Critical-Face Separation for Integer Circulant Determinants of Order 48: The Least Coprime Separator and Opposite-Direction Ray-Class Families

Abstract

This preprint studies integer circulant determinants of order 48 and comparestwo natural continuation regimes, represented by values of the form 64m and9m when m is coprime to 6. It develops an exact arithmetic model usingcyclotomic orders, Picard groups, ray-class data, and finite packetcomputations to determine when the two regimes agree and when they separate. The paper identifies 67 as the smallest coprime cofactor that separates thetwo regimes: ±603 occurs as an order-48 circulant determinant, whereas ±4288does not. It also establishes positive-density families of primes with thesame behavior. A complementary fourth-power phenomenon produces separationin the opposite direction, showing that the two continuation languages aregenuinely incomparable; the first numerical prime in that family is 241. The accompanying archive contains the paper, LaTeX source, an independentverification prompt, exact-arithmetic programs, expected transcripts, andSHA-256 manifests. The work is computer-assisted. Its scope is the comparisonof these two critical continuation regimes at order 48, not a completeclassification of all order-48 circulant determinants.