A Pointwise 32–p² Critical-Face Equivalence for Circulant Determinants of Order 8p: The Inert Case and Sharpness at p=5

Abstract

This preprint studies two natural critical valuation faces of integercirculant determinants of order 8p. It shows that when p is a primecongruent to 3 modulo 4, the complete cofactor sets on the 32-face and thep²-face agree point by point, with either determinant sign. The comparisonincludes every positive cofactor coprime to 2p, not only an asymptotic ordensity-one subset. The paper also identifies a concrete boundary of the theorem. At the splitprime p=5, the explicit cofactor 6,837,241 gives an order-40 determinant onthe 32-face but not on the 25-face. Thus the inert congruence hypothesis isessential. The paper does not claim a classification of all order-8pcirculant determinants or of the remaining valuation faces. The accompanying archive contains the paper, LaTeX source, a uniquely namedindependent verification prompt, exact-arithmetic programs, expectedtranscripts, independent audit notes, and SHA-256 manifests.