Frobenius minimality of the Huneke-Wiegand counterexample in symmetric numerical semigroup rings

Abstract

Son Pham publicly identified a counterexample to the Huneke-Wiegand conjecture in the class of two-generated monomial ideals over symmetric numerical semigroup rings, subsequently verified independently by Professor Craig Huneke. This preprint preserves that discovery priority and addresses a separate question: how small can such an example be by Frobenius number? We prove by proof-carrying exact computation that the least Frobenius number is 181, attained by Pham's example at shift 14. A complete theorem-tree implementation and 1,156 independently checked fixed-pair DRAT proofs reproduce the published positive range F<69. A one-hot selector CNF then gives accepted DRAT proofs for every odd F from 69 through 179 and recovers the exact public semigroup at F=181; a separately generated fixed-pair formula returns the same model. Independent theorem-tree enumeration at F=69,71,73,75 checks another 79,790 symmetric semigroups and 2,933,163 gap cases with zero counterexamples. The search audit rehashes 228 external files totaling 760,081,739 bytes, independently validates the SAT models, and recomputes the strict-order manifest with no mismatch. The result is minimality only within the symmetric numerical-semigroup, nonprincipal two-generated monomial-ideal class; it does not assert uniqueness, minimum multiplicity or embedding dimension, or minimality among arbitrary Gorenstein domains and modules. Code, compact manifests, verdicts and verification instructions: https://github.com/fsantibanezleal/CAOS_RESEARCH.