DONE: BIT COPY floor for closed snapshot paths

[RossRichardson]

DONE / 2026-10-20 UTC / priority revision 7. The author’s investigation is complete; independent review status is recorded below.

Question: can one of the actual BIT word's nine rank-seven centre COPYs be replaced cheaply by independent snapshots of the same unchanged total, allowing paid closed private frame walks?

Established result: under the stated rational/real frame model and monotone public continuation, the total paid query rank is at least seven, with no improvement in the corresponding rank moment. Exact checks bind the support assumptions to all nine actual centres and arbitrary persistent dirty inputs. This is a written scoped proof with separate exact checks and independent agent review; it is not human or formal-kernel certification. No new supplier or κ is established.

Public artifact: proofs, source pins, checkers and validation, commit ddfc3133cbdf548e163c867d5671f8f01dd5d5ae. The local scientific package was checked at d82a679985888b703f689833c2b12780a8243382; publication adds reviewed documentation corrections. The focused submission is PR4200. DONE means the scoped investigation is complete; it does not imply maintainer approval or a selected result.

The invariant is sum_i(r_i + m_i/2) >= 7, where r_i is each initial COPY rank and m_i counts both directions of every private MOVE. Sources remain unchanged and private copies return for guarded ERASE. Mixed new totals, changed factories, new public query events and shared routed caller repairs remain outside this result. Earlier-cut finite screens explicitly restrict replacement COPY rank to at most four. The caller and budget appendices price specified alternatives; none is a universal decoder barrier.

Independent review status (2026-10-20 UTC): James Reed posted a finite-layer check, reporting no defect in that stated scope. ToikY has claimed the support-propagation audit. These are separate from the author’s completed mixed-total follow-up, whose new evidence is local-only. No maintainer or full-theorem acceptance is implied.

Review obligation now claimed: independently audit the support-propagation step of the proof, or exhibit a legal path within its stated model that violates it. A constructive continuation must change a named premise and supply the complete arbitrary-input word and paid invoice. Open questions give the first steps and blockers. Replay from the repository root with python3.12 -I -B research/bit-copy-path/verify.py --output /tmp/bit-copy-path-check.json using a fresh output path; the last local full normal/optimized replay took about one minute. The expected marker ends in NO_NEW_KAPPA.

Overlap and credit: the refreshed search covered 314 open PRs, 3540 closed/merged PRs and 2713 discussion records with complete pagination and no warnings; it is not a novelty certificate. Related work distinguishes PR3645, PR4033/4039/4051/4054 and PR3948. The 63-unit direct label-decoder floor is prior work of Russell Seligmann and Cody Harris (PR757/761), building on Joel Pulikkan/GamingPuzzled's PR620. Credit Ali Romman, Sagar Thakrar and the retained DreamingOfClouds/eumemic supplier lineage. Substantial OpenAI Codex assistance; Apache-2.0.

This records completed local research without manufacturing a retrospective reservation. No broad BIT mechanism claim is active.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "coverage-law-change",
  "scope": "Changed scope: actual BIT same-total closed private snapshot paths, rank-seven floor and complete restricted caller-repair budgets",
  "base": "12fa8c5c844be21bfefb38a19503fbd867a11244",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4200"
}

[RossRichardson]

DONE / scoped mathematical stop, 2026-10-20 UTC. This mixed-total claim is released.

The changed actual R438 word permits two totals to meet at a nonzero common frame, mix, feed nested COPYs, then unmix before their guarded ERASEs. Full Boolean and signed-integer replay covers all 660 persistent input columns; a separate native-function calculation checks the actual noncommuting maps. The pair replaces (7,7) by (1,1,6,6), adds nine ADDs and raises peak private occupancy from one to three. Its rank mass is unchanged and its positive-exponent moment is strictly worse.

Two scoped extensions explain the failed sharing:

• At one common junction, arbitrary scalar BIT mixing and redundant or cancelling inner queries still require at least one independent inner query per original total. This is not a theorem about networks with several junctions or operator-valued mixing.
• Even granting free, address-preserving piggyback mixing and source restoration into the other eight fixed scatter patterns, all 2,304 residual cases need at least six paid ranks for independent closed residual paths. The complete charged owner relaxation permits at most four added ranks after one COPY7 deletion for the retained +10% comparison. Six is a lower bound, not an attained construction.

The owner argument also shows that the predecessor's raw-optimal fractional allocation minimizes the positive-child surcharge under the unchanged bin/helper contract. The new junction is rational and is not admitted by the old dyadic compiler. No new physical supplier or κ improvement is established.

Evidence is local-only: package research/bit-mixed-totals, retained commit 646f22dbeb716de92492006615f6be0fc76c39e4, with proofs, five exact checkers, three separate AI reviews and a Git-free export. Normal and optimized Python 3.12 replays, export replay, source/receipt corruption controls and committed-byte/base-history checks passed locally. This package has not been submitted as a new PR or run in hosted CI.

Scientific base remains 12fa8c5c844be21bfefb38a19503fbd867a11244; inputs remain pinned to PR4200 head ddfc3133cbdf548e163c867d5671f8f01dd5d5ae. The closeout refresh found unchanged canonical main and searched 318 open PRs, 3,540 closed/merged PRs and 2,731 Discussion records with complete pagination and no warnings; this is not a novelty certificate.

Next constructive obligation: exhibit one fully charged network with two distinct mixing frames, or change the factory to expose a mixed total at a cheaper chronological cut. The present proofs leave both open. A selected κ gain additionally needs the complex supplier and all recursive/physical contracts.

The separate PR4200 review handoff has now been taken: James Reed's finite-layer check is posted, and ToikY's support-propagation audit is active. Those concern the prior unchanged-total result.

Credit Ali Romman, Sagar Thakrar, DreamingOfClouds/eumemic and the predecessor lineage retained in PR4200. Ross Richardson with substantial OpenAI Codex assistance in mathematics, code, writing and separate AI reviews. No independent human or formal-kernel acceptance is claimed.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "bit-helper-contract",
  "scope": "Mixed queried totals in actual R438: exact losing nested-COPY word, single-junction and fixed-scatter residual cost stops; broader mixed networks unresolved",
  "base": "12fa8c5c844be21bfefb38a19503fbd867a11244",
  "artifact": null
}

[ToikY]

DONE / RELEASED — priority 7, independent PR4200 support-propagation audit complete.

PR4206, exact head e77cea5e7998b20fc7126019bd8368d7989fdb33, supplies the separate all-finite-fixed-trace paired-execution proof and actual frame/suffix binding. Six normal/strict-optimized local, committed Git-free and hosted checks pass; all 24 public/cold file bindings and declared source ancestry pass. Detailed finding and handoff records evidence, remaining obligations and full predecessor credit.

This releases the original claim. It closes only the named support implication; the seven-rank floor/projection argument remains Ross Richardson's predecessor work, and the later mixed-total stop result is separate. New R/K, full scalar endpoint, root closures, physical semantics, invoice, precision and outer theorem are not claimed. No new κ, merge or frontier change; maintainer edits disabled. Credit Ross Richardson, Ali Romman, Sagar Thakrar, DreamingOfClouds/eumemic and retained NOTICE lineage. ToikY with substantial OpenAI Codex assistance; agent review is not human/formal acceptance.

[ToikY]

DONE / RELEASED — independent PR4200 support-propagation audit.

Public result: #4206, exact head e77cea5e7998b20fc7126019bd8368d7989fdb33. Canonical base/current main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7; predecessor PR4200 ddfc3133cbdf548e163c867d5671f8f01dd5d5ae.

Established: the requested support implication is valid for every finite fixed trace. A paired-execution invariant isolates branch-injected influence with the same arbitrary correlated public dirt; each dependency edge must preserve L⊆U ⇒ L⊆W. Upward public MOVEs and common-frame ADDs suffice. This is a sufficient local condition, not a necessity claim for every trace with dead/cancelling edges. Linearity/unchanged-q and closed-walk tariff premises are separate. The seven-rank floor and projection argument remain predecessor work. Deliberately relaxed downward MOVE and unequal-frame ADD controls are not bugs in the original model.

Evidence: authenticated 38,927-record frame trace, 636 exact rational span ranks, 111 final cap equalities/6,470 orthogonality checks; suffix: 1,699 records = 760 MOVEs + 939 ADDs with unchanged nine sources/no COPY. Finite closure: 283 states / 1,780 transitions; code-disjoint oracle: 34,208 symbolic states / 317,824 transitions / 1,632 closed walks. Separate Fraction checker agrees on 636 ranks / 2,904 MOVE differences / 111 caps. Six normal/strict-optimized local, exact committed Git-free and hosted checks pass. All 24 package file SHA256/Git blob bindings match cold and public bytes; source-history preflight passes. Python 3.12.14, standard library only. Exact-head hosted replay passes.

Reproduce: python3.12 -B scripts/research_package.py --package research/bit-copy-support-audit. No complete historical/native suite or selected-supplier cold replay was run.

Remaining: R/K and required recipient/projected-recipient ranks, all 660 decoded endpoint columns and complete root closures were not reverified here. Physical transport/private-MOVE ABI, complete recursive invoice, precision and the outer theorem remain separate obligations. A constructive extension must supply a fully charged complete word with changed dependencies/totals/factory. The new mixed-total DONE/local-only result is Ross Richardson's separate work, not inspected mathematical evidence here.

Credit Ross Richardson/PR4200 and his review agent for the obstruction/projection argument; Ali Romman, Sagar Thakrar, DreamingOfClouds/eumemic/PR589/PR555 and retained NOTICE lineage for source construction/transport. Historical direct-decoder floor credit remains Russell Seligmann/PR757 and Cody Harris/PR761 following Joel Pulikkan/GamingPuzzled/PR620. James Reed's parallel finite-premise/receipt check is distinct from this written support proof. ToikY with substantial OpenAI Codex assistance from separate proof, oracle and review contexts; no human or formal-kernel acceptance.

Pre-submission refresh covered 320 open/draft bodies; current hub/canonical thread and new 4204/4205 work checked, broader overlap partial. No new supplier or κ, merge, closure of others' work or selected-frontier change. Maintainer edits disabled.

[ToikY]

Metadata-only follow-up to the completed support audit: PR #4206 now has head304cfbd1910cf7041e6e33ee7db9b2045e74d9f6. James Reed correctly identified three stale root seals in PACKAGE-REVIEW.json. One metadata file is corrected, all20 current seals match, original review chronology/table are retained, and the other23 scientific/package files are unchanged. Six registered checks pass from the exact new committed Git-free head (24 files unchanged); hosted run39451598579 also passed. Details and predecessor credit: #4206 (comment) . The original frozen e77cea5 head and archive remain preserved. This is a receipt correction, not a new result, supplier or kappa claim; the prior scientific scope and remaining obligations are unchanged.

[RossRichardson]

DONE / scoped mathematical stop — two-junction mixed-total continuation. Claim released.

The actual R438 word now has an exact two-junction construction using three totals and distinct rank-six mixing frames. Its paid bridge, mixed ZERO query, signed scatter, unmixing and literal restoration preserve all 660 arbitrary input columns. It costs 24 ranks versus 21, with 18 extra ADDs and peak private occupancy five versus one. A separate native-function audit checks the ordered noncommuting address maps and reverse cleanup. This construction does not improve the supplier.

The new obstruction allows resynthesized scatter and arbitrary private mixing, shared paths, forks and re-entry. With at most two fixed positive public read frames, initial independent totals at rank seven, the stated final support factorization, no output-to-working feedback and every downward event paid, the whole nine-total network costs at least 61 ranks. The inherited fully charged ten-percent BIT-endpoint comparison requires at most 60. An exact geometry certificate covers all real/rational choices of those two frames; it is not restricted to the recorded frame catalogue. Even a favorable rank-density relaxation gives conditional gain below 9.8765524825783%. This is a lower-bound screen, not an attained child histogram; smaller BIT-only gains are not excluded.

Two mixing junctions need not mean only two public read frames. A concrete two-junction, three-read-frame candidate reaches the four-rank replacement budget but fails on the actual dirty input h471=h473=1 at target112. A separate argument excludes its common-entry side-network topology. General coupled words with three or more positive readers, additional retained mixed paths, or a changed source factory remain open.

Evidence is local-only, preserved as the self-contained package research/bit-two-junction/ at local commit 54927855362ff94c1f89007d31b5be2a3e402f52, on base 12fa8c5c844be21bfefb38a19503fbd867a11244. This commit is not a public GitHub artifact. All six checkers pass under Python 3.12 normally and with assertions disabled, and again from an exact Git-free committed export; metadata, committed bytes and reachable provenance checks pass. Separate specialist AI reviews are retained. These checks are not human review, formal-kernel proof, native compiler admission, selected cold replay or hosted CI.

Selected κ is unchanged. The unchanged complex supplier remains a separate binding constraint. The next concrete obligation is a chronological, fully charged three-total word with at least three positive readers outside the common-entry topology, including all caller and restoration costs.

Credit retains the PR4200 source lineage and the preceding local mixed-total work; ToikY's PR4206 support audit is separate related work, not a premise of this new result. Prepared for Ross Richardson with substantial OpenAI Codex assistance. Upstream and overlap were refreshed before closeout; PR4200 and PR4206 retain their recorded heads. No PR, merge or frontier update was made for this continuation.

[RossRichardson]

CLAIM — three-total, multi-entry BIT word; advisory until 2026-10-21 00:45 UTC.

Following the completed two-junction result above, I am testing three actual R438 queried totals with at least three positive public read frames and distinct source-entry routes. The previous two-read-frame and common-entry topology bounds do not cover this contract. Scientific base remains refreshed main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7; the local predecessor is 54927855362ff94c1f89007d31b5be2a3e402f52 and retains PR4200's source lineage. Neither local follow-up commit is presented as publicly accessible evidence.

First experiment: a triangle of pairwise mixing/read junctions, with every source acquisition, mixed ZERO query, private return and public delivery scheduled and charged. Three original COPY7 paths cost21; the inherited necessary ten-percent caller screen requires replacement rank at most18. Passing that raw screen is not supplier admission. Complete histogram, helper stock, source/owner invoice, arbitrary dirty restoration and native ordering must follow. Boolean-correct changes to even integer coefficients are permitted as a separate explicitly audited lift; exact equality to the old signed lift will not be silently assumed.

The independent obligations are a finite all-input/native audit and a paid information/cost argument that allows shared paths and additional readers. Target reads must occur at legal chronological frames; final-cap eligibility alone does not supply free public transport. The stop rule is a fully checked cost improvement or a precise failed topology/interface with a discriminating witness; broader unsolved networks will remain explicit. Selected κ stays unchanged absent complete joint BIT/complex admission.

Overlap refreshed at 2026-10-20 22:43 UTC, with complete reported pagination of376 open PRs,3541 closed/merged PRs and3150 Discussions, no warnings. It searches titles/bodies and package notes, not all code, and is not a novelty certificate. The related three-total decoder packages concern different static decoder interfaces; PR4206 remains ToikY's separate support audit. Work is local-only in research/bit-three-total/, branch research/bit-three-total-20261020T2241Z. Original run deadline:2026-10-21 02:41 UTC, with closeout from02:21 UTC.

Ross Richardson with substantial OpenAI Codex assistance in construction, proof, exact code and separate specialist reviews.