A Pointwise Critical-Face Transfer for Circulant Determinants of Order 4p

For every odd prime p and every positive integer m coprime to 2p, thispreprint proves that 32m in the order-4p integer circulant-determinant spectrumimplies p^2m in that spectrum, with equivalence when p is congruent to 3modulo 4; the signed form is equivalent by sign symmetry. It also gives anexact reduction to two finite conductor/ray packet faces, constructs everynoncritical valuation region, and determines the uniform pure 2-primary andp-primary axes. The deposit includes LaTeX source, an independentverification protocol, and deterministic exact-arithmetic certificates.