An Inert Critical-Face Equivalence for Circulant Determinants of Order 4p

Let S(C_n) be the set of integer circulant determinants of order n. For everyprime p congruent to 3 modulo 4 and every integer m coprime to 2p, this paperproves the complete critical-face equivalence 32m in S(C_{4p}) if and only if p^2 m in S(C_{4p}). The proof is uniform in p and makes no class-number-one assumption. Itfactors the normalization conductor through two Milnor squares, identifiesthe full ray-ideal packet quotient with Pic(Z[C_{4p}]), and proves that allthree 2-primary sources and both p-primary sources have one common Picardtarget. The paper also proves an explicit cross-2-Sylow lambda-bridge forevery odd prime p, simultaneously constructing determinants for C_{4p} andC_2^2 x C_p from one cyclotomic ray condition. The result identifies twocomplete faces of an infinite cyclic family; it does not classify everyface of S(C_{4p}). Exact-arithmetic verification programs and independentproof audits accompany the preprint. The p=3 specialization is contained in the known complete classification ofS(C_12); the contribution here is the uniform transition and proof mechanismfor every prime p congruent to 3 modulo 4.