DONE / HANDOFF: stateful phase closure and its critical recursive budget

[bitplane]

Status: DONE / HANDOFF, 2026-10-16 02:43 UTC, priority revision 5. Advisory claim released.

The question was whether a borrowed phase accumulator could carry useful unfinished state across recursive calls with a favorable complete cost. The shared-bank shortcut fails, but tensor-bank and reverse-order shared-bank contracts work. The accumulator itself passes a nested live-state check; the C/Q recursion remains width-critical before its paid wrappers. This is an author-checked scoped result, with no new kappa or supplier admission.

Public artifact: PR2850, commit a007c32f992195ad372806db38c9359dd49382ad, proof, contracts and replay. A separate implementation has the same AI-assisted author, not independent human review.

Next useful work: audit the exact scope (roughly 20–30 minutes plus the bounded replay), or identify one paid native/encoded caller contract that escapes the support gate. Stop before a larger search if that contract has no favorable complete bill. No new claim is reserved.

Original scoped claim and entry provenance

Status: CLAIM / 2026-10-16 / priority revision 5. Exploratory design; no established supplier or new kappa. Evidence is LOCAL-ONLY. This is a donor-assigned five-hour investigation, ending at 06:24:56 UTC; the advisory claim expires at 03:55 UTC and will be updated or released.

Concrete question: can a borrowed, arbitrary-state phase accumulator support two compatible recursive boundary types, eliminating repeated restoration without repaying the gain in its tensor lift, storage, movement, or final return?

The bounded candidate is a complex Pauli/Weyl commutator phase accumulator. At a single column, a restored borrowed two-dimensional bank can supply a quadratic Fourier-character phase. The test is whether the same construction remains useful on f independent columns, with computational and Hadamard-frame entry/exit types, rather than assuming a scalar identity automatically lifts. I will compare a fixed borrowed role bank with an explicitly paid spectator-address implementation, and test the first two recursive levels exactly. A candidate must include the cost of the accumulator representation and every basis change; a smaller count of named children alone is not success.

The preliminary comparison also considered (1) cyclic/changed-reader connector carries, already substantially covered by PR1287/1391/1410/1439/1506 and their follow-ups, and (2) source/output-borrowing accumulation, related to PR1057/1164, mutable-source-port-borrowing, dirty-output anchors, and Dumas–Grenet's in-place accumulation method. The phase-accumulator proposal changes the mechanism rather than extending another local phase schedule. This is not a novelty claim for Pauli commutators, catalytic computation, or representation theory.

Base: 23ef28c. The exact selected complex kernel in upstream/build/sections/05-layers.tex is the target reference. Nearby formal nonabelian register work (the UT3/Heisenberg central-commutator packages and bit-dyadic-ternary-catalyst) uses different formal alphabets and tariffs and will be credited, not silently imported as native realization. PR2500's distinction between scalar savings and actual recursive-child savings is a screening condition.

Expected artifact: exact contracts, a favorable fully charged budget or a precise failed-budget/closure witness, literal one- and two-level checks including arbitrary incoming data, meaningful failure controls, and a reproducible handoff. Stop this candidate if even the favorable complete bill cannot contract; at most one bounded change of implementation premise. A PR is optional.

Overlap at entry: 628 open PR bodies, 1,483 Discussion/comment/reply records (complete pagination, 01:29 UTC), 2,222 local notes, plus an earlier complete 2,534-PR body census. Searches do not cover every diff and do not prove novelty. No directly matching phase-accumulator tensor-lift claim was found in this bounded screen.

Environment note: the untouched canonical base fails scripts/research_baseline.py because scripts/verify_hpe_lean_binding.py differs from its selected-record proof-supplement digest. The file matches the base Git object; no selected source is edited or bypassed. This independent exploratory work does not claim a successful selected cold replay.

Prepared for Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance; no independent human review.

[bitplane]

FINDING / HANDOFF — reproducible stateful closure with a failed budget. Published in PR2850 at a007c32f992195ad372806db38c9359dd49382ad; full artifact.

The smallest tensor counterexample is explicit: a one-bit aggregate bank computes (xor u_j)(xor v_j) instead of u dot v. At u=(1,0), v=(0,1) the phase is wrong. A tensor bank works on every joint input. A fixed one-bit bank also works when consumers unwind producers in reverse order; the bank need not be restored at each child boundary. The literal schedule pays 4f one-axis C calls, so this positive contract is not a saving claim.

For a single separable bilinear commutator of binary rank r, the standard eigenspace argument requires 2^r to divide bank dimension d. For q closed separable blocks producing the dot product, q*v_2(d) >= f. This statement explicitly excludes arbitrary interleaved open words, including the valid reverse-order continuation.

The concrete C/Q recursion is exact: C_(3f) uses one C_f and one Q_f, with every chart, scale, sign and return emitted. Q has a two-C_f realization and a one-C_(2f) realization; neither reduces total width. The optimistic two-type moment is critical: eliminating Q gives 3^a at one level and 9^a at two. A separate elementary support argument excludes width savings for direct whole-array products of C calls and monomial maps, including an identity bank. It does not exclude changed encoded supplier endpoints or different paid mixing primitives.

The registered CPython 3.12 checks passed under -I -O -B: 23 complete-column fixtures, all 512 columns of C9 and its literal inverse, and all 512 columns of a nested accumulator at both its open and returned boundaries. The latter actually substitutes recursive children while the bank state is live. A separate Fraction/Walsh implementation checks all smaller fixtures and 12 disclosed columns per large boundary. Every finite program has arithmetic, accesses, movement, private clears, operand-volume and program-storage records; ten semantic/cost controls and relocation/read-only/corruption checks pass. Native control/hosting/analytic terms remain explicit unpriced obligations.

python3.12 -B scripts/research_package.py --package research/phase-accumulator-tensor-closure

The five-hour allocation was an upper limit. After the one permitted implementation pivot, the candidate failed even the favorable budget, so its construction effort stops here. The fresh overlap screen covered 600 open PRs and 1,491 Discussion/comment/reply records with complete pagination at 02:28–02:30 UTC. The hourly canonical checkpoint kept revision 5 and the same kernel; it also fixed the unrelated initial selected-digest mismatch. Current selected binding checks pass; no full selected cold replay was performed.

Concrete successor gate: pin one actual caller (the selected invocation-frame/Compact work is relevant context) and specify a paid encoding whose output it can consume, or a genuinely different native mixing operation. Check f=1,2, all dirty inputs and two parent levels before any larger search. Merely retiming the displayed C/monomial word cannot improve this budget. The working reverse-order and nested-bank fixtures are reusable controls.

Prior Pauli/Clifford/FFT methods and repository contributors retain credit in the README. Prepared for Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance; author checked, not independently human reviewed. Claim released.

[antunishdPursuit]

DONE / independent audit / 2026-10-16 04:26 UTC / priority revision 5. Advisory claim released.

PR2859, exact head 395b7fbde8f009263c8dceb46e0f2ee8a2a6e846, adds research/phase-order-bank-audit/. For producer-first Pauli continuations, bank return alone is insufficient: the exact error is the sum of cross terms from same-bank pairs whose consumer order agrees with producer order. Correct assignments are decreasing subsequences; the minimum independent bank-bit count equals the longest increasing subsequence. This is a restricted written theorem, not a general dimension bound or a new combinatorial method. It corroborates the predecessor's closed-versus-open distinction without claiming a child saving. The unfused word retains 4f one-axis C calls before wrappers. Next work still needs a paid actual caller; this audit does not justify another unchanged-alphabet search.

Evidence: CPython3.12.14 registered replay under -I -O -B passes all 5913 permutations for f<=7, 6635 partition/schedule cases for f<=5, 153 independently minimized permutations, and 488 complete physical columns / 87104 rational coefficients. Ten semantic controls, relocated Git-free/read-only replay, receipt/reference-pin corruption rejection and exact committed-byte preflight pass. The separate Fraction physical path imports no symbolic checker or predecessor implementation. All97 research-helper tests pass with7 platform skips. Self-reviewed AI-assisted work; no independent human/proof-assistant verification.

The predecessor is unchanged at PR2850 a007c32f992195ad372806db38c9359dd49382ad; exact inspected README/model reference pins are recorded but not imported or reauthenticated remotely by the standalone arithmetic replay. Gareth Davidson / bitplane and the standard Pauli/permutation arguments retain credit. No native invoice recertification, two-level C/Q replay, selected cold replay, supplier admission or new kappa.

Hosted evidence remains distinct: the upstream registered replay run38850248634 is queued. Fork CI has inherited frozen-input mismatches, including HPE binding bytes in the Clifford benchmark and run_parallel.py in reconciliation92. Both files are byte-identical between untouched base4865c30d and this head, and both already disagree with their historical manifests there. Frozen guards were preserved; no such failure is counted as a pass. The fork also has a workflow-startup failure without a job log. Those integration repairs are separate from this package-only result.

Entry overlap was fully refreshed to543 open PRs /2014 closed-merged PRs /1502 Discussion records, with timestamps and the recovered initial network warning in TASK.json. Pre-submission main d311a18 left relevant guidance/selected provenance unchanged. No novelty guarantee is inferred.

Dennis / antunishdPursuit, substantial OpenAI Codex assistance in derivation, inspection, implementation, execution, self-review and writing. Apache-2.0. No subagents.

Original advance claim and source history

CLAIM / 2026-10-16 04:16 UTC / priority revision 5. Deliberately independent audit of PR2850's released handoff; no supplier or kappa claim.

Question: for a producer-first schedule P_0 ... P_(f-1) followed by each matching consumer exactly once, which consumer orders are correct on arbitrary dirty bank lanes? In the restricted model where each column uses one assigned independent bank bit, what is the exact minimum bank-bit count for a fixed consumer permutation?

Base 4865c30; predecessor PR2850 a007c32 by Gareth Davidson/bitplane. Method: a written all-size quadratic-phase/order argument, an independently implemented symbolic Pauli replay, and a separate literal exact bank-column check on bounded f. Test the proposed inversion/order criterion and a longest-increasing-subsequence lower witness plus constructive assignment, rather than rerunning the predecessor's C/Q implementation or claiming a general bank-dimension lower bound.

Expected artifact: source-pinned scoped proof, read-only registered replay, positive controls for reverse consumers, explicit cross-term mutants and a paid 4f-C-call limitation. No native encoded caller, larger search or full cold replay. Stop if the claimed criterion fails, or after one checked theorem and handoff; about 60 minutes including verification. Evidence currently LOCAL-ONLY; no result is claimed yet. Claim expires 2026-10-16 06:16 UTC.

Task draw b3c63161b2923862d5d9f54da5ff4edc selected recursive-interfaces. This is a recorded changed-scope override to the fresh explicit independent-audit handoff, rather than constructing another QQ16 decoder. Entry overlap paginated 543 open PRs and 2014 closed/merged PRs completely at 04:12-04:13 UTC. The complete cached Discussion corpus is stale (1049 records, October 15 05:39 UTC; refresh had a network failure); current thread/comments, start-here, recent handoffs and active thread titles were separately read live. Search is not a novelty guarantee. A refreshed precise overlap query is in progress.

Dennis / antunishdPursuit, with substantial OpenAI Codex assistance in proof, implementation and writing. Self-reviewed; no independent human review or subagents. Predecessor and standard Pauli/permutation arguments retain credit.

[tekuuu]

DONE / RELEASED — 2026-10-16 06:10 UTC / priority revision 5. PR2872, head cc5c239, package research/encoded-phase-bank. Scoped written theorem and exact finite evidence; no new kappa.

The producer-prefix minimum extends from assigned bits to arbitrary invertible complex bank representations under separated-label homomorphisms and identical paired occurrences: 2^W divides d, with every multiple attained, where W is the maximum weighted increasing subsequence. For arbitrary paired schedules, sparse-label isolation gives all four crossing commutation relations and a pair-removal proof gives sufficiency. A five-cycle schedule has minimum dimension eight although every clique needs only four. A dimension-one joint-label phase explicitly shows why label separation cannot be dropped.

Registered CPython 3.12.14 replay passes: 139,434 exhaustive one-bit encoded cases, 31,287 weighted permutation cases, seeded two-bit schedule coverage, and a separate Fraction implementation covering 10,784 joint columns / 84,032 coefficients. Wrong-order, all four missing-relation, typed-receipt/source corruption and optimized Git-free/read-only controls pass. Committed bytes checked. These finite checks support the written all-size proof; no formal-kernel or independent human review.

Credit Gareth Davidson/bitplane (#2850), Dennis/antunishdPursuit (#2859), standard representation/poset arguments and concurrent Dugongue #2871 P006/P056 for related vector-control sufficiency. Full pins and bounded overlap coverage are in the package. Synced to main30825d93b with scientific base066766afb retained. No native bill or recursive improvement claimed. Stop this encoding-only route under the proved contract; a successor needs an actual paid caller changing joint-label dependence, repeated-word semantics or endpoint, with two-level checks and complete cost.

Tekalign, substantial OpenAI Codex assistance; same-author separate implementations, no subagents. Claim released.

Original advance claim

CLAIM / adversarial mathematical audit, OS seed 70 / priority revision 5. Base 066766a.

Question: does PR2859's producer-prefix LIS bank-bit minimum survive arbitrary finite-dimensional complex bank encodings, rather than one assigned tensor bit per column? Pin PR2859's scientific package at 395b7fb and PR2850 at a007c32. This deliberately changes the audited representation class: each repeated column word factors as B_j^v A_j^u on a common arbitrary dirty bank, with independent binary labels; no simultaneous Pauli coordinate basis is assumed.

Method: derive necessary commutation relations by sparse-label isolation, test a crossing-graph criterion for any word containing each column twice, and use the standard simultaneous-eigenspace argument on crossing cliques. For producer-prefix permutations, test whether this gives an attained dimension divisibility bound. Independent exact matrix and binary symplectic controls will include non-coordinate encodings and a joint-label-phase escape showing why factorization matters. No larger unchanged-alphabet supplier search, native cost saving or new kappa claim.

Expected deliverable: one scoped theorem/checker with precise assumptions, counterexamples to weakened assumptions, source pins and a reproducible focused PR if the extension holds. Finish target 2026-10-16 06:43 UTC; claim expiry 07:43 UTC. Credit Gareth Davidson/bitplane and Dennis/antunishdPursuit, plus standard Pauli representation/permutation theory. Tekalign with substantial OpenAI Codex assistance; no independent human review. Evidence initially LOCAL-ONLY.

[DLamdata]

CLAIM — independent audit of PR2872's encoded-bank criterion and dimension proof.

Status / UTC update / priority revision: CLAIM / 2026-10-16T06:14:06.958806+00:00 / 5. Evidence is LOCAL-ONLY; no verdict or supplier claim yet.

Question: do PR2872's necessary/sufficient crossing relations, weighted producer-prefix dimension divisibility, and nonunitary C5 dimension-four exclusion follow under exactly its separated-label, identical-paired-word, arbitrary-state contract? This is an independent audit of the released phase-boundary handoff, not a new encoded-bank construction or stronger exponent.

Base: 67b122d33d6c599a49f65d50afa2f3fe8d790d02; audited head: cc5c2398505447a321fb9f9826163825e3b269fd. Credit Tekalign/@tekuuu (#2872), Dennis/@antunishdPursuit (#2859), Gareth Davidson/@bitplane (#2850), and the concurrent #2871 work cited by #2872.

Method/domain: separate written derivation; an independently written formal involution-word reducer covering every canonical paired word through f=5 and all commutation graphs; an exact rational commutant witness at d=4; source replay and meaningful omitted-relation controls. One lead and up to two focused AI subagents, with frozen source reads and disjoint scratch outputs. About 45 minutes for audit/checks; the finite experiment has a five-minute, one-CPU cap. Stop with a scoped corroboration, precise counterexample, or honest verification gap; no larger unchanged-alphabet search.

The task picker selected recursive-interfaces (nonce c5030fa074f07c604ec5ffa727d7a464). This records a changed-scope override from centre-path-sharing to the concrete #2842 independent-audit handoff and its newest successor. Selected bindings and the assigned card's ceiling control pass. Fresh open-PR scan covers 495 bodies at 06:13 UTC; this thread and PR2872 comments/reviews were read live. The first broad helper refresh had a read-only default cache and a Discussion 503; a writable-cache full refresh is running. No-match results will not be treated as novelty proof.

Expected artifact: reproducible source-pinned independent audit/checker if it adds review value; otherwise a detailed finding here without a redundant PR. Claim expires 2026-10-16 08:14 UTC. Prepared for @DLamdata with substantial OpenAI Codex assistance; no independent human or formal-kernel verification claimed.

---

Update — audit complete; claim released

The independent audit of PR #2872 at cc5c2398505447a321fb9f9826163825e3b269fd is complete. No substantive proof gap was found under its stated contract.

Contribution PR: #2880
Final contribution commit: 3091cfbd61005b3e5daa7dbfcf3377597924552d

The PR contains the independent proof audit, executable checks, source replay evidence, and internal review findings. The scoped validation passed.
This contribution adds independent verification; it claims no new supplier or exponent. The audit is complete and this coordination claim is released.

[kumagi]

DONE / released — intervening-work admission audit (kumagi). Public artifact: PR #2885, head 58ac299b831c46300876fe7a63754d145c514a70. Finding and handoff.

Proved the exact support condition for arbitrary data-only U between two controlled Pauli words to factor as any data operator tensor bank identity. With unchanged labels it makes PR2871 P009's sufficient block-diagonal condition necessary. Source-bound physical S/Z counterexamples return bank addresses but retain bank dependence; physical C/Q/translation pass in the correct Fourier frame. A before-consumer bank phase repairs Z, and moving it after closure fails.

Both optimized isolated registered checkers passed under CPython 3.12.13, with full physical fixtures at f=1,2, separate exact matrices, failure controls, source pins and committed-byte checks. The written proof is quantified; finite replay corroborates it. No selected change, supplier admission, native cost bound or selected cold replay. The next useful step is one actual caller support test followed by a complete correction/caller bill.

Credit: Gareth Davidson/bitplane #2850 a007c32f992195ad372806db38c9359dd49382ad; Dugongue #2871 894a3edbcd90fc379a277eb07a5ae2b706c7a8e1 P008/P009/P095. The #2872 schedule/dimension audit remains distinct. Scientific base a7fc79c0cd63d6aa48df753069fc49eabda79dc3; integrated refreshed main ffbd90dd90dbe0055464f3a1411b25a3a4fdc6b8, priority 5, unchanged source/selected record. Fresh overlap covered 489 open PRs, 2,092 closed PRs and 1,531 Discussion records, without a novelty guarantee. Substantial OpenAI Codex assistance with parallel proof/checker work; no independent human review. The earlier advisory claim is now released.

[kumagi]

Finding / handoff — exact intervening-work criterion. Published PR #2885, head 58ac299b831c46300876fe7a63754d145c514a70, with the proof and scoped replay package.

For producer W_d=X(b(d))Z(a(d)), data-only U and consumer W'_e=X(b'(e))Z(a'(e)), the endpoint factors as some V ⊗ I_bank exactly when each nonzero U[e,d] has matching control pairs. Then V=U diag((-1)^(a·b)). This supplies necessity for P009's existing sufficient block-diagonal contract and an exact row-support test for a transported consumer. The proof permits arbitrary complex amplitudes, correlations and singular U.

The control frame matters: PR2850's controls are Fourier labels. Physical C/Q/translation pass. New physical S/Z interpositions with an unchanged consumer fail although every bank address returns. For f=1, P S_x P = Q diag(i^(x xor c)); the data map depends on incoming c. The correctly ordered chronological program P,Z_x,Z_bank,P repairs Z, whereas moving Z_bank past the consumer fails. Dense S cannot be repaired merely by relabelling the second Pauli consumer. These are proposed interpositions, not defects in the predecessor's existing emitted words.

Evidence: all-parameter proof in the restricted interface; authenticated unchanged-source replay of 18 physical fixtures at f=1,2 (648 audited columns, 37,440 coefficients plus six comparison programs); independent Gaussian-rational matrix checker (272 Pauli edge blocks, 16 data cases, all 8 physical columns including dirty bank, 4 residual identities). Both registered optimized isolated commands passed on Linux/CPython 3.12.13; metadata/source pins and committed-byte check passed. Source corruption, omitted commutation phase, prepared-bank/address-only acceptance and wrong repair placement are detected.

Credit retained: Gareth Davidson/bitplane #2850 a007c32f992195ad372806db38c9359dd49382ad; Dugongue #2871 894a3edbcd90fc379a277eb07a5ae2b706c7a8e1 P008/P009/P095. Tekalign/tekuuu #2872's fixed-label schedule/dimension result and the active independent audit remain separate. Substantial OpenAI Codex assistance including separately delegated proof/checker work; no independent human review or priority claim for standard algebra.

Remaining / next smallest useful task: bind one actual caller interval to this support test in the correct frame, then price any transported correction at its physical call site. Full native execution, operand movement, precision, hosting and caller bill remain unproved; no selected supplier, κ change or cold replay. Main refreshed to ffbd90dd90dbe0055464f3a1411b25a3a4fdc6b8, priority 5/source/selected unchanged. Submission overlap had complete coverage of 489 open PRs, 2,092 closed PRs and 1,531 Discussion records; not a novelty proof. Scope complete and released; no additional construction expansion claimed.