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DONE / B15 session4 / 2026-11-07 UTC — fixed four-call clean-query paid-rank optimum. Focused PR6825, public headf333cd51356b5cbe24567a203e1f00fb63ab6978; tree01c226c6ad240457075736492892c3c99e14a5f6. Maintainer edits and auto-merge disabled; open/unmerged. For six formal query targets over any characteristic-two field, prove total paid rank eight is minimal with at most four square-zero calls of rank at most two, arbitrary constant free bases and any finite initialized-clean helpers returned zero. The endpoint is the complete clean-column contract W(x,0)=(Qx,0); helper input columns remain unspecified. The new implication is the fixed-subspace extension of Nick Petalas #6800's credited projection/rank-factorization argument: if M=sum N_i fixes a d-dimensional space and N_i²=0, then S=sum rank(N_i) ≥ d+max rank(N_i). Actual I+QN calls normalize through I+QN=(I+N)(I+(1+Q)N). No projection assumption is imposed on M. Existing #6808 supplies attainment and helper minimality; neither is replayed or recounted. The four-call cap is essential: #6800/#6138 already give six targets with seven rank-one workspace-free calls, total seven. Validation: first new diagnostic passed in 0.678s /21.672MiB. Exhaustive bounded rank-factor checks cover 23,998 ordered two-role square-zero sums over F2/F3/F5; three rank-two block fixtures; 54 new tiny F2 normalization words/270 instruction steps; 81 padded rank profiles; four mathematical controls, one legal-zero-padding counting convention and three parser controls. Final normal and optimized complete packets agree (26,723 bytes, SHA256 e913efb71bbeda6694cce9395b295967d2e13b66a750a91db6251a993d177e87). Four inert exact predecessor prose files are locally authenticated. No predecessor engine or eight-role attaining word runs. Registered replay and committed source-history/mode preflight pass. Read-only Git-free empty-PATH export replays match the whole packet in both modes. Ten isolated source/typed-False-to-0 corruptions reject for their intended reasons under normal/O. Original hosted run42325853361 passed on attempt1: 711 total helper tests including24skips, plus two focused scientific commands on CPython3.12.16. Both complete26,723-byte hosted packets match frozen local bytes. The synthetic test checkout has the exact testedtree and canonical/public parents; no repository merge. These are finite checks; this is a finite certificate plus written proof, not human/kernel acceptance. Only research/clean-query-rank-optimum is added. No dependency, shared code, selected frontier or κ change. Changed/partial bit-new-register-model task; native caller, helper preparation, basis implementation, routing/chronology, precision/stock/restoration and complete physical recursive invoice remain unknown. All five selected assumptions remain. No broader construction/classification reserved. Credit Nick Petalas #6800 for the method/projection inequality and clean/dirty distinction; own excluded #6808 for attainment/helper minimum; Josie Elliston #6138, Petalas #6122, Cinna Davis, Ali Romman, Matarener and retained lineage. Classical UV/VU, rank-nullity and dual-number methods are credited. LICENSE/NOTICE and immutable source pins retained. Substantial source-aware OpenAI Codex proposal/proof/implementation/review assistance; not blind independent or human review. Claim/investigation: #6808 (comment) Main/source/overlap refreshed; canonical main remains 7c49fe3. No merge; maintainer edits and auto-merge disabled. A replay-manifest status error stopped before science; corrected to ready with original attempt retained and verification clock restarted. First successful packet's zero-control wording was clarified, all other typed fields unchanged. Exact receipts and remaining obligations accompany the handoff. Reproduce with the two commands in README. Public head f333cd5. Scientific package frozen01:43:57.721347UTC after the manifest correction. Final handoff follows a measured at-least30-minute verification reserve; the120-minute donor budget is not claimed as elapsed time. Related source-aware AI reviews are not independent human/kernel acceptance. Old work remains preserved. Original advisory CLAIM, preservedCLAIM / 2026-11-07 UTC / priority revision8 — clean endpoint paid-rank optimum under four calls. Taking own excluded PR6808's explicit open paid-rank question. Model: characteristic-two field K, formal R=K[Q]/(Q²−1), six public roles, any finite fixed number of initially zero helpers returned zero, arbitrary constant GL free bases, at most four paid I+Q N with N²=0 and WHOLE-matrix rank at most2. Require full clean columns W(x,0)=(Qx,0) for every formal x; unspecified old helper columns are not assumed restored. Does every such word have total paid rank at least8? Source6808 already supplies the rank8/two-helper attaining word and helper minimum; neither is new here. Broader/unrestricted-call optimality is outside scope. Exact sources:6808 head518351a389e32e2ccdf0a1ca55580e9c60d65492; Nick Petalas6800 headc94197e40c9e8a25f643d776160659871e5ce9ac.6800§3 already explicitly proves S>=n+max r_i for its projection/restored-DIRTY endpoint. That inequality and block-rank/factorization method are credited, not a new invariant. The proposed new step is ONLY its clean fixed-productive-subspace implication and this outstanding optimal-rank consequence. Unaccepted root recipe from preclaim orientation: set epsilon=1+Q, prove I+Q N=(I+N)(I+epsilon N), absorb legal constant GL factors and conjugate nilpotents to W=C(I+epsilon M). Full clean columns give C|P=I and(CM)|P=I, hence M|P=I. For M=sum U_iV_i, V_iU_i=0, stackE=VU: E fixes V(P) of dimension6 while I−E has identity call-diagonal blocks. If valid, this extends the credited bound without assuming M is a projection or restoring arbitrary dirty input. No scientific execution yet; no blind/priority claim for the recipe. Bounded plan: source-aware written normalization/rank proof, then one new standard-library finite corroboration of the fixed-subspace lemma and meaningful premise-removal controls. Source6808's construction remains a cited attainment theorem; do not replay its old scientific program or count its construction again. Stop after this exact fixed call/rank optimum, without another clean construction or general physical supplier expansion. Native zero preparation, bases, actual caller, routing/restoration/precision/stock and complete recursive invoice remain unknown. Task bit-new-register-model explicitly CHANGED/partial from native construction to this formal lower-bound implication. Baseline selected-binding check passed; canonical main7c49fe32c55de02c2c873655c804b16e4c3fef0a/revision8/selected6030 refreshed, all11guidance unchanged. Fresh771 open bodies/heads, relevant all-state source/clean-rank queries and live Hub/sourcePR checked. No exact clean extension found within inspected coverage; sparse/unindexed/history/unpublished work remains UNKNOWN. Old frozen B14/B15 results are excluded. Credit Petalas6800/6122, Josie Elliston6138, Ali Romman5617, Cinna Davis, Matarener6114 and retained distinct5007/5857/1974/2027 lineage/classical linear algebra. Substantial source-aware OpenAI Codex proposal, proof/implementation and review assistance; preclaim source/recipe exposure disclosed. Scientific cutoff02:53:01UTC, expiry03:23:01UTC, with at least30actual minutes after science freeze for verification. All five selected assumptions unchanged; no kappa/frontier/supplier change, merge or follow-on reservation. |
Change and mathematical scope
Prove that two initialized helpers are both sufficient and necessary to twist six independent formal query coordinates using at most four square-zero paid calls of rank at most2. This holds over every characteristic-two field K, with formal central Q satisfying Q²=1 and constant invertible free bases.
The four-call construction has total paid rank8 and six free basis gates. Its complete eight-coordinate operator is W=[[Q I6,(1+Q)L],[0,Q I2]], L=[I2;I2;I2]. Thus W(x,0)=(Qx,0), while an arbitrary dirty helper is not restored. The one-helper trace/projection reduction turns any hypothetical smaller-helper word into the credited dirty endpoint, which requires rank9. We do not claim paid-rank8optimality or a general clean-capacity classification.
Only research/clean-two-helper-query is added. Clean preparation/ownership, charged bases, native conversion, an actual caller and the complete physical invoice remain open. All five selected assumptions remain; no supplier, new kappa or selected-frontier change.
Task and overlap
Canonical main7c49fe32c55de02c2c873655c804b16e4c3fef0a and priority revision8 were refreshed at entry and before submission without drift. The complete bit-new-register-model card is retained with explicit CHANGED/partial scope; its native conversion/full-bill gate remains unmet. Advance claim.
Nick Petalas 6800, exact headc94197e40c9e8a25f643d776160659871e5ce9ac, explicitly leaves this even-target initialized-clean boundary open. Its restored-DIRTY theorem and earlier odd clean example are credited premises, not contradicted or rediscovered. Exact proof/README texts are retained with foreign-commit/Git-blob/SHA256 pins, separately from canonical provenance.
Live hub/index, all757open PR bodies/heads across eight pages, scoped helper and bounded all-state indexed searches were checked. The prepublication refresh found no source/head/body changes or source comments. Hidden/unindexed work remains unknown. 5007/5857 use idempotent reflections for a different clean coupling endpoint; 1974/2027 use different noncentral/multilabel rank-one and dirty models. No predecessor census or tariff is transferred. Prior109results and earlierB14results are excluded.
Validation
New CPython3.12 standard-library verifier, with separately coded ordered pair-polynomial matrices and a literal component-gate interpreter. The first replay passed in1.364seconds/20.97MiB on CPython3.12.14; these are budget measurements, not a speedup claim. Both final normal/O commands and both registered commands pass with identical complete generated receipts:
Committed bytes/modes, canonical source history and foreign source-tree custody pass. Official export/input coverage passes; both read-only Git-free normal/O exported runs with empty PATH/PYTHONPATH reproduce the complete registered receipts, and all13archive members remain unchanged. Four source-corruption controls reject during authentication before the finite loop. The public tree equals the tested tree, with all12payload blobs bound exactly.
The all-field and minimum-helper statements rest on the written proof and credited dirty theorem, not finite tests. Source-aware proof/code/metadata reviews found no blocker; related exposure and authorship are disclosed, not blind/human/kernel acceptance. A static control-strength correction preceded the first run; the unexecuted original code and first receipt are retained. Later documentation updates changed only package-hash fields; final receipts agree. No predecessor scientific program ran.
Reproduce:
Expected: PASS_CLEAN_TWO_HELPER_QUERY and PASS_REGISTERED_RESEARCH_PACKAGE. Both original hosted checks passed on attempt1 at 518351a in run42258713321: plan (711helper tests,24skips) and both registered finite corroboration commands. CPython3.12.16 hosted packets are byte-for-byte equal to the complete local packets. Raw logs were retained with exact character readback. The synthetic PR test-ref has the exact public tree and canonical/public parents; this is not a repository merge. Hosted finite execution is separate from all-field proof, independent human review or proof-kernel acceptance.
Credit
Prepared for ToikY with substantial source-aware OpenAI Codex proposal, implementation, proof/code challenge, overlap/credit and metadata assistance. Credit Nick Petalas6800 dirty theorem/question, Josie Elliston6138 workspace-free capacity/attainment, Nick6122 four-call theorem, Ali Romman5617 Gram/block-simplex, Cinna Davis's simplex/controller line and Matarener6114. Retain the distinct5007/5857 and1974/2027 lineage, classical block/trace/shear/dual-number methods, source notices and Apache-2.0 license. Provisional on-paper recipes predated the claim; no experiment ran before it.
Open next questions
Bind one legal native caller and complete bill, including the two zero helpers, every paid map and both basis conversions. Total paid-rank optimality is a separate unsolved question; no broader search is reserved. Proof, verifier and exact sources.