Raw endpoints and projected cleanup: all-dimension range and sharp identity loops

[bitplane]

DONE — PR1420

For every k,q>=1 in the stated scalar XOR alphabet, the raw and projected matrices have complete joint range GL(k+q,2) × {[A B;0 I] : A invertible}. Explicit identity loops attain every projected effect without extra registers. A prescribed elementary projected effect costs exactly five gates, or six when its donor is the sole target, even with additional dirty source/helper registers. The written proof includes a general constructive upper bound.

The complete selected-at-base PR589 witness inserts a five-gate frame-zero identity loop before A. With original C, all 661 dirty registers have the original complete map. With compiled scalar projected cleanup, only helper222 is corrupted by input223. Raw U/J/K, all frame paths, 2,911 MOVEs, nine COPY/ERASE pairs and paid profile {1:1323,2:1342,3:79,4:44,5:102,6:11,7:19} (rank mass 5,129) are retained. ADD counts are 24,305 / 24,310 / 24,311. Both scalar 104/192-bit guards pass. The unchanged owning checker correctly rejects the bad word; this is not a production-checker defect. The compiled projected cleanup is frame-legal but explicitly fails the original literal H4.

Evidence: independent packed-row and literal-permutation engines agree on all 4,716 paired states and 28,248 edges; all 12,630 small projected matrices and 39 larger fixtures through 64 registers are constructed. Every selected-word dirty column is replayed forward and backward with literal COPY/ERASE, and every opcode matches the pinned exporter. Full registered Python 3.12 replay and seven semantic test groups pass; a separate read-only isolated Python 3.14 -O replay has the identical receipt 0afacad9740685e504cedfd9b81d85c702ae53ada78795a3662b4523f5a8e50d. No native 5/9/11-stage, physical ownership/hosting or outer assembly replay was run. Scalar guards do not provide a new packed-precision proof. No new κ or supplier admission is claimed; written proof, not Lean-checked.

Credit: @woklerx PR1105 already identified the underlying feedback-cleanup obstruction, four-block criterion and three-wire census. This adds the all-dimension range, sharp identity-loop costs and complete source-bound witness. PR1194's correct semantic-D theorem and Arham Tahir's PR1311 application are unaffected. Original word/exporter/helper-star lineage and exact predecessor heads are retained in the package.

Assignment: one OS draw 9092ff22c4d18f77e2b829eaae055120, bit-mechanisms / bit-helper-contract, scope key bfb61f829a81535adeae, priority revision 2. The required first example passed. TASK records the changed adjacent scope because PR1194/1311 already cover the original semantic-D question; no reroll. Research base bce54fb; submitted on main cd91ac8 with all four live source bytes unchanged. Mathematical commit dc6ba01; final submission head d19a944. Package: research/projected-cleanup-fibers/ (14 files). Final overlap snapshot covered 842 open PRs and 137 discussion records with no pagination warnings.

Handoff: derive a frame-aware chronological cleanup contract that retains both transfers, or use the small distance tables to study minimum loops for a general projected defect. Physical admission and a general minimum remain open. Reproduce with python3 -I -B scripts/research_package.py --package research/projected-cleanup-fibers.

Prepared for Gareth Davidson (@bitplane) with substantial OpenAI Codex assistance; Apache-2.0. No subagents or independent human review. Claim completed; hosted CI and maintainer review remain separate from local verification.

[an664]

HANDOFF / DONE: the projected helper-SWAP minimum is seven for every finite number of arbitrary dirty helpers, with exactly one target and unit scalar XORs. The full short proof and model boundaries are in the finding.

This credits PR1420's existing three-wire minimum, write-count observation and seven-gate witness. The new extension is the arbitrary-helper bound: at most six raw-identity gates force only the two swapped helpers and target to be written; four kept gates cannot produce a pure swap without a surviving external dirty-input column. The original witness embeds for all k. This does not settle other projected defects or change the selected supplier. Local normal/-O replay and omitted-hypothesis controls passed; universal scope comes from the written proof. Substantial GPT-6 Pro assistance with Codex coordination and local verification; Apache-2.0.

[an664]

HANDOFF / DONE: PR1546 resolves one further prescribed defect: a projected helper three-cycle costs exactly 13 unit scalar XORs for every finite k >= 3, with exactly one dirty target and full raw identity. The scoped completion gives the evidence and limits.

The written rank/write-count/source-only reduction limits a hypothetical length-at-most-twelve word to five registers. Local normal/optimized replay recomputes 1,421,983 radius-six paired states in each of two representations, checks exact agreement and finds empty translated intersections. A separate GPT-6 Pro review corroborated the universal reduction conditionally on that finite premise, without receiving the search code. Its noncommuting translation fixture is included in the final fully replayed checker.

PR1420's range, elementary minima, write-count observation and SWAP witness retain Gareth Davidson / @bitplane credit; @woklerx PR1105 remains earlier lineage. No frame-aware admission, paid supplier or kappa result follows. A bounded next task is formalizing the target-write rank bound; other projected defects remain open. Andrey Mas, substantial GPT-6 Pro assistance through ChatGPT and Codex coordination/local replay; Apache-2.0.

[an664]

HANDOFF / DONE: PR1598 supplies the concrete Lean row certificate requested after PR1546. For every finite legal scalar-XOR word with full raw identity and m > 0 target writes, computed coefficients generate each row of I XOR projected from the last m - 1 actual donor occurrences. Dirty feedback and repeated donors are included.

Local Lean 4.30.0 replay passed all 78 theorem axiom reports, using only propext, Classical.choice, Quot.sound; no sorryAx or custom axiom. The 1,170 external donor/mask cases and semantic fixtures are separate finite checks. The numeric rank inequality is not yet formalized: the next bounded task is a bridge from this explicit certificate to a pinned finite-field rank API. The three-cycle helper reduction/BFS/minimum and physical admission remain outside this formal artifact.

Gareth Davidson / @bitplane's PR1420 and @woklerx's PR1105 retain the raw/projected and feedback lineage credit; PR1546 supplies the written lemma. Andrey Mas with substantial GPT-6 Pro formalization and fixture-code assistance through ChatGPT, and Codex source coordination, execution audit, local kernel replay and packaging. Apache-2.0; no new supplier or kappa claim.

[an664]

HANDOFF / DONE: the finite-image successor extends PR1598's concrete donor certificate to the actual full Boolean matrix action. For every raw-identity XOR word with m > 0 target writes, the image of I XOR projected injects into Fin (2^(m-1)); the zero-write image is exactly {0}. Dirty inputs, feedback and repeated donors are retained.

Local Lean 4.30.0 replay accepts all 43 new and 78 dependency reports using only propext, Classical.choice, Quot.sound. The registered finite checks are separate. An independent Pro semantic review reports no material flaw; its authenticated independent finite checker passes local normal/optimized replay. The complete written review was recovered, authenticated and read locally. The conventional rank/dimension bridge is still a bounded next task; this does not formalize the helper reduction, BFS or physical admission.

PR1546's written lemma and the PR1420 / Gareth Davidson (@bitplane), PR1105 / @woklerx lineage retain credit. Andrey Mas with substantial GPT-6 Pro proof/checker assistance; Codex coordination and local replay. Apache-2.0; no supplier or kappa claim.