A Uniform Rational Critical-Face Theorem for Integer Circulant Determinants of Order 3p and the Pointwise Equivalence at Order 33

Abstract

For a prime p greater than 3, this preprint compares the first locallyadmissible but globally nonautomatic critical faces 9m and p^2m for integercirculant determinants of order 3p, with m coprime to 3p. An exactnormalization-conductor calculation proves the uniform pointwise transfer ±9m in S(C_3p) implies ±p^2m in S(C_3p). When p is congruent to 2 modulo 3, local source exhaustion makes the transferan equivalence. In particular, for every m coprime to 33, ±9m in S(C_33) if and only if ±121m in S(C_33). An independent arithmetic computation identifies Pic(Z[C_33]) with C_44 andplaces the two critical sources at its unique order-two element. For rationalprimes q not dividing 33, the common events 9q in S(C_33) and 121q in S(C_33)are classified by exact marked ray packets and have natural and Dirichletdensity 23/176. The proof treats the packet converse source by source andrecovers global membership by the exhaustive union over local primarysources. The accompanying archive contains exact finite-field, resultant,unit-image, Picard, ray-packet, Galois-action, witness, and densitycertificates, together with a clean aggregate runner.