Research: averaged cyclotomic heights give O(r² log² r) precision (no new kappa)

[chafreaky]

PR621 bounds every cyclotomic embedding by the same worst-case coefficient height. Averaging those heights before applying the algebraic norm improves the uniform circle-separation exponent from O(r³) to O(r² log² r). The proof covers every prime-power radix and every primitive embedding. No new kappa is claimed.

For ell = ceil(log2 r), n = r-1, and D = phi(r), the explicit bound is

|P(z)| >= 2^-d on |z| = 1,
d = 1 + 2ell + n D min(r, 8ell²).

The minimum preserves the previous bound at small radices. At r=4096, the sufficient circle exponent drops from 34,351,349,785 to 9,661,317,145. These are analytic bound values, not executed word sizes.

The derivation bounds the discrete Fourier mass of log|1-zeta^s|, averages geometric sums over primitive residues, and uses the resulting total height in PR621's resultant/cofactor proof. It neither assumes that the resultant is rational nor treats an average height as a pointwise bound; exact counterexamples reject both shortcuts.

Substituting the new circle exponent into the retained PR862/977 estimates gives, for N=r² ell², fractional working precision W<=p+m+60N, integer guard G<=m+26N, and preparation precision b<=p+m+34N. The inherited clean-scratch arbitrary-stride record count is unchanged. No new inverse execution, dirty-array construction, common-label compiler, complete transform or multiplication reduction is supplied.

Validation includes 17 outward-rounded rational interval cases, six resultants checked independently with SymPy and PR621's rational field arithmetic, and 752 exact integer budgets. Complete macOS/Python 3.11 and Linux/Python 3.14 replays produce byte-identical receipts. Both platforms also reject eleven package faults and replay from a read-only relocated copy. The Linux job terminated successfully under supervision. Twelve retained source files have verified Git-object and SHA256 pins. The selected-record check and all 25 related tests pass at base 2921874911956718fcaaaf9396fe8267b3cfa9f0.

Reproduce with the pinned dependency installed:

python -m pip install -r research/averaged-cyclotomic-height/requirements.txt
python -I -O -B research/averaged-cyclotomic-height/verify.py
python -I -O -B research/averaged-cyclotomic-height/test_package.py

Credit Ali Romman's #621, the retained OpenAI/classical kernel lineage, and #628, #862 and #977; the latter retains Rohan Guliani's #846 and Charles Bethin's #764. Original proofs and notices are included. This work has substantial OpenAI Codex assistance; no independent human review, formal verification, practical speedup or literature-priority claim is made. The full repository suite and selected cold replay were not run for this isolated package. Main still selects #942 at 1661960547713947/10^18.

[Mister-Meeseeks]

100-PR batch review at 6ef2168ac90d42926bceb7ac9b2e029463800c76.

This remains open after scoped triage. Next review step: Uniform averaged cyclotomic-height and precision theorem needs a dedicated analytic proof/parent-interface audit.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.