Prove uniform prime-power cyclic inverse bounds (no κ claim)

[aliromman]

The prior cyclic inverse notes establish finite invertibility at roots of unity. That does not give a conditioning bound independent of the cyclic length. This package proves such a bound for every prime-power kernel order, including dyadic orders, and every primitive root choice.

For n=r-1, D=phi(r), and H=2^r-1, it proves |P(z)| >= delta = 1/(2rn H^(nD-1)) on the entire unit circle. For every unit scalar tau, cyclic length L, and integer shift step w, it then proves ||P(tau Z_L^w)^-1||_infinity <= delta^-1 + 2nH/(r delta^2). Finite products and unit diagonal gauges retain the stated bound. Radix two has a separate bound of two.

The proof uses integral tail coefficients, prime-power residue reduction, a nonzero Sylvester determinant, paired field embeddings, cofactor bounds, and an absolutely convergent Fourier series. The logarithm of the inverse upper bound is O(r^2 phi(r)); no dependency on cyclic length remains.

Validation:

• The written proof passed a separate AI-assisted mathematical review.
• The standard-library engine checks eight exact cyclotomic cases; isolated optimized Python gives the committed deterministic record.
• An independent SymPy engine matches all eight determinants and norms, checks an extra primitive choice, and verifies 36 actual cyclic inverse products. Both engines reject the intended unit-root and omitted-conjugation controls.
• Both engines pass from a relocated standalone package with no historical workspace inputs.
• make verify passed on pinned main 5996f9043c27bafa90b23e1c937de92a738f1a76, including make gen4-bank-verify, make frontier596-verify, all remaining tests, and patch applicability checks. git diff --cached --check passed.

This is a scoped analytic contribution, with no new integer kappa, complex b, BIT c, or paid inversion algorithm. Efficient preparation, combined precision and rounded execution, restoration, and tape costs remain open. The scalar theorem uses standard mathematical facts; any multiplication application still needs the original algorithmic interfaces. No all-composite, full multiplication theorem, practical speedup, literature priority, or maintainer acceptance claim is made.

Credits and exact pins are in SOURCE.json and NOTICE.md: OpenAI's exact-DFT kernel, classical q-Pochhammer/Newton factorization (Alexey Kuznetsov), and prior unpublished local cyclic inverse work. Prepared with substantial OpenAI Codex assistance during research directed by Ali Romman. Apache-2.0.