A Torsion-Rank Obstruction to Turaev's Alexander Polynomial Realization Conjecture

We give counterexamples to the sufficiency assertion in Turaev's Conjecture 12.26 in Ohtsuki's problem list. If a closed connected oriented 3-manifold has first homology \(\mathbb Z\oplus T\) and its Alexander polynomial is nonzero modulo a prime p, its order of vanishing at t=1 modulo p is at least \(\dim_{\mathbb F_p}(T/pT)\). An integral square presentation from a primitive dual surface proves this bound. Consequently, whenever T has p-rank at least three, the pair (\(\mathbb Z\oplus T\), \(t+t^{-1}+|T|-2\)) satisfies the conjecture's symmetry and augmentation conditions but is not realizable. The concrete example \(T=(\mathbb Z/2)^3\) has polynomial \(t+t^{-1}+6\). A Smith-normal-form ideal condition strengthens the obstruction, and explicit torus bundles show that the same polynomial is realizable with cyclic torsion of the same order.

This is a complete negative answer to the sufficiency assertion in Conjecture 12.26, recorded as AMR-103-0231 in ulamai/UnsolvedMath v1.6.0. It is not a classification of all realizable pairs. The cut-surface and determinant methods are classical and credited. No absolute priority claim is made.

The five-page English preprint is by Alper Ferudun, Mercury Software GmbH. It is AI-assisted, self-audited and unrefereed; no independent peer review or proof-assistant certification is claimed. Portable exact regression checks accompany the source and do not replace the general proof. Version 1.0, 15 October 2026. CC BY 4.0.