Research: exclude two single-donor helper-star absorption corrections under fixed frames

[adamgall]

Exact restricted result

On the pinned PR1062 11,500-ADD R439 word, remove one terminal helper-star restoration a ^= b and toggle exactly one source/helper donor in each of two existing A writes to a, both before its first read. The complete domain contains 236 star edges and 414 eligible writer pairs.

The necessary-and-sufficient arbitrary-dirty compensation equation is r_i(s) XOR r_j(t) = e_b. Invertibility of the original state makes the second donor unique when it exists. Consequently 227,286 first-donor trials cover all 124,780,014 possible donor pairs without sampling.

314 exact dirty completions; 52 save one scalar ADD; zero satisfy the unchanged monotone frame contract. Every exclusion carries an actual source-label vector outside the proposed receiving frame's rational span. The verifier replays every exact completion on all 661 formal columns in both directions, with literal COPY/ERASE checks.

Concrete counterexample: omit 231 ^= 459; add donor109 at A[9]; remove donor459 at A[527]. This has the correct arbitrary-dirty endpoint and 11,499 scalar ADDs, but source109 would move from frame {109} into {4}. Exact rational ranks are one for the receiving frame and two after adjoining line109. Scalar endpoint success therefore cannot establish this proposed interface replacement.

Boundary and accounting

The package does not construct a legal two-writer supplier or claim new kappa. No gate frames, source word, other operations, COPY/ERASE lifetimes, cleanup, starts or final frames are changed. One donor toggle per writer is a material restriction; multi-donor sets, more writers, later compensations, rebuilt words, extra registers and changed frame itineraries are not excluded.

The baseline complete histogram is independently recounted as {1:1323,2:1342,3:79,4:44,5:102,6:11,7:19}, mass 5,129, including all nine COPY ranks. Illegal candidates are not assigned an admissible paid profile. Sagar Thakrar's actual PR1868 single-writer move 506 ^= 374 / A[288]→donor69 is retained as a credited positive control: correct dirty boundary, unchanged complete frame paths/profile, 11,499 ADDs, strict 104/192-bit bounds.

Novelty and provenance

This answers the two-writer / one-toggle-per-writer subcase of #1868's explicit multi-gate-split next question. Its single-writer census and twisted contract are predecessors, not claimed as new. Full bodies of #1868, #1580, #1592 and discussion1743 were inspected. Terminal-restoration/repair obstructions differ from this pre-first-read two-writer scope.

• Research base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• PR1868 targeted live head check: OPEN at 6b222d01fee072bcd3a680e6a96ae7d22e13bcac, unchanged.
• Source word Git blob: 9a17b4920334878d6c949191f6931bd97126fe3c; SHA256: b452d2c2b0b2a546d1105b1da9769aa78e5c54d6f00e97b30c77e3f54ce445ee.
• Exact source/checker pins, original task card, scope and predecessor heads are in SOURCES.json/TASK.json.
• Final cached overlap coverage: 543 open PRs including drafts at 13:23:53Z, 529 discussion records at 13:25:07Z, 2026-10-14; complete pagination. No arbitrary code-diff coverage; closed work through local records only. This is not a novelty guarantee.
• Coordinated scope: Ten-instance independent research session: interface and verifier audits #1884 . This is one focused result from that session.

Only research/r439-two-writer-absorption/ is added. Selected frontier, frozen inputs, upstream and shared workflows/indexes remain unchanged.

Observed verification

macOS arm64, CPython 3.12.12, standard library:

uv run --python 3.12 --no-project python scripts/research_package.py --package research/r439-two-writer-absorption
uv run --python 3.12 --no-project python -I -B research/r439-two-writer-absorption/verify.py

Final registered replay (optimized verifier plus seven focused tests) and separate direct smoke both PASS, 9.05 seconds total wall time:

PASS_R439_TWO_WRITER_ABSORPTION pairs=414 trials=227286 exact=314 scalar_saving=52 legal=0 columns=661 controls=7 no_new_kappa
Ran 7 tests ... OK
PASS_REGISTERED_RESEARCH_PACKAGE research/r439-two-writer-absorption outcome=completed

Seven failure controls include omitted/wrong corrections, erroneously retained restoration, illegal-path billing and source corruption. Tests also demonstrate clean-helper false acceptance and independently enumerate the three-bit identity. Default replay is read-only and compares every certificate field. No full repository suite was run.

No native multistage, hosted supplier, changed precision/tape realization, outer assembly or Lean theorem audit is supplied. The bit side is nonbinding. Local replay is separate from hosted CI and maintainer acceptance.

Open next question and credit

For a chosen writer pair, allow multi-donor correction sets, solve the constrained row-sum equation using only full-itinerary-legal donors, and charge every ADD. No favorable such completion or hosted invoice is presently supplied.

Credit Sagar Thakrar (#1868/#1062), woklerx (#1194), Arham Tahir (#1311/#1580/#1592), Sricharan Vagvala (#967), Gareth Davidson (#915/#926), Aaron Wang (#1015), Ross Richardson (#923), DreamingOfClouds (#589), eumemic (#555/#543 and checker), Zhihao Chen (#425) and the retained interface lineage. No predecessor executable is imported. New material prepared for Adam Gall with substantial OpenAI Codex assistance in mathematics, implementation, investigation, verification and writing. Apache-2.0; no independent human or proof-assistant review claimed.

[Mister-Meeseeks]

Batch review at 6cde2d14f0090f51672feae32a04f3145e70f595.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete414 writer-pair/227286-donor census yields314 exact scalar completions,52 saving an ADD, all rejected by explicit rational frame noncontainment witnesses. Full661-column forward/reverse replay, seven tests and credited1868 positive control pass. Restricted two-writer/single-donor corrections only; no general absorption obstruction.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.