# Primitive-root companion: precise uniform Hecke dependency

The factorization application consumes Theorem 1.2: a fixed width 10^-6 for finite-order Hecke characters over all cyclotomic fields containing mu_12, without conductor/height cutoff. This differs from the main all-base primitive-root infinitude statement. Selected introduction and zero-detection setup were read; the full analytic proof remains unverified.

[Detailed review, exact scope and evidence](../../../../prototypes/finite-field-factorization-2026-10-09-v1/report.md). [Updated opportunity 016](../../../../opportunities/016-deterministic-finite-field-factorization/2026-10-09-v2.md). Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. No source proof/kernel/program acceptance, customer study or new revenue claim.