[petalas]

Summary

Scoped mathematical proof with exact endpoint certificates. Lower ToikY #4029's sufficient many-axis prime-supply cutoff fourfold:

- ell >= 2^32 = 4294967296
+ ell >= 2^30 = 1073741824

Retain 2 <= d <= ell^334, both dyadic target scales, the same prime band and separate halo premise. The new rational logarithm estimate gives an error-log upper bound -2279/2 at the new endpoint; two derivative signs cover every larger width. This is an original parameter improvement using the cited predecessor reduction, not an independent audit. No new kappa or selected-frontier change.

Only five files in research/many-axis-prime-cutoff/ are added. Scientific base a5df48c55ea806d7952b65acc9c76d878ff09b13, priority revision 7. Advance CLAIM. The delivered scope matches the claim; research stopped at the first verified reduction.

Evidence

• Before: Prove source-bound prime supply for the many-axis tensor caller #4029 supplies the source-bound prime theorem at 2^32. After: the new endpoint proof establishes the same prime quota from 2^30 onward. Credit ToikY for the theorem, band, interval estimate, constructor and setup bill; unpaidmoon Review the encoded many-axis production_axis guard. #4142 for the guard/oddness boundary and cutoff question.
• Exact rational endpoint margins, both monotonicity signs and positive exponential-series bounds pass. A second integer calculation verifies 8^8192 > 480 * 2^10065 * 3^8192. Twelve rejection controls pass in normal and optimized Python. The generated stdout receipt binds all five package files; no receipt is handwritten.
• CPython 3.12.15/macOS arm64 direct normal and optimized commands passed in 0.028 and 0.064 seconds. The complete registered replay passed in 0.405 seconds. These are local check durations, not speedup claims.
• Metadata/pins, committed-byte/full-source-history preflight, official export integrity and both isolated read-only Git-free replays pass at ef5136a007437627d140698bdd1fd052489d8692. All six exported files, including the exporter receipt, retain identical hashes. Syntax and function-size inspection pass; the longest function has 24 lines. Ruff was unavailable; no lint pass is claimed.

Exact reproduction commands from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/many-axis-prime-cutoff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/many-axis-prime-cutoff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/many-axis-prime-cutoff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/many-axis-prime-cutoff --check-source-history

Expected markers are PASS_MANY_AXIS_PRIME_CUTOFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 installation may replace the local interpreter path.

Limitations and remaining obligations

Fiori, Kadiri and Swidinsky's explicit prime-counting theorem, Corollary 22 and #4029's interval reduction are imported mathematical premises. Neither is reproved or independently audited. The new proof supplies universal coverage of the smaller domain; finite execution does not prove the analytic input.

The cutoff remains enormous and is not claimed least. No giant prime search, encoded prime producer, complete physical caller, native supplier or multiplication theorem is supplied. Halo/growth constants, numerical tensor/recovery, precision/tape, stock and other supplier obligations remain. Selected metadata orientation passed, but selected scientific cold replay, predecessor scientific suites, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed.

Credit also Jose Manuel #3935, Tekalign #4024, Cinna Davis, Chafik Boukhalfa #1786, Josie Elliston and the retained source08/Harvey-van der Hoeven/Agarwal-Cooley lineage. This contribution was produced by Nick Petalas.

Related open work

The completion refresh examined this account's nine other open PRs created since the advance CLAIM at 2026-10-23 10:55:55 UTC. One studies the same prime band:

• #5152 sharpens the band's product-loss envelope and explicitly derives no prime-supply cutoff. This result lowers the sufficient cutoff and retains the predecessor's product bound. The quantities differ; no competing cutoff or extra numerical cross-check is claimed.

The other eight new account PRs study different objects. Merged #4029 and #4142 remain the cited supply and guard predecessors. Unpublished work and arbitrary unseen diffs remain outside this refresh.

Open next questions

1. Can the original endpoint errors yield a smaller cutoff than the weakened majorant? Begin with rational endpoint-logarithm bounds and a new monotonicity certificate. No further reduction or least cutoff is proved here.
2. Can an encoded producer meet the actual source-tape and head-return contract? Begin with Review the encoded many-axis production_axis guard. #4142's loading/oddness boundary and Prove source-bound prime supply for the many-axis tensor caller #4029's written trial-division algorithm. Literal arithmetic, execution and the complete physical bill remain absent.

CI notes

Both hosted checks passed at ef5136a007437627d140698bdd1fd052489d8692 in Registered research packages run 39943359734: plan in 4m8s and the focused registered package replay in 1m12s. The run is completed with conclusion success, and its head SHA matches the contribution. The registered replay runs both normal and optimized exact verifiers. All local checks listed above passed; no local obligation was deferred to CI. This records automated replay, not independent human or proof-kernel acceptance.

Merge danger

Two-way door. Blast radius: research. One new isolated opt-in package; shared code, frozen predecessors and the selected frontier are unchanged.