[ToikY]

Change and mathematical scope

Adds one fixed 130-row forward/inverse marked-tape program for the released negative-coefficient half branch: for every signed W-bit x=2q+p, W>=2, resident read-only coin u, output S(-x,2,u)=-q-pu=-S(x,2,1-u), retain original parity and return both heads/exteriors. A written all-width proof gives exact worst-case 6W+2 transitions per direction, attained at x=-1,u=1. The new modular negator costs exactly2W+1; credited positive transitions are composed with zero extra transition at each phase join. No global optimality claim.

The supplied uniform six-glyph alphabet, initially clean parity and stationary coin are premises: W+2 visited cells across two tapes, no additional clean work cells. An interior P1 tag is removed. This is not binary-native encoding, marker/coin preparation or a free return-to-entry/repeated-call compiler. Invalid inverse inputs have no rollback promise.

Task and overlap

Canonical main7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision8, selected6030 unchanged. The joint-query-controller card is explicitly CHANGED/partial: local arithmetic interface only; no excluded active-mode schedule, QQ16 or full supplier invoice. Fresh main, live Hub/index, all758open PR bodies/heads and bounded all-state searches found no direct competing negative table; unseen work remains unknown. Advance claim.

Nick Petalas #6801, exact headf24ba360da3950ea92d773b5e429949719be1060, already handles arbitrary signed inputs for the positive coefficient and explicitly releases the negative branch. Its positive TABLE is an executed credited component here, with physical coin read1-u. zxv #6092, head39df190f3a8a04cccb88ef1a6ca9aad39ed0364d, owns both-sign arithmetic/ABI under a different W-clean/unary-program interface. The contribution is this charged negative marked table, not new negative arithmetic.

Validation

Python3.12.14 stdlib, serial; first semantic replay2.442s/21.22MiB. Per invocation:16,376exhaustive negative input/coin pairs W2..12 with both directions;32,752standalone negator inputs plus their involutions;2,032complete inverse-boundary points W2..8 (1,016valid,1,016nonimage);424translated/larger calls throughW4096. Four healthy suites execute4,420,138literal transitions, including1,513,050source-derived positive steps. Eight intended-reason controls distinguish coin, signed minimum, omitted negation, tag release, incoming direction, local collision, inverse sign guard and a payload-correct/exterior-wrong excursion. The separate42control positive-phase steps include mutant rows; they are not all original source instructions.

Final normal/optimized, both registered commands and read-only Git-free empty-PATH exported packets match completely. Official16-member export/input coverage, eight early source-corruption rejections, all15committed blobs/modes/sizes, canonical provenance and foreign source custody pass. Source-aware proof/code/metadata/delivery reviews found no blocker; no blind, human or proof-kernel acceptance is claimed. The original6801checker and6092unary program were not fetched/executed; the credited positive instruction data WAS executed. Original hosted run42270962109, attempt1, passed711helper tests/24skips and both registered finite commands at public headd54a5f449a0e98c11255c635aadb1703d6bc6b50. Both complete5,376-byte hosted packets equal the frozen local packet byte-for-byte. Saved decoded logs retain exact tool-character readback; synthetic checkouta7cbc5dc391def320b3ae3ea446d3d5d59431037 has the exact public tree and canonical/public parents. This is a synthetic test ref, not a repository merge. Hosted CPython3.12.16 corroborates finite execution, not independent mathematical acceptance.

Reproduce: python3 scripts/research_package.py --package research/linear-negative-half-tape; direct normal/optimized isolated commands and exact source pins are in the README. Written quantifiers and resource premises are in PROOF.md; finite tests do not establish them by enumeration.

Credit and remaining obligations

Credit zxv6092/6066, Nick Petalas6801/6026, JoyboyBrian6095, AAAmirinu6244, Ali Romman5599 and retained stochastic/native lineage; distinct even-source dirty ADD work is not transferred. Classical first-one/two's-complement/ripple/local partial-machine methods are identified. Original source notices retained. Substantial source-aware OpenAI Codex proposal, implementation, proof/code review, overlap and metadata assistance; related exposure and pre-first counter/schema corrections disclosed. First receipts and later metadata-only corrections retained separately.

Marker/layout/binary conversion, native MOVE, coin freshness/acquisition/delivery, reached precision, actual caller, repeated-call compiler and complete physical supplier invoice remain open. All five selected assumptions remain; no supplier admission, kappa/frontier change or merge.

Open next questions

README next questions: smallest useful follow-up is an actual caller integration with explicit marker, coin and return-state ownership/cost. The present six-glyph/resident-resource contract blocks physical admission; reducing the6W+2constant is a separate unclaimed optimization.