Research: reframed two-writer helper-star absorption stays excluded (forced-carry closure; no new kappa)

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Summary

Scoped interface obstruction under the full reframing escape class; no new kappa. Prepared by Aaron Wang, with help from opencode (substantial AI assistance). Apache-2.0. Only research/r439-two-writer-reframing/ is added; certificates/selected-result.json, the selected bit supplier (#589) and upstream/ are unchanged.

On the pinned 11,500-ADD R439 word of #1062, omit one terminal helper-star restoration a ^= b and toggle one donor in each of two existing phase-A writes of a before its first read. #1913 found all 314 exact arbitrary-dirty completions of this class (52 with one fewer scalar ADD) and rejected every one at the unchanged gate frames, explicitly leaving changed gate frames and itineraries open. This package closes that escape for the useful class: with gate frames free to change (arbitrary monotone reframing, including shrinking below the original frames), all 52 scalar-saving completions remain inadmissible. Each has an exact rational leakage witness: a dirty target's forced carried span strictly exceeds its pinned rank-7 t-perp final frame. Of the 262 cost-increasing completions, 260 are excluded the same way; the 2 survivors cost one ADD more than the source word. No completion in the domain saves anything under any reframing.

Changed premise and predecessors

The changed premise is exactly one: the receiving gates' frame assignment, which all predecessors hold fixed.

• Research: exclude two single-donor helper-star absorption corrections under fixed frames #1913 (Adam Gall): the complete two-writer single-donor domain (314 exact completions) and its fixed-frame exclusion; its stated open question, changed gate frames and itineraries, is the object here.
• Research: certify complete multi-donor R439 star absorption linear system and gain-0 chaining (no new kappa) #1937 (Aaron Wang): the complete fixed-gate multi-donor/multi-writer linear system and gain-0 chaining on three words.
• Research: twisted helper contract absorbs three R439 helper-star restorations into A (10,492 -> 10,489 on PR1188's word; no new kappa) #1868 (Sagar Thakrar): the twisted helper contract H2-tw/H4-tw/K-tw, three single-writer absorptions (the positive fixture reproduced here), and the multi-gate-split question.
• Prove a semantic zero-frame D contract for a fixed R439 word #1194 (woklerx): the semantic-D replacement contract H2-sem.
• Word lineage: Finite bit construction: phase-B block programs on the R439 helper-star D, 11,783 -> 11,500 ADDs (no new kappa) #1062 (Sagar Thakrar, the pinned word), Finite bit construction: restored helper stars save 197 R439 ADDs (no new kappa) #967 (Sricharan Vagvala, helper stars), Gareth Davidson Bit supplier: save 11,097 R439 ADDs with guarded target-basis compensation (no new kappa) #915 and Finite bit construction: two-sided R439 compensation saves 1,228 more ADDs beyond PR915 #926 and Finite bit construction: compose R439 A/B/star D to 10,652 ADDs #1122, Ross Richardson Finite bit construction: R439 frame520 uses 29 ADDs, four fewer complete ADDs than PR844 (no new kappa) #923, Aaron Wang Finite bit construction: multi-write shear programs for all ten R439 phase-B blocks (24,305 -> 24,053 ADDs, no new kappa) #1015, DreamingOfClouds Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589, eumemic Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555 and xcheck_p.py, Zhihao Chen κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425.

Method: the forced-carry closure

The law is the pinned eumemic xcheck_p.py Part F nested-frame law: registers start at fixed start frames (sources at their own line, targets and helpers at frame 0); before each gate every operand climbs to the gate's frame, nested with a strict increase of dimension; final frames are pinned (sources and helpers at the full frame, target t at t-perp, rank seven).

Fix any scalar edit of the word. Process the gates in section order and set each gate frame to the span union of the current frames of its touched registers, starting from the fixed start frames. This forced-carry closure is the componentwise least legal frame assignment: every legal reframing's register currents contain the closure's register currents at every point, by induction over the gates. If the closure reaches a final step where a forced carried span is not contained in the pinned final frame, no reframing is legal, because the offending direction sits in every reframing's carried span. If the closure is final-clean, the closure itself is an explicit legal reframing. The closure therefore decides reframing-admissibility exactly, and its witnesses are domain-free.

Result

The domain is re-derived independently and matches #1913 exactly: 236 terminal star edges, 414 eligible writer pairs, 549 first donors per pair (227,286 trials, unique exact second donor), 314 exact completions.

quantity	count
exact arbitrary-dirty completions	314
completions with one fewer scalar ADD	52
completions with one more scalar ADD	262
legal at fixed gate frames (#1913)	0
scalar-saving completions legal under arbitrary monotone reframing	0 of 52
all completions excluded under reframing	312 of 314
closure-legal completions (both cost +1 ADD)	2

Every completion's exact 661-column arbitrary-dirty boundary passes in both orientations through the literal COPY/ERASE stream, so the obstruction is purely geometric. The witnesses have one shape: the added donor's forced join at the edited gate spreads through later gates to a dirty target in 111-221, whose pinned t-perp final frame has rank seven and cannot contain the leaked direction (joined rank becomes eight). Example rows (register, leaked labels, rank final, rank joined):

omitted restoration	toggles	witness
231 ^= 459	A[9] adds 109; A[527] removes 459	target 111 leaks 109, 7 to 8
234 ^= 241	A[12] adds 9; A[247] removes 241	target 112 leaks 9, 7 to 8
506 ^= 374	A[288] removes 374; A[1033] adds 69	target 116 leaks 30,46,54, 7 to 8
482 ^= 540	A[786] removes 540; A[787] adds 417	target 145 leaks 5, 7 to 8

The third row is instructive: it splits Sagar Thakrar's known legal single-writer absorption of 506 ^= 374 across two writers. The one-writer move is legal; the split forces the join into a rank-7 final and dies.

All 314 rows with complete witnesses are in CERTIFICATE.json. The two closure-legal completions omit 444 ^= 470 and insert donor 77 at A[1993] and donor 470 at A[1994] (respectively A[2038]); both toggles are insertions. They cost +1 ADD (11,501 versus 11,500), and their least legal reframing's paid histogram is 1494 at rank 1, 1308 at rank 2, 46 at rank 3, 45 at rank 4, 103 at rank 5, 10 at rank 6, 18 at rank 7 (rank mass 5,129; 3,024 children versus the source word's 2,920). More children at smaller rank means a worse moment bill at every exponent strictly between 0 and 1, by concavity. They are interface evidence that the closure is not vacuous, not savings.

Boundary contract and paid profile

The acceptance gate asks for a full boundary contract and paid profile, not scalar counts alone. All 314 completions and the positive fixture pass the exact 661-column arbitrary-dirty boundary in both orientations (every source holds its own column, every target holds its own target plus source columns, every helper holds its own column). No excluded completion has a legal frame assignment, so there is no paid histogram to bill; the verifier refuses to bill an excluded completion. The pinned source word's complete profile is 1323 children of rank 1, 1342 of rank 2, 79 of rank 3, 44 of rank 4, 102 of rank 5, 11 of rank 6, 19 of rank 7 (rank mass 5,129), with ADDs 11,500 and positive bounds F = 25,717 and B = 5,911,840; the strict 104/192-bit inequalities pass. The single-writer absorption of #1868 (Sagar Thakrar) is reproduced as a positive fixture: it stays legal under both closures, saves one ADD (11,499) and keeps the paid profile identical.

Reproduce

Standard-library Python 3.12, from the repository root:

python3.12 -I -O -B research/r439-two-writer-reframing/verify.py
python3.12 -I -O -B research/r439-two-writer-reframing/test_verify.py
python3.12 scripts/research_package.py --package research/r439-two-writer-reframing

Expected marker and registration output:

PASS_R439_TWO_WRITER_REFRAMING pairs=414 trials=227286 exact=314 saving=52 saving_excluded=52 reframing_excluded=312 legal=2 columns=661 controls=10 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/r439-two-writer-reframing outcome=completed

Default replay is read-only and compares every certificate field; --write regenerates the certificate. Both pinned inputs are checked by SHA-256; no predecessor executable is imported. Ten controls: baseline legality and pinned profile, domain counts, all-314 boundary replays, the 312 witnessed reframing exclusions, the single-writer positive fixture of #1868, a removal-only edit that is geometrically legal but boundary-rejecting (separating the two checks), a wrong-substitute boundary rejection, refusal to bill an excluded completion, source-corruption rejection and a star-domain mutation rejection.

Scope, limits and remaining obligations

• The class is two writers, one donor toggle per writer, one omitted terminal star restoration, on the pinned 11,500-ADD word. Multi-donor sets per writer (the class of Research: certify complete multi-donor R439 star absorption linear system and gain-0 chaining (no new kappa) #1937) are not replayed under free frames here; the closure decides that class too and it is the recorded next question. Post-read propagation and compensation, changed scalars, rebuilt A/B/C and other words are outside scope, as in Research: exclude two single-donor helper-star absorption corrections under fixed frames #1913.
• This is a finite exact exclusion with machine-checked witnesses plus a decision lemma, not a theorem about arbitrary words and not a supplier admission. No new kappa; the selected record is unchanged.
• Not run: native 5/9/11-stage C++/Boost replay, hosted emission/invoice, all-size compiler, precision/tape, analytic transfer, Lean. Physical realization of any reframing is not claimed; the exclusion is at the word level of the inherited nested-frame law.
• The lemma's model: free rational gate frames with pinned starts and finals and the strict nested-climb law. Models with frame descent, reset, clean allocation, free role permutation or new transport are outside, as in the predecessor packages.

Open next questions

1. Multi-donor splits at free frames: does the closure also exclude the two-writer multi-donor splits of Research: certify complete multi-donor R439 star absorption linear system and gain-0 chaining (no new kappa) #1937 once gate frames may change, or does a clean closure for one of its dual-witness targets yield a priced legal reframing? First step: replay its five solved same-path edges and its dual-witness targets through closure_walk.
2. Cheapest legal reframing: when a closure is legal, is its paid profile minimal? Larger joins can merge rank increments and lower the moment bill. First step: on the fixture of Research: twisted helper contract absorbs three R439 helper-star restorations into A (10,492 -> 10,489 on PR1188's word; no new kappa) #1868, enumerate coalescing joins and compare paid histograms.

Overlap and coordination

Claim and scope: #1317 (comment). Pair-collision check of discussions 1317 and 1166 through 2026-10-15 07:52Z found no fresh claim on this object. scripts/find_overlap.py on the refined terms (612 open PRs, 1138 discussion records, complete pagination, 2026-10-15 07:50-07:52Z) found only #1913 and #1937 as fixed-frame neighbors; no open PR or discussion claim covers reframed two-writer absorption. Provisional overlap evidence, not novelty certification.

Contributor: Aaron Wang. Substantial AI assistance from opencode is disclosed; no independent human review is claimed.