# Verification report

Manuscript: Quantum Integer Torsion in the Skein Module of a Connected
Sum of Two Solid Tori. Alper Ferudun, Mercury Software GmbH. Version 1.0,
6 October 2026.

## Accepted scope

For R=Z[q,q^{-1}], B=R[x1,x2] and the B-module presentation in
Bakshi--Le--Przytycki, arXiv:2604.09971v1, Theorem 3.1, the proof gives
explicit w_m, m>=2, with exact B- and R-annihilator ([m]). Their classes
embed the direct sum of B/([m]) into R-torsion. There is no nonzero
uniform annihilator in B or R. The element ([6]/Phi_24)w_6 has exact
annihilator (Phi_24), and Phi_24-q^6(q^2-q^{-2})=1 proves comaximality.

The relations G_n divided by g=q^2-q^{-2} span an unsaturated submodule
J. The quotient B[y]/J still has explicit [m]-torsion; identifying it
with its fraction-field image loses this torsion. This contradicts the
bounded-annihilator assertion in Corollary 3.12(a) and answers Question
1 of the cited v1. The skein-module interpretation uses its Theorem 3.1;
the full handle-slide calculation is not independently rederived.

## Mathematical self audit

The proof checks the integral Chebyshev basis, the two nonconstant
coordinates, the cancellation of the constant coordinate for both
parities, highest-index uniqueness in finite combinations, exact ideal
contraction from B to R, finite-support induction in the direct sum,
and exponent-width additivity in a domain. It does not assume a PID or
finite generation, nor does it treat the relation submodule as an ideal.

The m=2 example uses coprimality in the common-factor sense, not
comaximality over Z. The distinct Phi_24 example supplies the stronger
comaximal statement. The case m=1 is correctly excluded. Specializing
q is not used to infer integral nonvanishing.

No unresolved algebraic gap was identified within the accepted scope.
No independent reviewer, external-model approval or proof assistant is
claimed. The source presentation is an explicit cited dependency.

## Exact regression checks

The bundled checker uses SymPy 1.14.0 and exact integer Laurent
polynomials, without floating point. Two retained runs each passed
156 assertions with zero failures. The range m=2,...,12 covers the
doubling formulas, exact annihilation identities, nonvanishing,
failure of g-annihilation and torsion in the divided-relation quotient.
Cross-annihilator tests use m=2,...,5 and k=1,...,8. Additional tests
cover the Phi_24 factorization, Bezout identity and nonzero class,
mixed witnesses, wrong signs, an omitted factor g and the mistaken
polynomial-ideal interpretation.

The membership test is exact for the supplied finite vector because a
higher relation would have a unique uncancelled leading coordinate.
The sampled range does not prove the theorem for all m; the written
proof does. The code option named `saturated=True` selects J_n=G_n/g;
it does not assert J is saturated. The theorem proves it is not.

## Source coverage and prior art

The frozen record is AIM-TOPOLOGY-0043 in UnsolvedMath v1.6.0, commit
c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e. Its underlying AIM Problem
2.25 asks for the integral module of closed lens-space connected sums.
The present manifold has boundary. No survival under Dehn filling,
full torsion classification, splitting conclusion or full source
closure is claimed.

The presentation is credited to Bakshi, Le and Przytycki. Mroczkowski's
RP3#RP3 calculation, Bakshi--Przytycki's higher-genus counterexample,
Bakshi--Kim--Shi--Wang's double-S1xS2 calculation, and Belletti--Detcherry's
lens-space torsion theorem are prior work. The latter corrects an
overly broad unresolved-family statement in the inherited dataset
report. This attributed correction is not counted as our discovery.

A bounded primary-source search did not establish absolute novelty or
priority. None is claimed. AI assistance was used for derivation,
literature retrieval, exact checks and manuscript preparation. This is
an unrefereed preprint. Public availability or a DOI does not constitute
peer review or mathematical certification.