Frobenius minimality and minimum-layer uniqueness of the Huneke-Wiegand counterexample

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 two separate questions: how small can such an example be by Frobenius number, and what pairs attain the minimum? 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 projected enumeration classifies the complete minimum layer. After shift 14 is blocked, the selector formula is UNSAT with an accepted 45,867,741-byte proof. After the public membership vector is blocked in the fixed-shift formula, the result is UNSAT with an accepted 1,608,691-byte proof. An independent auditor reconstructs all formulas, validates both SAT models, accounts for 12 external files totaling 63,609,504 bytes, and freshly rechecks both terminal proofs. Thus Pham's pair is the unique normalized pair attaining the Frobenius minimum in this class. The result does not classify higher Frobenius values, minimize multiplicity or embedding dimension across all counterexamples, or apply to arbitrary Gorenstein domains and modules. Code, compact manifests, verdicts and verification instructions: https://github.com/fsantibanezleal/CAOS_RESEARCH.