[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. For fixed ambient parameter p and allowed dyadic axis count d, the inherited conservative CRT reservoir envelope fits the printed repair allowance for every accepted precision and width pattern exactly when

c = ceil(log2(p+2))
M = floor(p*2^(64c) / (19200*(p+2)^64))
d <= 6400*(M+1)-1

The new step proves that the maximum batch over the unbounded precision domain is exactly floor(d/6400), with an explicit finite attaining layout. Every larger axis count has an accepted failure witness, so no cutoff in p alone covers unbounded d. At p=131070,d=65536, every width pattern passes through precision 36 and the all-wide layout fails at 37. At p=2^24-2, the largest uniformly covered dyadic count is 2^22.

Only research/reservoir-universal-axis-cutoff/ is added. This contribution was produced by Nick Petalas. Credit Hiroki KUMAZAKI #4174 for the constructor; James Reed #4923/#4925/#4926 for the factor-3 envelope and individual comparisons; unpaidmoon #4646, Ryanson Jonathan #4328, Cinna Davis #4100 and the retained compact-switch/CRT lineage for the repair-room premises. Their results are cited inputs, not audit deliverables. Integer division and monotonicity are standard; no literature-priority claim is made.

Evidence

Before, the retained envelope comparisons treat specific layouts or retain L<=p as a sufficient premise. The new proof optimizes every accepted precision and width pattern at fixed p,d and gives an exact capacity criterion with necessity witnesses.

Local CPython 3.12.15 warning-strict normal, optimized and development-mode replays pass with identical generated receipts. The checks enumerate 74,880 precision rows, 1,152 attaining seams, 393,222 width-count patterns in six boxes, 1,464 dyadic-neighbor probes at 732 ambient parameters, four actual pinned Layout fixtures and fourteen controls. The unbounded theorem is the written proof, not finite extrapolation.

The exact registered normal and optimized commands took 0.1995 and 0.2233 seconds locally, below their 120-second limits. Package metadata and registered replay pass. Committed-byte/source-history preflight passes for all eight package/input files at 75b7bfe3635e593c9f2a4a4a97c039588ce3415c.

The official committed export and declared-input integrity check pass. Normal and optimized warning-strict replays from its read-only Git-free export with empty PATH pass. Changed allocator bytes and a Boolean-to-integer receipt mutation reject under optimization; an existing output is preserved. All nine exported regular files retain their bytes and modes. RESULT.json is generated and bound to package/live-input hashes.

From the repository root:

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

Expected markers: PASS_RESERVOIR_UNIVERSAL_AXIS_CUTOFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 executable may replace the absolute path. Standard library only.

Limitations

This classifies the conservative factor-3 majorant. Its failure is not failure of the actual repair algorithm, its smaller error bound, the real E polynomial or a complete supplier. No new kappa or selected-frontier change is claimed. Actual carry correctness, evaluator and repair execution, physical movement, ownership/restoration, compiler, native lifting and the enclosing multiplication theorem remain separate obligations.

No selected cold replay, historical descriptor census, repair engine, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. The actual finite Layout fixtures are source-bound evidence, not a new audit of their predecessor.

Open next questions

• Optimize the exact published window factor instead of 3, retaining both the batch staircase and the decreasing guard correction.
• Bind a complete caller's reached p,d,ell values and charged evaluator envelope to the new criterion. The physical caller and real E remain missing premises.

Related open work

• Audit conditional CRT last-K two-slice streaming tariff #4187 prices the conditional CRT last-K two-slice branch and retains evaluator/repair integration obligations. This result concerns a conservative reservoir repair-room coefficient and does not execute or certify that branch.
• Audit uniform Beneš compiler with charged sequential arena scans #4192 studies the uniform Beneš compiler and sequential arena preparation cost. This capacity theorem does not change that preparation tariff or supply a complete evaluator.

The current live open-PR and claim searches found no other PR computing this universal axis-capacity criterion. The overlap cache reports complete remote pagination but sparse local-note coverage; distinctive main README and merged-PR searches supplement it. Unpublished work remains outside coverage. The advance claim states the original scope. Canonical main remained 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8, at the publication refresh.

• #6185 implements an arbitrary-bit boundary around a completed CRT/Tensor Sandwich caller. It does not compute this reservoir majorant, axis capacity or precision threshold.

The completion refresh inspected all thirteen other open account PRs created since the CLAIM at 2026-10-29T13:06:54Z, through #6185. None computes the same universal reservoir-envelope quantity; no additional numerical comparison was required or run. The existing #4187 and #4192 relationships above remain unchanged.

CI notes

Both hosted checks passed at exact head 75b7bfe3635e593c9f2a4a4a97c039588ce3415c in run 40880771903: plan and replay (research/reservoir-universal-axis-cutoff). The completed check list was read once; run metadata confirms the completed successful conclusion and the same full head.

Hosted replay certifies the registered normal and optimized package checks at that commit. All recorded local replays, metadata, committed-byte/source-history, official export and read-only Git-free export checks also passed. The written theorem retains its conservative-envelope scope; actual repair, the real evaluator and the enclosing caller remain unrun obligations. Independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

Two-way door. Blast radius: the isolated new research package and its replay registration. No shared source, frozen predecessor, selected result or existing credit line changes.