[petalas]

Summary

Prove prime supply for subexponentially many axes in ToikY's PR4029 two-scale band. Claim type: scoped mathematical proof with exact endpoint certificates.

original: ell >= 2^32,  2 <= d <= ell^334
new:      ell >= 2^18,  2 <= d <= 2^floor(sqrt(ell)/4)

For either t=2^ell or t=2^(ell-1), the interval (t*(1-1/(3d)), t*(1-1/(4d))] contains at least D=d-1 primes. The new region contains the entire original region for ell>=2^32. At ell=2^32 it also permits d=2^12000, beyond the old cap 2^10688. The separate halo premise remains required.

The new step is a uniform dimension-domain proof using the cited interval reduction. It is not a predecessor audit or a prime-producer implementation. The result was packaged at the first verified improvement. No new kappa or selected-frontier change is claimed. Only six files under research/subexponential-prime-dimension are added.

Credit ToikY / PR4029, head 726c9a3, for the band, caller binding and interval estimate; Fiori, Kadiri and Swidinsky for their imported prime-counting theorem; Jose Manuel / PR3935, Tekalign / PR4024, unpaidmoon / PR4142, Cinna Davis, Chafik Boukhalfa, Josie Elliston and the retained source08, Harvey–van der Hoeven and Agarwal–Cooley lineage. Nick Petalas's PR5149 and PR5152 retain credit for the older polynomial cutoff and product envelope. This contribution was produced by Nick Petalas. Classical exponential and monotonicity methods retain their mathematical credit; no global literature-priority claim is made.

Advance CLAIM. The delivered scope matches the claim. Canonical base 7c49fe3, priority revision 8, selected PR6030.

Evidence

• Before: the cited prime-supply argument requires polynomial dimension. After: PROOF.md establishes the new uniform region, its inclusion of the original domain and a strict enlarged-domain witness.
• Exact error-log endpoint bound -25/2, quota-log margin 326696/5, both derivative signs and the containment margin 7097/10 pass. Four positive exponential partial sums and separate integer-power/exponent endpoint calculations pass. The checker also corroborates 24 dyadic endpoints and rejects nine invalid or unsupported certificate parameters. A rejected lower-cutoff certificate does not prove prime supply fails there.
• CPython 3.12.15/macOS arm64: both warning-strict normal and optimized commands pass. Each registered command was timed locally for timeout sizing only, 0.022 and 0.021 seconds rounded upward, within its 30-second limit. No performance improvement is claimed.
• Registered package replay, metadata/source pins and committed-byte/full-source-history checks pass at df48273. Official export integrity and normal/optimized Git-free replay pass with empty PATH/PYTHONPATH from a read-only export. All seven exported files retain identical hashes. Syntax and function-size inspection pass; the largest function has 28 lines. No static-analyzer pass is claimed.
• The verifier prints a generated receipt bound to all six package files. No receipt is handwritten; no predecessor executable is imported or replayed.

Exact reproduction from the repository root, using only the standard library:

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

Another CPython 3.12 executable may replace the local path. Expected markers are PASS_SUBEXPONENTIAL_PRIME_DIMENSION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

PR4029's interval reduction and Fiori–Kadiri–Swidinsky Corollary 22 are imported mathematical premises. The universal range follows from the written proof, not the finite dyadic probes. The coefficient and cutoff are conservative, with no optimality claim. PR5149's improved polynomial cutoff concerns a different domain comparison.

No giant prime search, encoded producer, complete physical caller, numerical tensor recovery, selected supplier cold replay, Lean or Docker was run. Native operators, source loading, routing, restoration, stock, compiler closure, precision/tape, analytic transfer and all five selected framework assumptions retain their status. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Can the unweakened endpoint errors yield a larger coefficient of sqrt(ell) in log(d), with an explicit cutoff? Start with a uniform endpoint-error and quota comparison; neither optimality nor a least cutoff is supplied.
2. Can a changed complete transform use the expanded region to reduce its dominant paid bill? Start with the actual caller's precision, halo, source-loading and full tensor cost laws. Prime supply alone does not move kappa.

Related open work

• PR6444 reduces a setup algorithm's complexity in the old abundance domain. This contribution changes the analytic dimension region and does not take that PR's encoded-implementation question or extend its complexity claim. No competing setup number is computed.

The overlap snapshot covered 539 fully paginated open PRs and 6,564 Discussion records, supplemented by main/merged object, formula and general-region searches and a fresh pre-commit open/claim check. The helper omitted 8,788 sparse note paths; direct whole-main README searches supplement it. Arbitrary historical diffs and unpublished work remain coverage limits.

The completion refresh screened thirteen other account open PRs created since the advance CLAIM: #6589 through #6594 and #6596 through #6602. Their title/body metadata identifies no additional work on this two-scale prime-supply region or a competing dimension bound. No additional comparison computations were run.

CI notes

Run 41625585653 at head df48273302be23c8912b42a132dccc87a94858ab finished without certifying the contribution. The plan job was cancelled at actions/checkout by the 10-minute job timeout before any repository code ran. The job started at 11:12:20Z and was cancelled at 11:22:24Z; checkout itself ran from 11:12:22Z to 11:22:21Z. Setup-python, the onboarding tests and package selection were all skipped. The dependent replay job was skipped because plan never finished.

The hosted replay has therefore not certified this head. Every local check listed above passed, including both registered commands, metadata/source pins, committed-byte/source-history checks and read-only Git-free export replay. The completed checks were read once with gh pr checks 6595 --repo CrocSwap/integer-mult-bounds; the run's job metadata confirms the cancelled checkout and skipped steps. This was an infrastructure cancellation, not a package-code failure. No retrigger commit was pushed.

Merge danger

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