Research: certify complete multi-donor R439 star absorption linear system and gain-0 chaining (no new kappa)

[navinor]

Summary

Scoped replacement-interface census and linear-algebraic exclusion; no new $\kappa$. Prepared by Aaron Wang, with help from opencode. Apache-2.0. Only research/r439-multi-donor-star-absorption/ is added; certificates/selected-result.json, the selected bit supplier (#589), and upstream/ are unchanged.

Under the twisted helper contract (H2-tw / H4-tw / K-tw of Sagar Thakrar's PR #1868, head 6b222d01fee072bcd3a680e6a96ae7d22e13bcac), omitting a nonempty subset $L_a$ of terminal D helper-star restorations on a centre $a$ and toggling source/helper donors across existing pre-first-read A writes $W(a)$ restores $a$ before its first read.

This preserves the exact 661-column arbitrary-dirty boundary

$$ (x,y,h)\longmapsto(x,y+x,h) $$

with unchanged literal C if and only if the binary indicator vector $c\in\mathbb{F}_2^{\lvert P(a)\rvert}$ over the frame-legal donor pool

$$ P(a)=\lbrace (i,r):i\in W(a),\ r\text{ legal at }A[i]\rbrace $$

satisfies

$$ \sum_{(i,r)\in P(a)}c_{i,r}\thinspace\mathrm{val}_i[r] =d_a \quad\text{in }\mathbb{F}_2^{661}, $$

where

$$ d_a=\bigoplus_{b\in L_a}e_b. $$

The exact net scalar ADD change is

$$ \Delta\mathrm{ADDs}(c) =-\lvert L_a\rvert+ \sum_{(i,r)\in P(a)}w_{i,r}c_{i,r}, $$

with

$$ w_{i,r}= \begin{cases} -1 & \text{if }r\in S_i,\\ +1 & \text{if }r\notin S_i. \end{cases} $$

Open PR #1868 tested single-writer substitutions and left open multi-gate splits of $S$ and chaining the gain-0 moves. Open PR #1913 (Adam Gall, head 6cde2d14f0090f51672feae32a04f3145e70f595) excluded two-writer single-donor-per-writer corrections ($\lvert D_i\rvert=\lvert D_j\rvert=1$) on the 11,500-ADD word and left open arbitrary multi-donor correction sets over full-itinerary-legal donors.

Exact results across three pinned R439 star words

The package evaluates three pinned candidate words:

• pr967: research/r439-helper-stars-seed19/candidate.json.gz (11,783 ADDs, 221 edges, 157 centres).
• pr1062: research/r439-bsection-stars-seed01/candidate.json.gz (11,500 ADDs, 236 edges, 162 centres).
• pr1122: research/r439-abc-semantic-bridge/inputs/candidate.json.gz (10,652 ADDs, 236 edges, 162 centres).

For each word, the package solves the complete unconstrained-size multi-donor, multi-writer $\mathbb{F}_2$-linear system across all pre-first-read writes, with $1\le\lvert W(a)\rvert\le5$, in two modes:

• same_path: Full-itinerary-legal donors, keeping all 661 register frame paths and paid profile {1:1323, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:19} identical.
• monotone_all: All fixed-gate monotone-climb-legal donors.

Donor-pool census

Word	Mode	Star edges	Two-leaf centres	Pool pairs	Zero-bit $b$	Nonzero-bit unsolvable
pr967 (11,783)	same_path	221	64	1,287	147	69
pr967 (11,783)	monotone_all	221	64	1,744	146	70
pr1062 (11,500)	same_path	236	74	1,412	158	73
pr1062 (11,500)	monotone_all	236	74	1,894	157	74
pr1122 (10,652)	same_path	236	74	1,484	134	97
pr1122 (10,652)	monotone_all	236	74	2,013	133	98

Pool pairs are counted as $\sum_a\lvert P(a)\rvert$.

Solvability and exclusion results

Word	Mode	Solvable single edges	Solvable two-leaf pairs	Extra from chaining	Dual witnesses
pr967 (11,783)	same_path	5	0	0	—
pr967 (11,783)	monotone_all	5	0	0	280
pr1062 (11,500)	same_path	5	0	0	—
pr1062 (11,500)	monotone_all	5	0	0	305
pr1122 (10,652)	same_path	5	0	0	—
pr1122 (10,652)	monotone_all	5	0	0	305

1. 890 canonical dual-hyperplane witnesses

Every unsolvable single-leaf target $e_b$ (216 on pr967, 231 on pr1062, 231 on pr1122) and every simultaneous two-leaf target $e_{b_1}\oplus e_{b_2}$ (64 on pr967, 74 on pr1062, 74 on pr1122) carries an explicit dual vector $u\in\mathbb{F}_2^{661}$ in CERTIFICATE.json.

Each witness satisfies

$$ \langle u,d_a\rangle=1\pmod 2 $$

and

$$ \langle u,\mathrm{val}_i[r]\rangle=0\pmod 2 \qquad \text{for all }(i,r)\in P_{\mathrm{mono}}(a). $$

All witnesses are independently checked by a frozenset oracle in test_package.py.

2. Complete affine solution spaces on the 5 solvable edges

Exhausting all $2^{\dim\ker}$ solutions proves:

Edge	Minimum $\Delta\mathrm{ADDs}$	Result
506 ^= 374	-1	No single-edge multi-donor or multi-writer split reaches -2 ADDs
507 ^= 355	-1	No single-edge multi-donor or multi-writer split reaches -2 ADDs
514 ^= 367	-1	No single-edge multi-donor or multi-writer split reaches -2 ADDs
264 ^= 536	0	Writers [38, 346]
326 ^= 530	0	Writers [98, 334]

3. Why early monotone_all gain-0 moves worsen the paid profile

Under same_path, 264 ^= 536 and 326 ^= 530 have unique solutions:

• 264 ^= 536: A[346] at rank-2 frame 153.
• 326 ^= 530: A[334] at rank-2 frame 147.

Both preserve all 661 frame paths and the paid profile.

Their additional monotone_all solutions at A[38] (rank-1 frame 21) and A[98] (rank-1 frame 51) split a 0 -> 2 climb into 0 -> 1 -> 2.

This changes the paid profile by $-H_2+2H_1$, raising child count from 2,920 to 2,921 at fixed rank mass 5,129.

4. Cross-centre chaining and 5-edge composition

Adjoining all star-leaf vectors

$$ \lbrace e_{b'}:(a',b')\in E\rbrace $$

of any unread centre $a'\in P_{\mathrm{mono}}(a)$, with i < first_read(a'), yields zero additional solvable single-leaf or two-leaf targets. The chained pool adjoins only these original star-leaf vectors; intermediate defect vectors of in-flight multi-write corrections (twist plus partial donor-toggle sums) are generally different vectors and are not modelled, so the exclusion of cross-centre chained absorptions covers star-leaf-vector injection only.

Composing all five same_path edges (composed_all_5) passes:

• The 661-column forward/reverse literal COPY/ERASE boundary.
• H2-tw / H4-tw / K-tw.
• xcheck_p.py.

The five-edge composition has identical ADDs (11,780, 11,497, 10,649), F / B bounds, and paid mass 5,129 as the three-edge composition.

Task card and overlap

• Track / task / scope key: bit-mechanisms / bit-helper-contract / bfb61f829a81535adeae (chooser OS nonce a1c179f07c59db9e5b8720002bb16ab1, priority revision 3, catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075), scope refined.
• Base commit: 0b1cccf86e2f5132ed286a26ef0a681a495740ea (selected-result.json SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32).
• Claim: Discussion #1166, comment 18925672.
• Overlap refresh: scripts/find_overlap.py refreshed at 2026-10-14 14:22–14:24 UTC (583 open PRs including drafts, 563 discussion records, complete pagination) found no competing multi-donor / gain-0 chaining package. Predecessor heads PR #1868 (6b222d01fee072bcd3a680e6a96ae7d22e13bcac) and PR #1913 (6cde2d14f0090f51672feae32a04f3145e70f595) were rechecked unchanged.

Reproduce

python3.12 scripts/research_package.py --package research/r439-multi-donor-star-absorption --check-only
python3.12 scripts/research_package.py --package research/r439-multi-donor-star-absorption

Expected output:

PASS_R439_MULTI_DONOR_STAR_ABSORPTION words=3 edges=221,236,236 pr1062_pools=1412/1894 solvable=5 two_leaf=0 chained_extra=0 duals=305 ADDs=11500->11497 controls=10 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/r439-multi-donor-star-absorption outcome=completed

Remaining obligations and credit

No new $\kappa$ is claimed.

The package does not claim a selected-supplier change, native 5/9/11-stage C++/Boost replay (covered for the three-edge words by PR #1902), hosted emission/invoice, all-size compiler, precision/tape, or analytic transfer.

Edits that cross a read of a twisted centre and simultaneously compensate downstream reader gates remain open and are recorded in README.md and TASK.json. Corrections that consume intermediate defect vectors of in-flight multi-write corrections are likewise not covered.

Credit

• Sagar Thakrar: PR #1868, PR #1062.
• Adam Gall: PR #1913.
• Hanson Sun: PR #1902.
• woklerx: PR #1194.
• Arham Tahir: PR #1311, PR #1580, PR #1592.
• Gareth Davidson: PR #915, PR #926, PR #1122.
• Sricharan Vagvala: PR #967.
• Ross Richardson: PR #923.
• DreamingOfClouds: PR #589.
• eumemic: PR #555, xcheck_p.py.
• Zhihao Chen: PR #425.

[Mister-Meeseeks]

Review follow-up for frozen head 379cddb9a2a0e88a99e6dfa1b035da08c4d1c716.

The 890 duals have an independent positive check, but one advertised negative control is an unconditional throw: run_controls/check_bad_dual raises ValueError without actually testing donor orthogonality. Please run both a valid and corrupted dual through the same real target-parity/orthogonality predicate. Also narrow the exhaustion statement or prove the required superset: the cross-centre pool adds original star-leaf vectors, whereas intermediate defects from multi-write corrections can be different. This is not a counterexample to the 890 individual checked duals.

Recorded scope, attribution and execution evidence. We are leaving this open with the outstanding work recorded; previously completed checks remain credited. Thank you for the contribution.

[navinor]

Corrective follow-up for the review comment at #1937 (comment), on the frozen head 379cddb9a2a0e88a99e6dfa1b035da08c4d1c716. The correction is commit 8b4a4d59 on the same branch research/agent-35-bit-helper-contract-a1c179f0. Both findings were reproduced first with independent exact checks before any claim was edited.

1. Reproduced: the corrupted-dual control was an unconditional throw

Independent checks of run_controls/check_bad_dual at the frozen head:

• AST inspection of the shipped closure body: it references only bad_bits, dmask, need, target_mask, no donor pool at all, and ends with an unconditional raise ValueError("corrupted dual witness rejected"). Donor orthogonality was never tested.
• Driving the real run_controls pipeline with three payloads (a copy of verify.py with only the payload line replaced) records a "rejection" for every one of them, including the valid dual, so the control could not fail:

payload=[47]        -> ['corrupted_dual_hyperplane_witness', 'bad dual fails target parity']
payload=[47, 494]   -> ['corrupted_dual_hyperplane_witness', 'corrupted dual witness rejected']   (the VALID dual)
payload=[0, 1, 2]   -> ['corrupted_dual_hyperplane_witness', 'bad dual fails target parity']      (garbage)

• The chosen witness is {"center": 512, "leaves": [494], "dual_support": [47, 494]} on pr1062. Running the real target-parity/orthogonality predicate over the real P_mono(512) pool (9 donors, writers [300, 1276]) separates them correctly: [47, 494] passes both checks; [47] fails target parity; the parity-preserving corruption [47, 61, 494] fails donor orthogonality on (300, 61).

2. Reproduced: the chained pool models only original star-leaf vectors

The cross-centre construction adjoins exactly the original star-leaf vectors 1 << b', while in-flight multi-write corrections carry different intermediate defect vectors (twist plus partial donor-toggle sums). Concrete, from the complete affine solution spaces:

• pr1122, 507 ^= 355 (writers [290, 844, 848], first read 852): 63 of 64 monotone_all solutions toggle at least two writes. The genuine solution toggling donors at A[290], A[844], A[848] (toggle vectors sum to e_355; full word simulation matches the analytic trajectory) leaves 507 carrying bits {32, 46, 56, 85, 319, 344, 408, 489} on writes 845-847. Its only star leaf is {355}, so this intermediate defect is not a star-leaf vector.
• pr967/pr1062, 506 ^= 374: 32767 of 32768 solutions are multi-write, with intermediate defects such as bits {19, 97, 265} live between A[1033] and A[1034] versus star leaf {374}; 507 ^= 355 gives bits {45, 73, 383} live between A[989] and A[990] versus star leaf {355}.

I also enumerated every solution of every solvable multi-writer centre (both frame modes, all three words) against every consumer write that is a pre-first-read correction write of both the donor centre and the consumer centre with frame-legal donor status: on these three pinned words no consumer write ever observes a live non-leaf defect. So the gap is structural but not instantiated on the pinned data inside the pre-first-read discipline. That enumeration is not part of the package and proves no superset, so per your instruction the claim is demoted to exactly what is checked rather than extended.

3. Old vs new wording of the exhaustion claim

Old (README.md "Why it matters" and TASK.json why_it_matters):

All pre-first-read multi-donor, multi-writer, two-leaf, and cross-centre chained absorptions under fixed frames are exhausted by the five edges certified here; any further twisted-D absorption into A must cross at least one read of the twisted centre.

New (README.md "Why it matters"; TASK.json why_it_matters is worded the same):

Within the checked model — star-leaf defects $d_a = e_b$ or $e_{b_1} \oplus e_{b_2}$ corrected by pre-first-read frame-legal donor toggles at one centre at a time, with cross-centre injection limited to original star-leaf vectors — the five edges certified here are the only solvable absorptions. Intermediate defect vectors from in-flight multi-write corrections at other centres are generally different vectors and are not covered (see Scope limits), so this census does not prove that every further twisted-D absorption into A must cross at least one read of the twisted centre.

Also demoted to what is checked: the README affine-exhaustion sentence now reads "single-edge corrections built from multi-donor and multi-writer splits never reach ..." with an explicit note that multi-centre corrections consuming intermediate defect vectors are outside the census; a new Scope-limits bullet "Cross-centre chaining coverage" records the gap; README finding 4 and the verify.py docstring state that only original star-leaf vectors are adjoined; TASK.json bottleneck_link now says "single-edge multi-donor and multi-writer splits ... within the checked single-centre donor systems"; stop_rule, evidence and VERIFY.json say "cross-centre star-leaf-injection chaining"; the README title and verify.py docstring now say "star-leaf-injection gain-0 chaining census".

4. Repaired control

One real predicate, dual_witness_verdict(dual_bits, target_mask, pool), runs over the actual P_mono(a) donor pool (rebuilt by build_mono_pool): target parity first, then orthogonality to every pool vector, raising at the first failed check. Control 7 runs three duals through it: the certified valid dual, the shipped corruption (drop the last support bit), and a parity-preserving corruption (add a pool-carried non-leaf bit). Recorded entries at the pushed head:

['valid_dual_hyperplane_witness', 'pass: target parity 1; orthogonal to 9 pool donors']
['corrupted_dual_hyperplane_witness', 'dual fails target parity']
['corrupted_dual_orthogonality_witness', 'dual not orthogonal to donor (1276,262)']

So (a) the valid dual passes the real predicate, and (b) the corrupted duals fail the same predicate at specific checks: target parity for the dropped-bit corruption, and donor orthogonality itself for the parity-preserving corruption. No unconditional throw remains. The controls list grows from 8 to 10 entries (9 negative plus 1 positive valid-dual check), and the README expected-output line changes controls=8 to controls=10; test_package.py now asserts the three recorded verdicts.

5. Verification at the pushed head

PASS_R439_MULTI_DONOR_STAR_ABSORPTION words=3 edges=221,236,236 pr1062_pools=1412/1894 solvable=5 two_leaf=0 chained_extra=0 duals=305 ADDs=11500->11497 controls=10 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/r439-multi-donor-star-absorption outcome=completed
Ran 4 tests in 41.210s
OK

A structured diff of CERTIFICATE.json against the frozen-head certificate reports every section except controls identical, so all retained results still reproduce: the 890 dual witnesses and their independent frozenset oracle pass, the five solvable edges with their affine minima, both composed words with F / B bounds, and paid mass 5,129.

6. What remains unproved, and remaining obligations

• No superset proof is given: multi-centre corrections that consume intermediate defect vectors of in-flight multi-write corrections remain outside the checked pools. This is not a counterexample to the 890 individual checked duals, which remain valid as certificates against P_mono(a); what is withdrawn is the exhaustiveness reading only.
• Corrections that cross a read of the twisted centre while compensating downstream reader gates, edits that rebuild B/C, and altered gate frames remain open, as before.
• Unrun obligations unchanged: native 5/9/11-stage C++/Boost replay (covered for the three-edge words by Verification: native replay of PR1868 twisted-helper words (no new kappa) #1902), hosted emission/invoice, all-size compiler/common-label realization, precision/tape rounding, outer analytic transfer. No kappa or frontier claim.

7. PR body

The correction made two things in the PR body wrong, so I edited it via gh pr edit --body-file: the expected-output line controls=8 became controls=10, and the three section-2 rows now read "No single-edge multi-donor or multi-writer split reaches -2 ADDs", with one added sentence under section 4 noting that the chained pool adjoins only original star-leaf vectors.

Prepared by Aaron Wang, with help from opencode.