A quadratic family of integral two-torsion classes in a Huneke-Wiegand Koszul presentation

Abstract

We study an explicit integral Koszul presentation arising in the graded-syzygy analysis of a previously constructed Huneke-Wiegand family. For every integer p>=8, its cokernel contains a direct summand isomorphic to (Z/2)^floor(((p-2)^2+3)/12). The classes are indexed by strictly increasing nonnegative triples of sum p-2. Reflection-symmetric interval potentials give explicit original integral sources for twice each class, while relative parity functionals detect them independently. For a semantic row projection retaining four of six recurring row types, we prove a new uniform constructive identity: two endpoint triangle rows vanish for every p>=8, each by a single explicitly labelled source column of coefficient height one. The proof classifies every face before contraction. A locked p=11 frozen-carrier comparison and independent exact sweeps through p=300 validate the implementation without replacing the symbolic proof. A tracked four-row vector is integrally congruent to one selected triangle class. We also give its full twice-source, two short kernel-domain corrections, and an integral potential basis of rank binomial(p-1,2). The complete cokernel, a matching upper bound, and a uniform identification with earlier isolated subpresentations remain unresolved. The family-to-presentation identification is an imported premise. Son Pham retains priority for the original public Huneke-Wiegand counterexample. Source code, computation records and archived sources: https://github.com/fsantibanezleal/CAOS_RESEARCH .