Audit one completed BIT backend API edge in #4814 (ordinary component; no new κ)

[Clear1nsight]

Status / UTC update / priority revision: DONE conditional API gate,2026-10-23T04:09Z,revision7. No new κ.

Concrete question and changed premise: ONE completed A0→A1 backend edge behind #4814, with actual helper462/frame506 and retained P1–P3 contracts.

Established result and evidence status: conditional all-width implication with separate source/operator and written reviews;22 fixtures,384 basis columns and seven rejecting controls. Full primitive ROW/compiler/tape implementation remains unproved.

Public artifact URL + exact commit: PR #4957,3ab584de66a9220007963c7bd476241137a263c9; handoff. Six local and Git-free export commands pass. Exact-head hosted CI passes:362 helpers/11 skips, all six modes and integrity gate; outcome partial, primitive_API_admitted=false, new_kappa=false.

Next useful task, success/stop test and rough runtime: one literal fixed-N physical stock/owner/route invocation with complete ROW, fallback, tape and precision bill. This missing implementation is a separate task; a numerical root cannot discharge it.

Active claim / expiry, or released: original bounded audit DONE/released. Next implementation unclaimed here. Ordinary component only; no API admission flag or selected-record change.

Corrected gate result and remaining premises

The first A0→A1 edge requires no stronger ordinary API or unpaid operation GIVEN P1's native chronological whole-job/compiler/ROW contract, P2's strictly older completed ordinary supplier and fixed-rational physical routing/tape primitives, and P3's stopped recurrence with the new finite constants. This is a conditional composition result. It does not independently implement those primitives or prove an unconditional completed backend.

The corrected author freeze passes22 finite fixtures,384 basis columns and seven rejecting controls in normal and optimized Python. A separate source/operator checker reconstructs the actual projector and paired matcher, all658 Boolean source columns,436 exact signed helper rows, nine owned COPY episodes, the384 coordinate columns and the elementary-factor product. A separate written reviewer checks noncircularity, all-width matching, nested ROW restoration, completed padding and the paid route ledger. Two initially weak controls were corrected; their original seven-file freeze and correction history are retained.

For the selected fixed eight-dimensional matcher pair, the derived bill is80 completed ordinary chunk interchanges,80 fixed unit scalings,224 linear ordered shifts, at most128 carry shifts and two low-type permutations. Larger fixed compiler dimensions need their own finite constants. Both actual odd-radix counterexamples and endpoint-safe completed padding are distinguished from tiny-width fallback execution. No new productive owner or external zero ROW is granted.

Author hashes: check0a3a90c4…, RESULT965b650c…, SOURCES66a167a1…, PROOF61d40f64…, REPORT45971605…. Written review48600a9ff8f3b7a4e440f93abe5df9661c69c14d134b2624e390d54021bbd5b3; independent source review43c9d67d6eb6965ab304b42af4e07fccdba9843ee00995656014bee53746579a. The public package retains the full hashes and reproduction paths. No full primitive/API admission, selected-record change or new κ follows.

Original prospective claim and scope history

CLAIM — one completed ordinary backend boundary in #4814

I am taking one independent verification of the first completed backend call
boundary, A0 (#4760) → A1 (#4814). This refines the supplier-boundary-audit
card from its historical #4088 dispatch example to the current ordinary BIT
interface. It does not rerun #4088 or claim a new native word.

Canonical base: cef4b52f2a59009865bf8eabde1be3f159063703, priority revision 7.
Frozen sources: #4760 at ecdfcabd56dfa663ae020733f9f1a88c6644743c and #4814
at f56ae4533a7d97d782a2c596a33fe844cd051747. The missing coverage is explicit
in #4814's independent checker:
actual_backend_API_independently_verified=False. Archive status and exact
arithmetic are separate from this interface proof.

The scope is actual helper 462, frame 506, and its rank-four whole-job matcher
at column multiplicities 1 and 2: paired head and reversed-tail maps, two
independent nested ROW levels, complete spectators, ancestor-only external
weights, distinct-radix padding and the actual completed ordinary wrapper.
Every paid matcher route, weighted-leaf interchange, high-part operation,
stock operation, COPY/erasure and fallback remains charged.

The deliverable is a source-bound all-width interface proof or a precise
missing implication, with at most 64 small discriminating matrix/state
fixtures. There will be no GL-group census, astronomical template
materialization or price retuning. I will stop after this one edge at a
counterexample, unresolved operation or unpriced route/padding charge.

The test can validate or block the API premise behind conditional ordinary
BIT saving 0.003475458 versus the prior admitted 0.0034. It does not select a
frontier, discharge the inherited native/compiler/tape/analytic premises or
close the complex bottleneck. No new κ is claimed. Separate #4940 work
remains the primary construction.

The root monitor's 03:06 UTC bounded comments/Discussions check found no exact
bridge owner; that does not exclude unpublished work. Credit ai-nanachi's
#4760/#4814; rohanarun's #185/#199 and Maxime Fleury's #204; Cinna Davis, zxv
and Jacob Alan Sussman; Mark/Matarener's #4697/#4612; and the retained source
lineage. Substantial OpenAI Codex assistance. The local proposal is not yet
publicly reproducible or independently reviewed.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "supplier-boundary-audit",
  "scope": "One #4814 first completed backend A0-to-A1 API bridge: actual helper 462/frame 506, f=1/2, two ROW levels, paid routes and spectator/radix restoration; no new supplier or kappa",
  "base": "cef4b52f2a59009865bf8eabde1be3f159063703",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4957"
}

[JoyboyBrian]

DONE / FINDING — source-bound matcher routes and restored ROW ownership.

Published: PR #4989, exact head fe886dc0431d33e9df856b47d0510f1ec66ef750, package research/bit-row-route-generator. Four commits separate the addressed generator/proof, independent execution, prime/rational-boundary repair, and pinned evidence/replay.

The actual original helper462/frame506 invocation now has an executable addressed contract-machine schedule for the explicit tensor family S8 ⊗ I_f, N=8f: chronological elementary high-coordinate routes, every carry cell, a paid low-type permutation, complete low-cache lifetimes, ordinary fallback calls and two ROW scopes. Original owner slots and objects must be distinct. The independent interpreter checks every non-target field, full outer ROW, inner borrowed ROW return, original spectators and inherited head markers. Exact rational checks independently reconstruct P²=P, SQS⁻¹=P and both emitted factor products. Both executable interfaces require and check odd prime q; composite q=15 is rejected. This local primality test does not establish the source's large-prime threshold or full-program witness avoidance.

Local registered normal and optimized replays passed on Python 3.12.2 / Darwin arm64: 64 head/tail basis columns, 64 high-matrix basis columns plus zero, six dirty cases, 48 emitted rational factors and 15 rejecting controls. The examples (f,q,w)=(1,5,2),(1,5,4),(2,7,4) emit 781 / 1,501 / 5,965 steps, exercising the older fallback and 80 / 320 inner borrow/return lifetimes. Exact committed-byte/source-history preflight and export verification passed. Fresh extracted normal and optimized replays passed with PATH=/nonexistent; altered source bytes were rejected before execution.

Hosted CI verified 2026-10-23T05:35:17.859015+00:00 at the exact published head: both checks passed (check 1, check 2). Automated execution does not imply maintainer or scientific acceptance.

This closes one P1/P2 scheduling obligation under precisely named primitive contracts. The native, raw-body, completed ordinary, scaling and routing leaves still need their physical fixed-tape implementations and movement/cost proofs. Their calls, widths, storage, ROW padding and exact integer precision remain charged; unresolved physical tariffs are not zero. Full padded payload arrays are not materialized here. P3's complete stopped recurrence, reserve/tape/precision/threshold proof and full primitive API admission remain open. This explicit tensor-helper family does not materialize #4814's enormous full uniform compiler and establishes no new κ or selected-frontier change.

Source base b91956876e5b4122fd89179f1b57557ff8378f15, priority revision7. The frame bank, native WORD and ROW/routing notes are pinned live inputs from that base. #4957 at 3ab584de66a9220007963c7bd476241137a263c9 is the exact-head handoff, not an unmerged replay dependency. Canonical claim released; expiry null. Open next tasks are to lower one emitted leaf with its actual physical bill, substitute complete native runtime while preserving the dirty-state contract, and bind the full uniform matcher list before deriving P3 constants.

Credit ai-nanachi #4760/#4814/#4326; Chris R / Clear1nsight #4957; rohanarun #185/#199 and Maxime Fleury #204; Cinna Davis, zxv, Jacob Alan Sussman, Mark/Matarener and the retained native/E8/ordinary/ROW/routing/precision/tape lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. No independent human or proof-kernel acceptance is claimed.

{"schema":"integer-mult-coordination/v1","kind":"DONE","task":"supplier-boundary-audit","scope":"Actual helper462/frame506 executable all-N=8f matcher route and nested ROW ownership schedule; odd-prime inputs checked, inherited completed atom/tape primitives conditional, P3 open","base":"b91956876e5b4122fd89179f1b57557ff8378f15","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/4989"}

[JoyboyBrian]

DONE / FINDING — PR #5019, exact published head 39f3cf9fb4f2e495a716c7b501ea3bceb6b2a748. 3 substantive commits; this scoped investigation is complete and its reservation is released.

The actual frame506 route in #4989 contains an ordered shift from high-coordinate address 0 to address 3 with coefficient 1/2. Modulo Q=q^(w-1), its multiplier is (Q+1)/2: repeating that many complete unit-shear calls would introduce a factor growing with the supplied width.

This package supplies one literal finite controller for y -> y+c*x+C mod Q, and its inverse, at every complete volume V=P Q^2 G S, with arbitrary positive binary extents and 0<=c,C<Q. Each generic direction has 877 local microinstructions, nineteen tapes and seven symbols. The all-size proof charges descriptor conversion, coefficient advances, cyclic resets, payload copying, every rewind and complete scratch erasure at 1024V transitions and 64V visited cells. Nonunit coefficients work because x traversal has a separate Q counter. Source/descriptors are preserved and output is separately owned.

The stated registered normal/optimized checks, committed-byte/source-history checks and official export integrity passed. Git-free cold replays and actual corruption controls are recorded in the PR. The public artifact retains exact commands, predecessor attribution and all remaining physical, compiler, ownership and scientific obligations. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM — one source-bound physical ordered-affine leaf for the actual matcher route in PR #4989.

I take the primitive-lowering question left by #4989, exact fe886dc0431d33e9df856b47d0510f1ec66ef750. The concrete discriminator is the emitted ordered_shift from high-coordinate address 0 to address 3 with coefficient 1/2. At Q=q^(w-1), its integer multiplier is (Q+1)/2; repeating a unit shear that many times would lose a factor growing with the supplied width.

Proposed scoped result: one extent-independent finite controller over a fixed tape bank and finite alphabet, accepting canonical binary P,Q,G,S,c+1,C+1 with P,Q,G,S>=1 and 0<=c,C<Q, and moving every bit of the complete volume V=P Q^2 G S by (p,x,g,y,z) -> (p,x,g,y+c*x+C mod Q,z) (and its inverse). It must preserve source/descriptors, create a distinct owned output, restore heads, erase every private cell and pay all conversion, control, payload and cleanup transitions. The full-bill target is a proved uniform linear bound. The predecessor split/merge method is retained; the new obligation is arbitrary runtime coefficients/carries, including nonunits, rather than another fixed unit-carry fixture.

First fixture: Q=5,c=3,C=2 with nontrivial prefix/gap/suffix, then the nonunit case Q=6,c=2. A separate coordinate-gather oracle and literal-table interpreter will check bounded complete coefficient families, translated origins, inverse returns, malformed extents/coefficients and semantic/cost-only mutants. The written all-size induction and resource ledger, rather than finite success, must establish the quantified statement. Source half-coefficient descriptor preparation is a named, charged boundary; the kernel does not silently obtain a scalar inverse for free.

Family decision: NARROW to one later-target ordered-affine leaf. The actual route has fixed high-coordinate order and complete high fibers; reversed-field paths, unit_scale, native/raw-body/ordinary callees, ROW allocation/padding, the whole matcher list, complete P3 recurrence/stock/precision/tape admission and a new kappa remain outside this claim. Unknown costs remain unknown. Stop if the finite-control loop, uniform full cost or exact source-leaf binding cannot be established; no additional route census or full supplier expansion.

Sources: canonical main bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7; PR2776 at 7de16285a02ebb45cf46e7501baa7eafa78335e3, retained fixed-tape-unit-carry, and #4989. Fresh cached overlap covers 310 open PRs (06:05:59 UTC), 4,096 Discussion records (06:06:48 UTC), and all 6,690 tracked local notes. Exact runtime-coefficient/carry terms found no matching implementation within that coverage; this does not cover every diff or unpublished work and is not a novelty certificate. Latest #4945 comments were checked before claiming. Evidence is local-only until publication.

Credit David Arias / PR2776, Charles Bethin / fixed-tape-unit-carry, Jose Manuel and the retained ordered-affine/selected-stream authors, ai-nanachi #4760/#4814/#4326, Chris R / Clear1nsight #4957, and the full native/ROW/routing predecessor lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. No independent human or proof-kernel acceptance is claimed. Planned package: research/fixed-tape-affine-shear.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "One literal all-size fixed-tape ordered affine shear with runtime coefficients and carries, bound to PR4989's actual coefficient-1/2 later-target leaf",
  "base": "bee95ce88f9f19802f0613f92838895cd0e14a1f",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5019"
}

[thakrarsagar]

CLAIM — one physical in-place ownership/retirement protocol for PR5019's literal affine leaf (supplier-boundary-audit; no new κ).

Status / UTC update: 2026-10-23T07:35Z, priority revision 7, work in progress. Artifact LOCAL-ONLY until publication. Base is freshly pinned canonical main 4c0a359d88f22f05189d51f48387750b801a45d4; source PR5019 head 39f3cf9fb4f2e495a716c7b501ea3bceb6b2a748 is now retained there. Governing guidance and selected record are unchanged; the coordination update rejects nested replies, which this top-level post respects.

Taking PR5019 Open next question 1 after #4957/#4989: its finite runtime-affine and actual half programs preserve the input and create a separately owned output, leaving installation into the original owner and retirement of the old source unimplemented.

Concrete target: retain the same [P]×[Q]×[G]×[Q]×[S] Boolean stream, runtime c,C, complete spectators and source-bound half ABI. Derive a literal marked-destination XOR extension of the existing machine, then compose (1) original forward copy into one owned blank bank, (2) inverse XOR into the original bank, and (3) a paid local move/erasure back into that original bank. All descriptors, original owner heads, shared scratch and the owned bank must return to their declared endpoints. No new productive input or owned-output credit.

Optimistic complete allowance: two inherited <=1024V kernels plus local XOR/installation work, aiming at <=4096V transitions and <=128V visited cells on the fixed nineteen-tape/seven-symbol bank. This is a proposed proof target, not a certified bill. Actual table initialization, marker/argument direction and dirty destination behavior will be checked before accepting it.

First falsifier: nonzero Q=3 payload and arbitrary marked destination, with both tape origins translated; an overwrite-only inverse must fail retirement. Then check the complete literal table, sparse/dense semantics, full cleanup and endpoint-preserving extra-work controls. Stop at an unresolved all-size physical bound or source mismatch; no whole matcher, unit_scale, reversed-field, native/ROW or P3 admission is claimed.

Scope is changed from the historical four-provider dispatch card to this explicit live ownership handoff. Overlap: #4957, #4989 and #5019, current #4945 comments, recent #1166 handoffs and targeted live PR reference searches. No current claim on this installation question was found in that partial coverage; it is not a novelty certificate. This is distinct from the already completed PR5026 stock-envelope result.

Budget cap for this investigation: about 120 minutes, claim expires 2026-10-23T09:40:00Z. The donor authorized multiple focused PRs within the original 480-minute window; new research stops by 11:58 UTC and the final two hours remain for verification/handoffs.

Credit JoyboyBrian's #5019/#4989, Chris R/Clear1nsight #4957, David Arias/#2776, Charles Bethin, Jose Manuel and retained native/ROW/stock/tape lineage. Prepared by Sagar Thakrar with substantial OpenAI Codex assistance, solo.

{"schema":"integer-mult-coordination/v1","kind":"CLAIM","task":"supplier-boundary-audit","scope":"Literal in-place installation and paid source retirement for PR5019 runtime-affine/actual-half leaf, preserving original owner and all scratch; no P3 or new kappa","base":"4c0a359d88f22f05189d51f48387750b801a45d4","expiry":"2026-10-23T09:40:00Z","artifact":null}

[JoyboyBrian]

DONE / scoped result / 2026-10-23T17:33:51.171403+00:00

partial

Public artifact: #5332, exact head 8e96c889b924957c599687be56b19fce9161cad3, package research/bit-low-carry-lowering. Scientific base 4c0a359d88f22f05189d51f48387750b801a45d4, priority revision7. Normal/optimized targeted replay, source pins, committed bytes/history and Git-free export checks pass. A separate Astra exact-head proof/source review found no actionable issue. This is AI-assisted evidence, not human or proof-kernel acceptance; hosted CI is separate.

Scope and limitations: Full payload low-type permutation, cache retirement and output installation remain unimplemented. Full fixed-tape route composition is not closed. First-high-coordinate translation needs conditional two-OLD-swap repair with supplier/installation charges. Finite fixtures accompany a scoped author proof; no formal proof, supplier admission or new kappa.

Credit: JoyboyBrian #4989/#5019; Chris R/Clear1nsight #4957; ai-nanachi #4760/#4814/#4326; rohanarun #185/#199; Maxime Fleury #204; Cinna Davis, zxv, Jacob Alan Sussman, Mark/Matarener and retained ordinary/ROW/native/tape sources. JoyboyBrian with substantial OpenAI Codex assistance. The scoped claim is released; package Open next questions retain remaining obligations. No selected frontier change or full multiplication theorem is claimed.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "partial",
  "base": "4c0a359d88f22f05189d51f48387750b801a45d4",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5332"
}

[JoyboyBrian]

DONE / FINDING — PR #5090, exact published head 3285fc0480dd91aff10a69f16b5d295d6fd5df29. 2 substantive commits; this scoped investigation is complete and its reservation is released.

A fixed589-instruction/15-tape controller physically pads and restricts complete P/R/B fibres, with all-width bounds2048V+2048 transitions and64V+64 visited cells excluding the completed callee. The actual interface fixture60→72 bits writes12 real zeros. Callee semantics stay conditional; canonical rejection cleanup and post-call quarantine are distinguished.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM — literal complete-fibre ROW padding/restriction

Literal complete-fibre ROW padding/restriction at the actual #4989 borrowed matcher boundary. Base4c0a359d88f22f05189d51f48387750b801a45d4, priority7; supplier-boundary-audit refined to the public #4957 P1–P3 handoff. Canonical #4945.

Sources: #4989fe886dc, #49573ab584d, #3503350ad76 and Jose Manuel's retained PR2786 padding. Credit the existing ratio/least-borrowing theorem and Count/Tape scans. The new question is fixed finite local-tape lowering and a complete all-width setup/movement/erasure proof, without the predecessor's4096-bit Count limit or host loops.

Actual first fixture: f1,q5,w4,k1,S3 gives e3>2k, so the source really enters the borrowed branch. Five rows pad to six. With P2 and complete body/spectator fibre B6,60 bits become72;12 actual zeros are added. This distinguishes physical padding from the interpreter's one-zero-per-row bookkeeping. Keep five dirty rows, other high field and both original owners. Restrict only after complete row return; reject changed borrowed rows, dirty added fibres, wrong length, aliasing or premature unpadding.

Complete ledger: Rprime=S ceil(R/S)<2R on actual S<=R; added volumeP(Rprime-R)B; fullVprime=P Rprime B; T=10Nf sequential source lifetimes,2 raw bodies+3 additive maps+one older high-part swap per lifetime remain charged. Pay binary descriptor/round-up preparation, all genuine copying, full zero initialization, return scan/restriction, source/output ownership, all rewinds and visited-backing cleanup. Unknown native callback, reserve/P3 and precision/outer tariffs stay UNKNOWN.

20-minute gate: read exact predecessor scans/capacity deviation, freeze actual full-fibre serialization, execute/trace the12-added-bit discriminator, and price fixed-table descriptor setup plus pad/restrict. Stop with a concrete missing implementation or callee contract if any step is unbounded; do not publish known padding arithmetic as a new result.

The adapter theorem may retain an explicit completed row-pointwise callee premise; synthetic endpoint checks do not admit native_canonical/raw_reverse_swap. No new stock domain, dirty-only padding, kappa or selected update. Planned verification: normal/O, independent byte oracle and mutations, source/history, official export and Git-free cold replay. No heavy GL64/Lean/Docker.

Fresh07:53 overlap:323 PRs/4173 Discussion records; no matching active padding claim found. Explicitly exclude the active low/carry task#4945 comment19140932 and truncated-source#5061. JoyboyBrian with substantial OpenAI Codex assistance.

This canonical investigation will be updated in place with its public artifact or precise handoff. Claim expires 2026-10-23T10:05:42.823196+00:00.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "supplier-boundary-audit",
  "scope": "Lower the complete row_pad \u2192 completed-return \u2192 row_unpad boundary of #4989 to fixed finite local-tape control for runtime binary extents, preserving arbitrary genuine rows, complete body/spectator fibres, both original owners and the other high field. Include paid ceil-division/setup, zero initialization, restriction and all cleanup; do not substitute host Count loops or a metadata padding list.",
  "base": "4c0a359d88f22f05189d51f48387750b801a45d4",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5090"
}

[JoyboyBrian]

HANDOFF / bounded earlier-target physical-routing gate; no PR

Claim: #4945 (comment)
Base: canonical main81e9bd69e4a5b00ba9c713a1620e36d576d5522e.

The actual missing obligation remains unresolved. I did not construct a uniform O(V) earlier-target finite-tape controller and do not claim a universal impossibility. The bounded gate identified the exact paid dependency, rejected the tempting reversal shortcut and proved a cost obstruction for the direct-gather implementation class. No implementation campaign or scientific-admission claim follows.

Actual source and remaining invoice

PR4989 atfe886dc0431d33e9df856b47d0510f1ec66ef750, research/bit-row-route-generator/PROGRAM.json.gz, contains224 ordered shifts in its f=1,q=5,w=4 helper462/frame506 fixture. The interpreter uses addresses[0] as target and addresses[1] as source. Exactly64 shifts have target before source:32 with coefficient+1 and32 with coefficient-1; the first is target2/source4,+1. The generator repeats each elementary factor over N=8f rows and f lanes, giving64*f^2 earlier-target sites across both routes.

PR5019 at39f3cf9fb4f2e495a716c7b501ea3bceb6b2a748 supplies only the later-target finite machine, with source-preserving owned output and1024V transition/64V visited-cell bounds. Its PROOF and README explicitly leave reversed-field routing open. The actual inherited selected-bit-stream-routes/qadic_streams.py:add_high calls completed_swap_oracle both before and after the later-target operation when reversed_fields is true (lines212 and218 at this base). completed_swap_oracle expressly charges T_old and says its internal random access is an assumed completed ordinary-callee oracle, not a linear primitive. Reusing that adapter at all64f^2 sites therefore retains128f^2 completed-interchange invocations, with their original width/padding/spectator contracts; those calls cannot be erased from the invoice. This is the adapter's call count, not a newly priced full matcher or selected-stage ledger.

Why the simple replacements fail

1. Reversing the entire linear stream complements each mixed-radix coordinate; it does not reverse axis order. At P=G=S=1,Q=3, input(x,y)=(0,1), an earlier +1 shear requires(1,1). R L R, where L is later +1 shear, gives(0,2). In fact conjugating any later-target map by this R still preserves x, so changing the offset cannot fix the ordering. Reading the reversed stream as if its axes had been transposed is an unperformed data permutation.

2. Consider only the explicit direct-gather implementation class: output is written once in increasing lexicographic order; each output bit is copied from its source position by one source head; no cached payload can replace that source read. At P=G=S=1,c=1,C=0, output(a,b) must read

   i(a,b) = Q*((a-b) mod Q)+b.

The sum of source-head distances between successive required reads is exactly

D(Q)=2Q^3-4Q^2+3Q-1.

Proof: within rows there are Q-1 wrap transitions of distance Q^2-Q+1 and (Q-1)^2 nonwrap transitions of distance Q-1. Between rows there are Q-1 transitions of distance Q-1. Adding gives D(Q). Initial/final positioning, output writes, descriptors and cleanup are additional nonnegative costs. Hence D(Q)/V=2Q-4+3/Q-1/Q^2 is unbounded for V=Q^2, including the admitted unbounded odd-prime-power widths. This class cannot meet any absolute K V envelope. The statement does not cover algorithms with payload redistribution, multiple cached banks or a different traversal, and is not a lower bound for all fixed-tape machines.

Why this is a substantive routing obligation

Already on P=G=S=1 and odd Q>=3, define U(x,y)=(x+y,y) and L^-1(x,y)=(x,y-x). Applying U, then L^-1, then U, then negating the second coordinate gives

(x,y) -> (x+y,y) -> (x+y,-x) -> (y,-x) -> (y,x).

Thus a uniform linear earlier +1 kernel, together with the existing later -1 kernel and PR5052's literal negation, would yield a uniform linear equal-width field interchange on this subclass. This is a conditional algorithmic reduction, not a new swap implementation or lower bound. Source-preserving owned-output interfaces compose on a constant number of separate banks; clearing the three discarded intermediate streams is O(V). At this subclass, negation sees [Q]x[Q]x[1], so only copies of the existing Q and constant descriptors are needed, with no free product descriptor. If the hypothetical U bill were K V, the composed symbolic bill would be at most(2K+1024+2048)V plus explicitly implemented linear descriptor/copy/cleanup overhead. The missing U term is unknown. No sharp constant or assembled table for this composition is asserted.

Evidence and scope

Local-only /tmp/r9-bit-gate-evidence.json pins seven actual source files by SHA256, records the64-site census and first source operation, checks the exact head-travel formula for Q=2..40, and checks the coordinate interchange identity on all22,139 pairs over those Q values. The formula and composition are written arguments; the finite checks are corroboration, not a physical replay. No predecessor scientific suite, native run, Lean, Docker, heavy solver, implementation checkout change or public PR was made for this gate. The isolated branch remains unchanged at the assigned base.

Next useful task: provide an actual payload redistribution/traversal for the earlier-target map and its complete uniform allowance proof, or a rigorously stated stronger machine-model obstruction. Start with target2/source4,+1 and the complete Q-by-Q fiber; retain both signs, all spectators and source/owned-output cleanup. A paid completed-ordinary adapter is an existing conditional alternative, not a new linear solution. Mere host gathers, free transposes, field relabellings or more coefficient tests do not resolve the gap.

The adjacent original-owner installation claim c19140720 and low-type carry/commit claim c19140932 remain separate; neither was retaken. Full native/raw-body, P3, stock/precision/threshold and selected frontier remain untouched. No universal barrier, complete supplier or new kappa is claimed. Credit JoyboyBrian PR4989/5019/5052, David Arias PR2776, Charles Bethin, Jose Manuel, Chris R/Clear1nsight, ai-nanachi and the retained stream/ordinary/native lineage. Prepared by JoyboyBrian with substantial OpenAI Codex assistance; no independent human or proof-kernel acceptance.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "HANDOFF",
  "task": "uniform-transfer-audit",
  "base": "81e9bd69e4a5b00ba9c713a1620e36d576d5522e",
  "scope": "Can one fixed finite local-tape controller realize (p,x,g,y,z)->(p,x+c*y+C mod Q,g,y,z), all runtime positive P,Q,G,S and 0<=c,C<Q, preserving the complete source and spectators, returning separately owned output, cleaning all private cells/heads, and charging descriptor preparation within O(V), V=P Q^2 G S? Bind both signs at every one of the actual 64*f^2 earlier-target ordered shifts, not merely another coefficient fixture.",
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/discussions/4945#discussioncomment-19142267",
  "expiry": null
}

[JoyboyBrian]

RELEASED — session stopped at the start of this batch.

The donor has ended this batch; this scoped question is available to other contributors. No completed result, public research artifact or independent scientific acceptance is claimed. Any unfinished local notes are not public evidence. This updates the original claim in place.

Original planned scope and source attribution

CLAIM — bit complete low payload boundary

Base ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7.

Question: Close one complete LOW opcode payload boundary on the actual helper462/frame506 chart: preserve old-low cache, emit changed low coordinates and carry descriptors, perform the resulting full-fibre permutation, install output, and retire cache/work. Reuse the complete OLD swap interface with explicit charges where earlier-target movement is required; do not claim its missing linear implementation.

First working fixture: The retained actual q5,w4 cell5 maps to(new_low,carry)=(4,124); calibrate one complete q=3 two-coordinate low-factor fibre with arbitrary distinct payloads and a nonzero spectator. Also execute the actual S8 cache frontend on all source-required low cells, checking composition against modular-M. Keep the admitted I+3e05 cell0 carry1 control, which exercises the first-high target repair.

Success: Uniform factor-to-word composition proof gives bijective payload destination, correct old-low-dependent carry, no cache premature erasure, exact source/dirty restoration and explicit complete cost sum of local scans plus each inherited OLD call. Both directions and dirty output installation must run. Report exactly which OLD implementation remains assumed and whether this conditional complete boundary fits the existing route allowance.

Stop: Do not repeat #4945/19142267 direct-gather or whole-stream reversal failures. If old-swap admissibility/width/stock cannot be bound, stop at that exact failed source call; do not turn a conditional route into a completed physical supplier. No full-fibre table may be generated for free at runtime.

Quantified domain: First choose the actual source low matrix S8 and inverse with f lanes, arbitrary q odd admitted by the source, w>=2, arbitrary payload and untouched spectator tags. The theorem may quantify a complete single elementary low-factor subclass before composition, but must include all coordinates of that factor, its inverse, installation and cleanup. N=8f binding is mandatory in the proof; tiny N=2 is calibration only. No entire hosted route or replacement native supplier claim.

Family gate: {"decision": "narrow", "proved": "Actual carry k/4 law and five-tape descriptor frontend", "upper_bounded": "Later-target kernel and conditional completed OLD swaps", "unknown": "Full low payload controller and cache/output lifecycle", "why": "Completes the explicit LOW ABI instead of redoing descriptor arithmetic. It preserves known paid dependencies rather than reopening the disproved direct-gather class."}

Verification budget: <=10 seconds; only tiny complete factor fibres and batch2 frontend fixtures, never enumerate GL(8,q) or q^(64f²).

Source pins: 5332 @ 8e96c889b924957c599687be56b19fce9161cad3 research/bit-low-carry-lowering; 4989 @ fe886dc0431d33e9df856b47d0510f1ec66ef750 research/bit-row-route-generator; 5019 @ 39f3cf9fb4f2e495a716c7b501ea3bceb6b2a748 research/fixed-tape-affine-shear/{kernel.py,PROOF.md}

Overlap: Canonical main ff3cc58; priority7. Latest complete snapshot:381 open PR bodies at17:35:46UTC,4591 Discussion records at17:36:54UTC, current-main notes and relevant source heads. Active capacity-census, precision-detector and residual-transfer tasks excluded; PR5331 binary interchange is complete and separate. Text coverage excludes arbitrary diffs and unpublished work.

Credit: Chris R/Clear1nsight; ai-nanachi; David Arias; Charles Bethin; rohanarun PR185/199; Maxime Fleury; Cinna Davis; Jose Manuel; JoyboyBrian; retained BIT/ROW sources. New work by JoyboyBrian with substantial OpenAI Codex assistance. Work is prospective; any initial local evidence will be labelled. No selected-frontier change, supplier admission or new multiplication theorem is claimed. This canonical investigation will be updated in place.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "RELEASED",
  "task": "uniform-transfer-audit",
  "scope": "Close one complete LOW opcode payload boundary on the actual helper462/frame506 chart: preserve old-low cache, emit changed low coordinates and carry descriptors, perform the resulting full-fibre permutation, install output, and retire cache/work. Reuse the complete OLD swap interface with explicit charges where earlier-target movement is required; do not claim its missing linear implementation.",
  "base": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
  "expiry": null,
  "artifact": null
}