The proposed first-zero exclusion using only the positive prime-chain
operator bounds is stopped by the existing NS57 control. Changing only
w_2 by -log(2)/100 preserves the original chain bounds for every prime,
yet NS57 supplies a complete nonnegative first-zero window with a nonzero
nullvector. The report gives a two-case all-frequency bound, not an
extrapolation from samples. NS57’s inherited analytic dependencies are
explicit; no first-zero location or new Weil certificate is computed.
Wall check: Known wall for this bounds-only inference; Same open gap for original KERNEL-API and arguments retaining the exact Euler recurrence or arithmetic radical identity. No new research row, node, thaw or version.
From a Git checkout retaining base commit
9db64fdfdc1237005c071f777088291df254ac72, use Python with
mpmath==1.3.0 and Maxima:
.venv/bin/python audits/api-prime-chain-screen-2026-09-29-v1/screen.py --dps 50 --output /tmp/api-chain-50.json
.venv/bin/python audits/api-prime-chain-screen-2026-09-29-v1/screen.py --dps 80 --output /tmp/api-chain-80.json
maxima --very-quiet -b audits/api-prime-chain-screen-2026-09-29-v1/check_identities.mac
pyright --project audits/api-prime-chain-screen-2026-09-29-v1/pyrightconfig.json
Choose fresh output paths; the script refuses to overwrite files. The Davenport–Heilbronn interface is pinned by commit and source hash; its not-applicable result is a hypothesis mismatch, not a passed control. The matched NS57 control preserves the literal prime/gamma/pole family needed by the bounds-only inference, but not the exact Euler recurrence.