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

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 all strictly increasing nonnegative triples of sum p-2. Reflection-symmetric interval potentials give explicit original integral sources for twice each class. Relative parity functionals, supported on ten to twelve original rows, detect the classes independently and vanish on every complete connecting image; extension over the remaining target gives a nonconstructive retraction. A previously tracked four-row vector is integrally congruent to one selected class, with an explicit source for the difference. We also give its full twice-source, two short kernel-domain corrections, and a neighboring integral potential basis of rank binomial(p-1,2) with at most seven-row images. Independent finite computations check the signed identities and complete-sector parity certificates, preserving all tested sources in lossless archives. A failed local search yields an exact locality obstruction. The uniform theorem follows from signed reconstruction, exhaustive source classification, and an elementary counting recurrence, not finite rank extrapolation. The full cokernel, an upper bound, and 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 .