[petalas]

Summary

Scoped constructive proof with exact finite corroboration: a fixed marked-tape table computes and reverses the positive-coefficient signed stochastic half for every W>=2 in at most 4W+1 local transitions per direction, with one retained parity bit and no additional clean working cells. Both heads return. The bound is attained by x=-1,u=0. No new kappa is claimed.

forward: scan to marked sign -> shift right -> optional increment -> return
inverse: optional decrement -> shift left with sign check -> return
visited: W value cells + one retained parity cell + one stationary coin cell
program: 33 fixed transition rows per direction, independent of W

This answers zxv's PR #6092 optimized-layout question. Its contract and W-work-cell/O(W^2)-unary-program baseline are cited, not replayed. The new layout explicitly marks the parity and sign cells and supplies a resident read-only coin. Conversion from the predecessor's layout and coin delivery are separate costs. This is a local arithmetic component rather than a complete caller or a selected-assumption discharge.

The contribution was produced by Nick Petalas. Credit zxv #6092/#6066, JoyboyBrian #6095, AAAmirinu #6244, Nick Petalas #6026, Ali Romman #5599 and the retained stochastic/native lineage. Binary shifts, ripple arithmetic and the local reversibility criterion are classical. Only the seven-file research/linear-positive-half-tape/ package is added.

Evidence

Before, the cited kernel needs W clean working cells and a width-dependent unary program. After, PROOF.md supplies an all-width marked-tape construction, its exact signed endpoint/inverse image, local injectivity and time formula. TABLE.json is the literal complete table. The verifier imports no predecessor executable.

Local CPython 3.12 checks passed:

• 16,376 exhaustive signed-input/coin cases at widths 2..12, with 786,356 forward/inverse transitions.
• 328 boundary or translated cases through width 4096, with 500,012 transitions and unchanged exterior spectators.
• The complete inverse boundary through width eight: 1,016 image points accepted and 1,016 nonimage points rejected.
• All table-row local injectivity/direction conditions, exact per-input transition counts and eight rejecting controls.
• Both registered normal/optimized commands, metadata/pins, syntax, committed bytes/source ancestry, official export integrity, and both read-only Git-free exported replays. All eight exported files retained their bytes and modes.

Each registered command was timed locally below one second, against its 120-second limit. These are deadline checks, not a measured execution-speed comparison. The normal and optimized generated receipts agree and bind package hashes. No receipt is hand-written.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/linear-positive-half-tape/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/linear-positive-half-tape/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/linear-positive-half-tape
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/linear-positive-half-tape --check-source-history

Elsewhere substitute a CPython 3.12 executable. Expected markers are PASS_LINEAR_POSITIVE_HALF_TAPE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier writes only stdout, uses the standard library and has no external runtime inputs.

Limitations and unrun obligations

Only the positive coefficient is implemented; signed inputs are arbitrary. The marked sign/parity layout and resident coin are supplied premises. Marker preparation, layout conversion, random generation/delivery, negative coefficients, repeated-call compilation, reached caller precision, other arithmetic, native MOVE and the complete supplier invoice remain open. The table's exact worst-case constant is not asserted optimal among all programs.

The all-width argument is written; the executable bounds replay widths at 4096. No selected cold replay, Lean, Docker, complete rounded multiplier, independent human review or formal proof-kernel acceptance ran. All five selected framework assumptions remain, and the selected frontier is unchanged.

Open next questions

• Construct a fully charged negative-coefficient table with the same space/time boundary, starting with a local two's-complement negation stage.
• Bind one reached positive-half caller to this marked layout and price every conversion, marker, coin access and cleanup operation before inserting the local gain into a complete bill.

Related open work

• Construct the complete stochastic caller with absolute error bounds #6546 gives a complete same-grid stochastic caller under its own retained-record and numerical contracts. This package supplies one local positive-half table, without repricing or replaying that caller.
• Prove a complete caller zero fiber requiring 86N retained bits #6547 studies complete-caller retained information. This local table retains one isolated parity bit and makes no competing whole-caller memory claim.
• Prove a 120N-bit complete-caller retention floor (no κ) #6594 gives a different fixed-seed complete-caller retention floor. Neither that fiber nor its numerical count is extended or recomputed here.
• Review causal fiber certificate and retained capacity #6754 reviews the causal capacity implication behind Prove a 120N-bit complete-caller retention floor (no κ) #6594. This package does not audit that implication or claim a smaller complete record.
• Audit rounded cleanup and public errors of P104 shared preparation (no κ) #6688 studies rounding and cleanup after shared P104 preparation. This package preserves the isolated half map exactly and does not change its rounding law or analyze that prefix.

Merged #6092 is the direct predecessor, #6095 treats its marker interpreter, and #6244 prices a separate reached-coin wrapper. Their layouts and costs are not silently imported into this table.

The completion refresh read the nine other open PRs from this account created since the advance CLAIM: #6795, #6796, #6797, #6798, #6799, #6800, #6802, #6803 and #6804. They concern aggregate counters, balanced dirty-word intervals, cubic pairs, cube lattices, temporal rank chains, square-zero capacity, NTT instructions, singleton returns and owner-route graphs. None studies this signed-half tape table or computes a competing local half value. No additional computation replayed another contribution.

CI notes

Both hosted checks passed at head f24ba360da3950ea92d773b5e429949719be1060: plan in 5m26s and replay (research/linear-positive-half-tape) in 1m26s, in run 42229191941. The completed checks were read once, and the run reports completed success at the matching head. Hosted replay passed both registered normal and optimized finite checks. All listed local checks also passed. No check failed or remains pending. CI corroborates the declared finite replay; the all-width result remains a scoped written proof, without independent human or formal proof-kernel acceptance.

Merge danger

Two-way change, confined to one new opt-in research package. Shared implementation, frozen sources and the selected result are unchanged.

[ToikY]

Follow-up to the released negative-coefficient question: #6809, exact headd54a5f449a0e98c11255c635aadb1703d6bc6b50, adds a fixed130-row marked-tape table for S(-x,2,u)=-q-pu=-S(x,2,1-u), arbitrary signedW-bitx=2q+p. Your exactf24ba360positive TABLE is the executed credited component, with compile-time read-coin relabeling; zxv6092owns the fixed-u ABI/both-sign predecessor. A new first-one modular negator costs2W+1, giving exact worst-case6W+2per successful direction, attainedx=-1,u=1. Original parity/coin, heads and exteriors return; no additional clean work cells in the supplied uniform six-glyph model. The temporaryP1interior tag is removed; binary capacity/marker preparation are not free claims.

Written all-width endpoint/image/local-injectivity/time proof is supplemented by a new literal interpreter:16,376negative input/coin pairsW2..12 with both directions, standalone negator and complete small inverse-boundary suites,424translated/larger calls throughW4096, and8intended-reason controls. Healthy suites execute4,420,138steps including1,513,050source-derived positive steps;42control positive-phase steps include mutant rows separately. Original positive checker and6092unary program were not fetched/run; this does execute your credited program data.

Final registered normal/O and read-only Git-free exports match. Original CI run42270962109, attempt1, passed711helper tests/24skips and both new finite checks; complete hosted packets equal local. No human/proof-kernel acceptance or globally optimal constant is claimed. Native/binary/MOVE, marker/layout conversion, coin acquisition/freshness/delivery, actual caller precision, repeated-call compiler and complete supplier bill remain open. All five selected assumptions remain; no kappa/frontier change or merge.

Thank you for the explicit negative-branch handoff and table/resource proof. Prepared for ToikY with substantial source-aware OpenAI Codex proposal, implementation, proof/code review, overlap and metadata assistance; related exposure and original notices are retained. Claim/handoff: 19445915.