[kumagi]

Change and mathematical scope

Adds research/crt-two-target-stencil/: a paid producer for two ordered disjoint CRT guard windows, consuming seven protected unary geometry words. Its actual stencil feeds the unchanged dense scanner, and that mask feeds the retained reverse-key repair. This answers PR4644 open question 1: the previous joined producer covered one target, although the scanner and record repair already accepted externally supplied multi-target data.

For every admitted G>=1, H>=2G, K>=G+4, J1>=J0+K, Y>=J1+K, the written proof derives the exact stencil, union-of-failures hole mask, cleanup and producer invoice:

46H+20K+38Y-24G-J0+12J1+12B+438 <= 70L+438,  L=2H+Y+B.

The producer/scanner uses 21 fixed tape roles and at most 8ML+15M+83L+456 read/write/unit-move actions, M=2^L. The caller erases its owned stencil for another 5L+7 actions. Both source functions must omit both active target bits. Borrowed geometry remains unchanged; private words, scanner counter, consumed mask and repair scratch are erased and heads returned.

This is a local interface/cost proof with finite implementation evidence. Existing reverse algebra and repair tariff remain cited; the opaque-record repair model, host key arithmetic and finite-control transitions do not become a unified binary invoice. Caller descriptor construction/transport, variable target count, last-K boundaries and complete supplier admission remain open. No new κ or selected-frontier change; conditional κ=.00302015 remains selected.

Task and overlap

• Card uniform-transfer-audit, refined to the two-target producer boundary. The family decision is narrow: finish this handoff and stop before another implementation expansion.
• Canonical base and final refreshed main: b91956876e5b4122fd89179f1b57557ff8378f15, priority revision 7. All thirteen live inputs are pinned at this base in SOURCES.json; the catalogue and selected record remain historical provenance.
• Canonical claim/result: Discussion4985, with a short pointer in the handoff hub.
• Nearest work: PR4644 supplies the one-target join; PR4208 the one-target paid producer; PR4201 the generic scanner; PR4328 host-mask multi-target repair. Their multi-target consumer/algebra capabilities are not claimed as new.
• PR4904 checks two-target sharp-window macro algebra and excludes tape repair. PR4716 audits current-state key queries; JoyboyBrian's newly published PR4982, head 67154122b2867e6dee5b9b25f2f1c862a3e81e95, lowers a fixed-geometry binary key port and does not supply this stencil producer.
• Final overlap at 2026-10-23T05:35:46Z searched all 6,636 tracked local note paths, 333 open/draft PR bodies and 4,085 Discussion records with complete pagination and no warnings. Targeted recent CRT handoffs/PRs were also read. No competing producer claim appeared; arbitrary diffs and unpublished work are outside that coverage.

Validation

Validated package head: ea2b6fa4d6479813c36ca0c0396c8d06437fb486. Python 3.12.13, standard library only:

python3.12 -B scripts/research_package.py --package research/crt-two-target-stencil --check-only
python3.12 -B scripts/research_package.py --package research/crt-two-target-stencil
python3.12 -B scripts/check_research_commit.py --package research/crt-two-target-stencil --check-source-history

• Registered ordinary and optimized replays pass, source-bound to thirteen inputs and the frozen receipt.
• 972 small and one long producer geometry check interval shape, opaque tape ports, exact actions/visited cells, preservation and cleanup.
• Four complete caller cases exhaust 180,224 addresses and repair 177,408 holes, checking every mask bit, forward/reverse macro, good-set invariant and final payload. Full repair fixtures have G=1; larger G has producer tests and the quantified proof only.
• Nine grammar/arithmetic rejections clean up. Removing the second digit test misses 768 holes; removing the upper guard misses 256. These controls check mask false negatives, not mutated-mask repair endpoints. The paired source choice f0=y[J1], f1=y[J0] is rejected and yields a concrete ideal-map collision.
• A separate read-only agent independently checked 28 geometries, interval endpoints, action/space derivations and cleanup; this is not independent human or formal-kernel review.
• Committed-byte and ancestor-source checks pass for all 21 package/input files. Two committed exports are byte-identical and pass offline integrity checks. A Git-free -I -O -W error -B replay is read-only; source drift and frozen-receipt drift controls reject.

No native/Lean/Docker build, historical suite, full binary repair layer or selected supplier replay was run. Hosted CI is separate from these local results.

Credit

Cinna Davis/#4100: compact caller. ToikY/#4208 and #4201: paid unary helpers and scanner. unpaidmoon/#4644: one-target join and handoff. Ryanson Jonathan/#4328: repair schedule and tariff. Hiroki KUMAZAKI/#4169: arbitrary-source inverse. Douglas Colkitt and John-Michael Smith/#763: inherited compact and exceptional-repair argument. Frozen predecessors remain unchanged.

Prepared by Hiroki KUMAZAKI (@kumagi) with substantial OpenAI Codex assistance in implementation, proof, verification, coordination and writing; Apache-2.0.

Open next questions

README.md and TASK.json give the two bounded follow-ups: identify an actual caller that pays for descriptor construction/transport/restoration, and specify a variable-count descriptor admission contract before implementing it. Related key work should first check PR4982 and Discussion4709. This contribution neither claims those tasks nor authorizes a full-family expansion.