# Verification report

**Manuscript:** Polynomial Unit-Distance Lower Bounds over Prime Fields.
**Author:** Alper Ferudun, Mercury Software GmbH.
**Version:** 1.0, 5 October 2026.

The complete mathematical scope is an existential polynomial lower bound
for exactly p points in F_p^2, along infinitely many primes in each of the
classes 1 and 3 modulo four. It is not the full extremal answer to the AIM
question. The stronger ordered-pair bound has exponent
1 + eta/(70 log lambda), with 0.005868357018 < eta/(70 log lambda) < 0.005868357019.

The proof uses Sawin's published field and norm-fiber construction and
Zaman's field-uniform Chebotarev counting theorem. The finite reduction,
local normalization, collision bound and exact-p packing are written in
detail. The norm-reduction principle is credited also to the related
sum-product literature. The full reasoning and its originating-researcher
self-audit are retained in `reproducibility/complete_proof.md` and
`reproducibility/proof_audit.md`.

The portable Python checker passed **440 checks**, including:

- exact source-parameter hypotheses and Golod-Shafarevich budget;
- rigorous rational enclosures of logarithms, pi and the exponent;
- split and inert finite-field normalization identities;
- disjoint translation packing and padding to exactly p points;
- fractional-ideal models even when the denominator is divisible by p;
- failure witnesses when H is divisible by p, the strict index bound is
  lost, or only one split residue coordinate is retained.

Integer and rational arithmetic determine every comparison. Logarithmic
tails and arctangent tails have explicit analytic bounds. The finite tests
are sanity checks, not a formal verification of the general proof or the
cited number-theoretic results. No practical large-degree coordinate witness
or explicit first prime is provided.

The manuscript is AI-assisted, self-audited and unrefereed. No external
referee report, proof-assistant certification, novelty certification or
absolute priority claim is made. The author remains responsible for the text.