[kumagi]

Change and mathematical scope

The literal exceptional key T(S^-1(q)) queries CRT controls at three current states. Reusing its entry offset can produce the wrong repair key, including when the inverse origin itself is correct and only the final T query is stale. This package answers the caller handoff in #4409 with exact transported CRT witnesses and a quantified cache identity: m+1 fixed modular inverses suffice; subsequent reconstruction needs no inversions. Three query sites are not an inverse-count lower bound.

A conditional seven-term allowance pays for three reconstructions, six conservative networks and aligned validity comparisons. Under the explicitly retained local tariff/operand premises, its sum remains at most 2^(4 ell)(p+2)^16, with the original guard amortization. The maximum corner ratio is exactly 3162627/536870912.

This is a scoped audit/proof and reduced finite caller fixture. It does not price coordinate materialization, complete modular address updates, cache erasure, extraction/sorting or the full tape program. No new κ or selected-frontier change.

Task and overlap

Refined uniform-transfer-audit, priority revision 7, from canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. All eight live inputs are pinned at that base by SHA256 and Git blob; no unmerged dependency is required. Claim: Discussion #4709.

#4377 prices reconstruction from one offset; #4409 explicitly asks about the inverse caller. #4169/#4174 supply the reused inverse/placement. #4650's one-axis arena and #4637's repair-mask bridge leave this caller arithmetic distinct. #4644 joins a one-target unary producer/key/repair program but explicitly leaves reverse-shear bit-step pricing open. The new result binds current-state source queries after actual reservoir transport, not another generic repair proof.

Validation

Python 3.12.13, standard library:

python3.12 -B scripts/research_package.py --package research/crt-repair-query-budget --check-only
python3.12 -B scripts/research_package.py --package research/crt-repair-query-budget
python3.12 -B scripts/check_research_commit.py --package research/crt-repair-query-budget --check-source-history

Passed normal and optimized registered replay, committed-byte/source-history checks, deterministic package export and optimized Git-free direct replay without file changes. Corrupting either a pinned source or the frozen receipt was rejected.

Evidence: 32,768 middle states; all 131,072 dirty word addresses; 114,688 repaired exceptional addresses; two stale-offset wrong-key witnesses; 548 padded fixed-modulus vectors; 7,960 arithmetic grid points. Other controls cover invalid-left disabling, right-prefix omission and noninvertible setup. Finite checks support the separate quantified proof, not a complete multiplication theorem. The reduced fixture uses H=G=1, K=5, below production constants. No selected native/Lean/Docker/full frontier replay was run; selected-record binding baseline passed.

Credit

Cinna Davis (#4100); unpaidmoon (#4377); Ryanson Jonathan (#4409); ToikY (#4192); John-Michael Smith and Douglas Colkitt (#763 repair lineage); retained #4169/#4174 inverse/placement. Prepared by Hiroki KUMAZAKI (@kumagi), with substantial OpenAI Codex assistance in research, implementation, agent review and writing. Apache-2.0; no independent human or proof-assistant acceptance claimed.

Open next questions

Bind the three state epochs to physically copied descriptors in one literal key evaluator, joining the #4650 arena and #4637 mask interfaces, and meter updates/cache restoration before extrapolating. The precise question, first step and dependencies are in TASK.json and the README. The complete tape tariff remains open.