Critical-Face Transfer and Prime Separation for Integer Circulant Determinants of Order 21

Abstract

Let S(C21) denote the set of determinants of integral 21-by-21 circulantmatrices. This preprint analyzes the first nonautomatic local faces, 9m and49m, for cofactors m coprime to 21. A normalization-conductor calculationgives Pic(Z[C21]) isomorphic to C4 and determines the complete markedcontinuation targets of the rational and primitive critical sources. The main theorem proves that +/-9m in S(C21) implies +/-49m in S(C21) forevery cofactor m coprime to 21. The converse is false: for every prime qcongruent to 8 modulo 21, 49q belongs to S(C21), whereas +/-9q does not. Thisstrict prime family has natural and Dirichlet density 1/12. The paper alsogives the exact away-prime truth table, with densities 5/16, 1/12, 0, and29/48 for the four possible membership patterns. The proof separates source-specific realization from global face membership:the latter is obtained by the exhaustive union of all locally admissiblesource targets. The accompanying reproducibility archive supplies exactfinite-field, unit-image, source-orientation, resultant, determinant,ray-packet, and density certificates, together with a complete SHA-256manifest and a clean release runner.