing exact source-selection binding, are disclosed in https://github.com/the-omega-institute/trureturing/issues/16322 under the repository's registration exception. Registration completion is unverified; no declared_validated status, admitted freeze or merged delivery is asserted.