[petalas]

Summary

Scoped counting proof with exact finite corroboration. The complete shared-arena CRT batch already fits the inherited allowance for every p>=3, while retaining the schoolbook product beside every tape action.

B = 8(p+2) U, E = 2^(4 ell)(p+2)^16
B < E for p>=3 and 1<=h<=ell<=p
p=2 exceptions: (ell,h)=(1,1),(2,2)
least uniform B/E bound: 9702312163/8589934592

The sharp constant is attained at (p,ell,h)=(2,2,2). An excessive majorant is not a machine lower bound. This settles the large-parameter part of unpaidmoon's PR4650 batch question without a tighter scalar inventory. The full batch retains the CRT and all four networks for every axis.

Only research/crt-arena-batch-threshold/ is added. Credit unpaidmoon PR4650/PR4377 for the one-axis bill, majorant lemmas, question and tariffs; ToikY PR4192 for the arena census; and Cinna Davis PR4100 for the construction. Their results are cited inputs, not audit deliverables. This contribution was produced by Nick Petalas.

Evidence

Before, PR4650 supplies a universal one-axis bound and one failing full-batch corner. The new proof classifies the full unbounded domain, and its three p=2 cases determine the least uniform constant.

Local CPython 3.12 normal and optimized warning-strict replays pass all 45,759 triples with 2<=p<=64, a separate polynomial bill expansion, 63 height checks, 64 extension steps and eight rejection controls. Each registered check takes about 0.13 seconds. The written inequalities establish the quantified theorem; the grid corroborates it.

Exact reproduction from the repository root:

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

All pass locally. The direct output is PASS_CRT_ARENA_BATCH_THRESHOLD. Package metadata/replay, eight-file committed-byte/source-history preflight, official export/integrity check and a Git-free isolated optimized replay with empty PATH also pass. Changed-source and boolean-valued receipt mutations reject; all nine exported files retain their bytes. RESULT.json is generated and bound to source hashes. Standard library only; another CPython 3.12 executable may replace the local interpreter path.

Limitations

The arena/reconstruction tariffs and maximum batch size are inherited premises. Actual arena execution, tighter scalar inventory, reservoir placement, exceptional repair, native lifting, all-size caller composition and outer multiplication remain separate. No supplier admission, complete multiplication theorem or new kappa is claimed. No Lean, Docker, native or historical heavy replay was run. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Open next questions

• Price the actual per-record scalar inventory at the two excluded p=2 corners. The current excess concerns only a majorant.
• Compose this batch allowance with a complete entry/call/exit program and paid repair-key service, preserving source-state queries, physical movement and restoration. The structured package handoff states the remaining premises.

Related open work

Rechecked this account's open PRs created since the CLAIM. No additional PR studies this full arena-batch bill.

• Audit uniform Beneš compiler with charged sequential arena scans #4192 supplies the shared-arena compiler census inherited through PR4650. This result adds the unbounded full-batch threshold; it does not execute or review that compiler.
• Execute paid CRT key service at the actual repair port #4795 implements a paid fixed-geometry repair-key service. Its physical caller composition with this arithmetic batch theorem remains open.
• Audit conditional CRT last-K two-slice streaming tariff #4187 prices the conditional last-K two-slice branch, which assumes a uniform evaluator. This package concerns the evaluator's full arena batch, not that suffix schedule.

CI notes

Hosted CI passed for 884f96d112e0002c24000b3d89b82fca142e0fc9: run 39860626769. Both plan and registered package replay passed. The run head matches the contribution commit. This is hosted replay evidence, not independent mathematical acceptance; the local checks and limitations above remain applicable.

Merge danger

Reversible package-only addition. Blast radius: this research package and its replay registration. No shared code or selected result is changed.