[unpaidmoon]

Change and mathematical scope

Scoped counting proof. No new κ. The selected conditional result stays 60403/20000000 = 0.00302015. PR CrocSwap#4100 is not accepted.

research/crt-uniform-evaluator proves the per-table bound E(p, ell) <= 2^{4 ell}(p+2)^{16} with constant 1, and leaves open a root-axis query whose every logical access scans one shared arena. This package prices that schedule for one axis.

For every integer p >= 2 and 1 <= h <= ell <= p, the axis pays the pinned one-axis CRT bill 65536 q^3 + 1024 q ell^3 + 1024 q^2 (q = p+2) and compiles four rotation networks on the PR CrocSwap#4192 arena (N = 2^h wires, record width w = 4q). Each tape action may also pay one schoolbook product of two w-bit words. The sum is at most 2^{4 ell} q^{16}.

The same charge on all 8q axes exceeds the allowance at p = 2, ell = 2, h = 2: 1241895956864 > 1099511627776. The per-table proof already covers that batch by a different scan.

At the one-axis corner p = 2, ell = 2, h = 2, U = 38809248652 and E = 1099511627776, so 28 U <= E < 29 U.

Reservoir placement, the ordinary callee, sparse repair, native execution and the outer theorem remain open. The arena simulator is not executed.

Task and overlap

• Task: uniform-transfer-audit, scope refined to the shared-arena question left by research/crt-uniform-evaluator.
• Base: canonical main fbc9ea0249746acca4f504d4fda2b84e0e36cf0d, priority revision 7. This fork PR uses cursor/upstream-main-pin-65d8 (that same commit) as its base so the diff is the package only.
• Ordered eligible list, the first still-open successor of each row in the Map independent audit coverage of #4100's obligations (navigation; no new κ) CrocSwap/integer-mult-bounds#4311 shortest unaudited list: (1) the PR2809 continuation-stack space bound, (2) the row gap after alias removal, (3) the row-field ledger substitution, (4) this single-arena CRT evaluator replay, (5) the free-word shadow of the PR Improve conditional κ with a mixed fourth staged entry (+8.696e-9) CrocSwap/integer-mult-bounds#4088 global program. Item 4 is this package.
• PR Audit uniform Beneš compiler with charged sequential arena scans CrocSwap/integer-mult-bounds#4192 (head 519d811cc48693ad9969d6d7994bef3940167005) prices the arena after the permutation is encoded. Its proof is vendored because that commit is not an ancestor of main. This package adds the CRT bill and the explicit schoolbook constant.
• research/crt-uniform-evaluator prices per-table scans and does not run the arena. Fork PR Prove the uniform CRT evaluator bound with constant 1 #70 is that evaluator. Fork PR Join the one-target CRT hole mask with reverse-word repair keys #73 joins a CRT repair mask with reverse-word keys and is a different bill.
• The leading scalar term 8 w^2 A (26 w J) equals (3328/5) A B q^2, the shape of Audit uniform Beneš compiler with charged sequential arena scans CrocSwap/integer-mult-bounds#4192's O((A+J+1) B (p+2)^2) envelope with an explicit coefficient.

Validation

Python 3.12, standard library only, from the repository root:

python3.12 -B scripts/research_package.py --package research/crt-arena-root-axis --check-only
python3.12 -B scripts/research_package.py --package research/crt-arena-root-axis
python3.12 -B scripts/check_research_commit.py --package research/crt-arena-root-axis --check-source-history

All three passed locally. verify.py prints PASS_CRT_ARENA_ROOT_AXIS. It checks the inductions, the vendor SHA256, the corner witnesses 38809248652 and 1099511627776, the 32-axis excess, and a dropped-sentence control. The optimized (-O) replay also passed.

The universal statement is the induction in PROOF.md, not the finite box. The box h = 1..8 only checks that the corner has the largest ratio there.

Not run: the PR CrocSwap#4192 simulator, the retained Beneš compiler, Lean, and Docker.

Credit

Cinna Davis supplies the CRT construction and the shape of E(p, ell). research/crt-uniform-evaluator supplies the one-axis bill, the rotation tariff and the open question. ToikY's PR CrocSwap#4192 supplies the arena census and tape invoice; the vendored proof is unmodified. PR CrocSwap#4311 records the evaluator row this question continues.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Does a word-level scalar inventory bring all 8(p+2) axes inside the same allowance? The schoolbook-per-action majorant does not, at p = 2, ell = 2. First replace 8 w^2 per tape action by the per-record comparisons in PR Audit uniform Beneš compiler with charged sequential arena scans CrocSwap/integer-mult-bounds#4192 §2. Blocked here because that inventory is not this theorem.
2. With one arena axis inside E, do reservoir placement and the ordinary callee still fit? First step is the H >= G comparison named by the evaluator package. Placement remains PR Audit CRT reservoir placement and split-axis coordinate descriptors CrocSwap/integer-mult-bounds#4174.

Workflow feedback (optional)

The CrocSwap#4311 shortest list is the ordered source for this card, and each closed row's open question is the next eligible item. A one-line successor on the map would keep that attachment after the row itself is audited.