Frobenius Minimality, Minimum-Layer Uniqueness, and an Infinite Family of Huneke-Wiegand Counterexamples

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 by Professor Craig Huneke. We preserve that discovery priority and answer two different questions: how small can such an example be, and does the same semigroup/ideal class contain an explicit infinite family? We prove by proof-carrying exact computation that the least Frobenius number is 181, attained by Pham's semigroup at ideal shift 14. First, an implementation of the complete Blanco-Rosales tree exhausts all 48,954 symmetric semigroups with Frobenius number below 69 and all 1,503,391 associated gap cases. An independent Boolean route checks 1,156 fixed-pair UNSAT certificates for the same range. We then introduce a one-hot shift-selector CNF that asks one existence question per Frobenius number. Its DRAT certificates exclude every odd value from 69 through 179, while the formula at 181 is satisfiable and decodes exactly to the public semigroup and shift. A separately generated fixed-pair formula returns the same model. An independent verification rehashes all 228 files in the minimality search, totaling 760,081,739 bytes, rechecks both SAT models, and reproduces the strict-order manifest with no mismatch. Complete theorem-tree enumeration at F=69,71,73,75 provides an object-level cross-check on another 79,790 semigroups and 2,933,163 gap cases. A second projected enumeration proves that shift 14 is the only feasible shift at F=181 and that the public membership vector is the only semigroup there. An independent verification of the 12 files of this classification rechecks both terminal DRAT proofs. Thus Pham's pair is the unique normalized pair attaining the Frobenius minimum in this class. Turning to the second question, for every integer p ≥ 4 we construct a symmetric numerical semigroup Γ_p of multiplicity 24p, Frobenius number 78p-1, conductor 78p, and embedding dimension 11p, for which I_p=(t^(24p),t^(30p)) is nonprincipal and rigid. This family theorem is deductive: seven exact interval-sum identities prove closure, symmetry, and D_s=E_s+E_s; a finite sweep through p=300 and an independent re-implementation are supporting checks. For every family member we also determine the value semigroup of the endomorphism ring exactly. It has multiplicity 24p, Frobenius number 54p-1, conductor 54p, genus 38p-1, and embedding dimension 12p. It is nonsymmetric, and the endomorphism ring is strictly larger than the original ring. Recent endomorphism-center criteria then show uniformly that the ideal remains rigid over its endomorphism ring but is not reflexive there, with the adjacent Ext and Tor obstruction groups nonzero. We further prove that this endomorphism semigroup has Cohen-Macaulay type and reduced type 10p, hence maximal reduced type, and that its completed semigroup ring is not almost Gorenstein. The traces of the ideal and its endomorphism ring both equal the conductor of the finite birational extension. We determine that common value ideal exactly and prove the identity ℓ_(R_p)(R_p/(R_p:E_p)) = ℓ_(R_p)(E_p/R_p) = p+1. The equality of the two lengths is a general consequence of one-dimensional Gorenstein local duality, while the exact common ideal and its value p+1 are the family-specific data. We further prove that this conductor is nonstable and compute its complete one-step stability defect: ℓ_(R_p)(T_p^2/t^(4s)T_p) = 14p. The full reduction sequence is also exact: t^(4s)R_p is a minimal reduction of T_p with reduction number four, successive Sally-quotient lengths 23p-1, 14p, 2p, 1, 0, and Hilbert-Samuel coefficients e_0(T_p)=24p and e_1(T_p)=39p. The associated graded ring of T_p nevertheless has depth zero: its complete Valabrega-Valla module is concentrated in one degree, where it has length p. We compute the exact Hilbert series and show that its numerator has only positive coefficients despite this failure of Cohen-Macaulayness. We then determine the complete zeroth local cohomology: it is k^p, concentrated in degree zero and annihilated by the full homogeneous maximal ideal. Hence the tangent cone is Buchsbaum but not Cohen-Macaulay, with unbounded Buchsbaum invariant p; its quotient by the finite-length torsion is Cohen-Macaulay with an exact Hilbert series. Over the polynomial Noether normalization induced by the minimal reduction, we determine the entire graded module: the torsion consists of p exponent-one cyclic summands, while the free quotient has rank 24p and explicit shifts. This yields the complete minimal resolution, projective dimension one, regularity four, top-local-cohomology a-invariant three, and the exact parameter-section identity ℓ(G_p/x_pG_p) = 25p = e_0(T_p)+I(G_p). The exact identity T_p^2=m_pT_p identifies the special fiber canonically with G_p/H^0(G_p). It is Cohen-Macaulay of type 10p+1, but its Artinian socle lies in degrees two and four, so it is neither level nor Gorenstein. We then determine its defining ideal: in the polynomial ring on its 10p minimal degree-one generators, it has exactly 50p^2-17p minimal quadrics and the single additional cubic X_0^2X_(3p)-X_p^3, with no other minimal equation. Thus its relation type is three and it is not Koszul. The all-parameter component calculation is an exact Presburger verification, supported by a separately encoded graph check; it relies on the soundness of the solver, whose UNSAT answers carry no independently checked proof. From the same exact h-vector, Artinian socle, and first Betti row, we then determine the homological edges over the full presentation ring. Its projective dimension is 10p-1, its regularity is four, β_(2,3) = 2p(500p^2-330p+31)/3, the complete last row has ranks 10p in shift c+2 and one in shift c+4 for c=10p-1, and β_(c-1,c+3)=8p. The canonical module has 10p minimal generators in degree -1 and one in degree -3. We then determine the first previously unknown interior strand. With the offset grading deg X_a=(1,a), β_(2,4)=8p and β_(3,4) = p(5p-1)(500p^2-440p+47)/2, and the 8p second syzygies occur with multiplicity one exactly at offsets {3p+a : a ∈ ℰ_(p,1), a ≥ 6p}. Relative squarefree-divisor chains and an integral unit matching give the upper bound; an exact quadratic colon and minimal mapping cone give the matching lower bound. We then complete the second Betti row. The only remaining possible entries are β_(2,5)=p(2p-3) and β_(2,6)=0. The degree-five support is the union of three explicit intervals, with pair-sum multiplicities on the outer blocks and a plateau profile on the middle block. An integral offset matching gives a unit Smith form, so the result again holds over every field. This completes one homological row, not the remaining Betti table. We next determine the degree-five third syzygies. If H_p={a ∈ ℰ_(p,1) : a ≥ 6p}, then β_(3,(5,b)) = #{{a,c} ⊂ H_p : a