Atomic Critical-Face Transfer and Semiprime Separation for Integer Circulant Determinants of Order 57

This preprint studies the two first nonautomatic critical faces for integercirculant determinants of order 57. It first proves uniformly that, for everyprime p>3 and every integer m coprime to 3p, occurrence of ±9m at order 3pimplies occurrence of ±p²m; for p congruent to 2 modulo 3 this is anequivalence. At order 57 the paper computes Pic(Z[C_57]) ≅ C_27972 and all markedvaluation-two target fibres unconditionally. It proves that ±361q occurs ifand only if ±9q occurs for every prime q not dividing 57. The reverse transferfails for composite cofactors: an explicit squarefree semiprime m satisfies361m in |S(C_57)| but 9m not in |S(C_57)|, with nonmembership certified by aquotient obstruction modulo 28. Two ray-class orbits give infiniteChebotarev families of such separators. The accompanying archive containsthe exact ideal, unit, Smith-form, ray-class, target-fibre, and packetcertificates used in the proofs.