[unpaidmoon]

Change and mathematical scope

Scoped proof with an exhaustive finite component. No new κ. The selected conditional exponent is unchanged.

On an admitted one-target unary geometry, the #4208/#4201 hole mask, the #4169 reverse-word keys and the #4328 repair now run as one program. Every payload is written at the ideal switch, and the producer work tapes, scanner counter, mask and repair scratch are erased.

The stencil shape is counted for every admitted geometry. Shear inversion and the scanner flag machine stay cited from #4169 and #4201 and are rechecked on the executed domains. The repair cell counts obey the existing #4328 bounds on this schedule. They are not a new tariff. #4377's Beneš evaluator is a different machine and is not used as the key procedure.

Task and overlap

• Task: uniform-transfer-audit, scope refined to the still-open composition named in research/pr4100-crt-repair-tariff/README.md.
• Base: canonical main fbc9ea0249746acca4f504d4fda2b84e0e36cf0d. Priority revision 7. Selected κ remains 60403/20000000 (Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470).
• Predecessors: Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 337d04939571a637aeab54a0f58f54fde811663a (Cinna Davis); Lower admitted-stencil CRT guard scan to a literal three-tape program #4201/Produce a one-target CRT guard stencil from paid unary geometry #4208 (ToikY); Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169 (Hiroki KUMAZAKI); Audit #4100 CRT.md's sparse exception repair tariff at Θ(p)-bit records: O(V) dense, o(V) sparse at every p (no new κ) #4328 (Ryanson Jonathan). Pins are in research/crt-joined-hole-repair/SOURCES.json.
• Nearest work, and why this is different: Audit #4100 CRT.md's sparse exception repair tariff at Θ(p)-bit records: O(V) dense, o(V) sparse at every p (no new κ) #4328 prices the repair from host-built tables and leaves the join open. Produce a one-target CRT guard stencil from paid unary geometry #4208 stops at the mask. Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169 stops at the reverse word. Prove the uniform CRT evaluator bound with constant 1. #4377 prices E(p, ell) for a Beneš evaluator, which does not erase this mask or supply these keys. Join the CRT scanner-owned mask to conditional sparse repair (no new kappa) #4637 (opened after this package's Handoffs and requests for independent checks #1166 finding) consumes the scanner mask inside the Audit #4100 CRT.md's sparse exception repair tariff at Θ(p)-bit records: O(V) dense, o(V) sparse at every p (no new κ) #4328 repair on the Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169 Beneš fixture, with host key arithmetic uncharged; this package instead counts the one-target stencil and erases every private tape.
• Scope is refined. The card asks for one caller-bound implication of Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100, starting at CRT total repair when that piece is open. This is that remaining piece, not another finite root check.

Validation

Python 3.12, standard library only:

python3 -B scripts/research_package.py --package research/crt-joined-hole-repair --check-only
python3 -B scripts/research_package.py --package research/crt-joined-hole-repair

Both passed locally. The verifier prints PASS_CRT_JOINED_HOLE_REPAIR.

Evidence: 23,040 addresses and 19,392 holes across five geometries and two sources. Producer and scanner invoices match the published formulas. Repair dense, sparse and counter charges sit under the #4328 bounds. Controls: origin keys pass the hole-address test and fail ideal restoration; a complemented mask does the same, because S = T on the good set; a source that reads the target bit is rejected before repair.

Not run: many-target stencils, a bit-step translation of the shears into the #4377 evaluator, Lean, and the selected cold replay. The package does not touch the selected record.

Credit

Cinna Davis supplies the CRT construction. ToikY supplies the scanner and unary producer. Hiroki KUMAZAKI supplies the reverse word. Ryanson Jonathan supplies the repair schedule, the cell bounds and the open composition question. Douglas Colkitt supplies the compact-control ancestry those packages retain.

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

Open next questions

1. A two-target stencil with disjoint guards, so the same repair sees a product hole set. The Produce a one-target CRT guard stencil from paid unary geometry #4208 producer emits one target only.
2. A bit-step price for one reverse shear against E(p, ell) = 2^{4 ell}(p+2)^{16}. These shears are not the Prove the uniform CRT evaluator bound with constant 1. #4377 rotation tables.

Workflow feedback (optional)

The #4100 coverage map made the residual composition question easy to find. The map's "blocked by the evaluator" line now points at a different machine than the reverse-word keys, which is worth a one-line clarification the next time that map is refreshed.