[ToikY]

Change and mathematical scope

Resolve PR1805's explicit open question 2 for one fixed pair, 277→434, on PR1062's exact 11,500-ADD R439 star word. Three literal splices are fully billed. A naive entry/exit wrap with target-only edits restores helpers but changes 68 target rows (first witness: target111, dirty input434). Correct whole-prefix conjugation costs +9 ADDs; the prepared-middle wrap costs +3. Their local/nine-stage precision bills are 96/182 bits, within the inherited 104/192 guards, and the paid profile/rank mass 5,129 is unchanged. No new κ, cheaper supplier, all-cut/all-pair exclusion or optimality claim.

The new value is the exact composed boundary, cut argument, counterexample and complete invoice. PR1805 already establishes its 65-ADD baseline-pair saving; PR1194 already supplies H2-sem. Those are credited, not claimed anew. The naive case deliberately violates the original constructor's target-only-D precondition; it is not a PR1805 software bug. Exactly seven new files in research/r439-copyshare-star-composition/; selected construction and frontier are untouched.

Task and overlap

Base c4e06b3. Catalogue card bit-helper-contract, bit-mechanisms; scope refined from deriving a replacement lemma to its explicitly open, paid composition obligation. OS nonce 4d62b03e9b7b05873417ecb768fe21f2, scope key 3475f46c8358c6a9211b. Entry 367 and submission 360 open PR bodies, including drafts, paginated through empty pages; repository overlap helper and direct current Discussion1163/1166/active-index reads used. Snapshots are non-atomic and do not cover all diffs, closed history or unpublished work.

Claim: #1166 (comment)

Exact predecessors: PR1805 b34e062; PR1062 cd016e5; PR1194 2e39c98; PR848 5104410; PR1782 1cfe5e1. PR1047 tests a different prepared-middle pair on PR967. PR1868/1937 pre-read absorption and Cycle18's post-read one-writer-per-helper screen are different domains; PR2018 four-omission and PR2020 partial-observation work were checked. All source blobs and hashes are in SOURCES.json. Current-main guidance and selected-result blobs remain unchanged at c676548.

Validation

Python 3.12, standard library only:

python3 -I -B research/r439-copyshare-star-composition/verify.py
python3 -I -O -B research/r439-copyshare-star-composition/test_verify.py
python3 -B scripts/research_package.py --package research/r439-copyshare-star-composition

Registered replay, normal/optimized seven-group suites, metadata/pins and all 24 onboarding tests pass locally. Fresh checks cover all 661 dirty columns in both directions, literal whole-word COPY/ERASE endpoints, all 636 ordered frame ranks and complete positive-envelope invoices. Separate dense GF2 conjugation, set-based column and Fraction rank oracles agree. Controls include omitted wraps/helper conjugations/ERASE, wrong middle correction, nonzero D frame, neutral target→helper feedback, damaged sources/Git blobs/base/receipt and no-overwrite rejection. RESULT is checked against fresh computation.

The exact finite checks and elementary conjugation proof are distinct from physical supplier admission. Root-lattice geometry, chronological realization and all-size common-frame/compiler assumptions are reused predecessor obligations. Full native multistage, hosted/outer assembly, selected-frontier replay and Lean multiplication were not run. Hosted CI and human acceptance are not claimed.

Credit

Victor Li / Wentao Li (PR1805), Aaron Wang (PR848/1782), woklerx (PR1194/1047), Sagar Thakrar (PR1062), Sricharan Vagvala (PR967), Gareth Davidson (PR915/926), Aaron Wang/Ross Richardson (PR1015/923), DreamingOfClouds (PR589), eumemic (PR555/checker), Dylan Crow (exporter) and retained lineage. Prepared for ToikY with substantial OpenAI Codex assistance, including separate source-cut and code review; no independent human review. Apache-2.0.

Open next questions

README lists the exact remaining questions. Smallest useful next task: specify a different legal cut or cancellation rule for this fixed pair, prove its complete dirty boundary, and charge every preparation/restoration. These three cases do not exclude unrestricted optimization. Only a genuinely cheaper legal word would justify native multistage, complete hosted emission/invoice and supplier/outer admission; none is supplied here.