The full signed Laurent second moment in A391810

Paul D. Hanna's OEIS A391810 revision 39 formula (4) states:

(4) Sum_{k=0..floor((n+1)/2)^2} k^2 * T(n,k) = -(2*n-5) for n >= 2 (conjecture).

The source defines T(n,k) as the coefficient at zero of x^k times the bilateral series Sum_{j=-oo..oo} x^j*(1-x^j)^(n+j), expanded formally at zero. Equivalently it is the series coefficient at -k. All integer outer indices, the constant coefficient, and both finite-sum endpoints are included. Negative integer exponents are Laurent-field inverses. The bound is natural division (n+1)/2 followed by squaring. The right side uses signed integer subtraction and negation, then a rational cast.

This is the explicitly posed formula (4), distinct from formula (2)'s zero row sum and formula (3)'s first moment. The exact prospective source event, forecasts and strict preregistration are associated with https://github.com/the-omega-institute/trureturing/issues/24520 and https://github.com/the-omega-institute/trureturing/issues/24520#issuecomment-6537206052. Neither earlier moment is counted as this question's resolution.

The official revision 39 export has SHA256 49bbf5e8e42f91b6a3d838b2ce5c311eca58af74a83a4f6f684b09e1ddf0da83. Its formula (4) contains the exact quantifiers and signed expression above. The bounded source qualification reused the earlier official OEIS, Crossref, SeqFan, arXiv:2607.24832v1, JIS Article 26.1.8 and public Lean collection searches. The subsequent source delta inspected 79 changed research paths, including 60 D5 paths and 156 private declaration starts, at dev b1a09a0eb65b72a5e37289484e41c28aa0fa100d relative to 66c45b655013f43de728fcab36d99df1e61f9da3; it found no exact supplier in that scope. Eight available ownership states identified no formula (4) owner. These bounded observations are not a proof of global absence.

The inherited HannaLaurentRowSums.series is the full formally summable bilateral Laurent series. Its series_coefficient theorem identifies every nonpositive coefficient with the integer coefficient of P_n(y)=-1+Sum_{q=1..n} y^q*(1-y^q)^(n-q), where y=x^(-1). The inherited row_degree, rowBound and rowEntry supply the exact inclusive support and integer polynomial interface. No new series or T is defined. Only the finite polynomial is evaluated at one.

Euler-operator calculation

For E=yd/dy, the coefficient at k of E^2(P) is k^2 times that of P, and E^2 does not increase polynomial degree. Evaluation at one is therefore the full coefficient moment through the published endpoint. For y^q*(1-y^q)^m, that evaluation is q^2, -3q^2, 2*q^2, or zero according as m is 0, 1, 2, or at least 3. For n>=3 the final three terms give n^2-3*(n-1)^2+2*(n-2)^2=-(2*n-5). For n=2 the two terms give 4-3=1. The constant -1 contributes zero. The exact coefficient bridge and integer-to-rational cast yield the original Laurent statement.

Any natural n>=2 for which the literal full coefficient sum differs from -(2*n-5) would refute formula (4). A finite ceiling on n, a missing constant or inclusive endpoint, natural subtraction on the right, or a substitution into the bilateral series at one would change this question.

D5/S3/ArithSums/HannaLaurentSecondMoment.result has parameters (n : ℕ) (hn : 2 ≤ n) and conclusion

(∑ k ∈ Finset.range (((n + 1) / 2)^2 + 1),
  (k : ℚ)^2 * (HannaLaurentRowSums.series n (by omega)).coeff (-(k : ℤ))) =
  (-(2 * (n : ℤ) - 5) : ℤ)

The declaration is conservatively proof_shape: bind-only, with admission_basis: open-problem-resolution for the exact external question. Its consumed private helpers organize polynomial differentiation, coefficient weighting and the boundary calculation; they make no separate escape-witness or mathematical novelty claim. The matching Blueprint contains the mathematical statement and derivation.

The result addresses formula (4) for every natural n>=2 in the formal Laurent expansion at zero. It claims neither optimality of the support bound nor another OEIS formula. Its canonical frozen declaration and typed resolution link identify this exact source statement. There is no justified successor claim here.

What the settlement shows

• Proved (D5/S3/ArithSums/HannaLaurentSecondMoment.result): the Euler operator reduces the full second moment to the last three finite polynomial terms for n>=3, with n=2 included separately. The proof covers all natural n>=2 without a parity restriction or finite ceiling.
• Proved (D5/S3/ArithSums/HannaLaurentRowSums.series_coefficient, D5/S3/ArithSums/HannaLaurentRowSums.row_degree): the inherited bridge and support bound identify every coefficient needed by the original question. These prerequisites are reused rather than counted as a new resolution.
• Open: the support bound's sharpness, other moment identities and other source conclusions are outside this result. Formula (4) supplies its exact premise to arguments that require it; no such argument's other hypotheses or conclusions have been reverified here.

Global solvedness and priority remain unverified. The bounded searches do not close the SeqFan, revision-history, citation-chain or global-literature gaps. The mirrored Reg source compiles its full-telescope equivalence, actual positive law, whole-family variation and sensitivity, and nonconstant coefficient observation. Its source binding remains declared_unresolved, with no certificate. The current diagnostic is IE-C050 ClosedTruthReadout, reason=unclassified_form, rule=source.coordinate_dependency, missing=1;selected=#[0, 2];scope=3: the selected original coefficient also depends on the positivity proof derived from binder hn. Missing evidence is an accepted exact source-coordinate binding of this proof-dependent coefficient observation. Assessment beyond that first boundary is unverified. The unfinished registration is disclosed in audit issue 24562. No declared_validated, CI success, merge or dev publication is asserted here.