Quantum Integer Torsion in the Skein Module of a Connected Sum of Two Solid Tori

We extract an explicit infinite family of torsion elements from the two-solid-torus presentation of Bakshi, Le and Przytycki. Over R=Z[q,q^{-1}], put B=R[x_1,x_2] and [m]=(q^{2m}-q^{-2m})/(q^2-q^{-2}). For every m>=2 we construct an element whose exact B-annihilator is ([m]), and these elements embed the direct sum of B/([m]) into the torsion submodule. Thus no single nonzero Laurent polynomial annihilates all torsion. A further explicit element has annihilator (q^8-q^4+1), comaximal with (q^2-q^{-2}). The calculation answers the coprime-torsion question in arXiv:2604.09971v1 and contradicts its bounded-annihilator corollary; the gap is an identification of an unsaturated quotient with its fraction-field image. Our argument is algebraic and uses the cited presentation for its skein interpretation. It does not compute the skein module of arbitrary closed connected sums of lens spaces. This is a self-audited, unrefereed, AI-assisted preprint. No independent review, formal verification or absolute priority is claimed. The source package contains the complete proof, detailed audit and exact-arithmetic checker with 156 passing regression assertions.