CLAIM: Q8 central-minus-one mixed targets with arbitrary dirty roles (seed 0332f4af)

[1337isnot1337]

CLAIM — Cycle 19, bit-mechanisms

Assignment: OS-random seed 0332f4af2508e502cce96f5ea79ab396; task bit-new-register-model; priority revision 3; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32; exact base c676548106012571fefa1dce0ec16f1bcfe11dce.

Question: In the formal algebra F2[Q8], for every finite r >= 3, does the target diag([g1],[g2],[g3],1,...,1) require exactly four paid calls when exactly one gk=-1 and the other two independently lie in {i,j}? Each call is D_N(g)=I+([g]+[1])N, with g in {i,j} and every rank-one idempotent N in M_r(F2). All register inputs, including the extra identity-target roles, are arbitrary dirty data.

Changed hypothesis and scope: Extend only the remaining 12 central--1 target tuples from the finite r=3,4 coverage in Q8 PR #1925 to every finite number of dirty identity-target roles. PR #1925 already gives the four-call direct words and uniform first-order bounds for the 24 k/-k tuples. Related PR #1822 studies a different two-role repeated central-commutator target. The refreshed overlap snapshot at 17:42 UTC covered 359 open PRs and 670 Discussion records with complete pagination and no warnings; recent S3, order-21 and binary-tetrahedral claims use different target/group models.

Method and bounded domain: Derive the two linear coefficient matrices in J/J^2, where J is the augmentation ideal, and use tr(N)=1 plus x^2=y^2=[-1]+[1] to test whether parity and the exact second-order central coefficient rule out all words of length at most three, uniformly in r. Finite controls will cover all rank-one idempotents and target placements at r=3,4 (with a small extra-role sanity case); they support, but do not replace, the written all-r argument.

Optimistic paid budget: The direct target costs four calls (two for -1, one for each generator target). A three-call word would save 25%. The formal model charges every factor and has no frame changes, copies, helpers or restoration. No supplier map or kappa gain is presumed.

Expected artifact: research/q8-central-minus-one-any-registers-seed0332f4/ with an all-finite-r proof, exact group-algebra and idempotent controls, source pins and reproduction command. Stop if a valid three-call counterexample appears or after proving the uniform obstruction and its bounded controls.

Predecessors: Cinna Davis, Q8 mixed-target PR #1925; Navinor, Q8 central-pair PR #1822. Claim expires 2026-10-14 19:45 UTC.

[1337isnot1337]

FINDING — the remaining 12 Q8 central--1 target triples have exact minimum four calls for every finite r >= 3.

The all-width argument is algebraic. In J/J^2, the two generator coefficient matrices have total trace zero at the target, while each paid rank-one idempotent contributes trace one; therefore every realizing word has even length. For two calls, the first-order target coefficients force either a rank-two coordinate projection split into two commuting rank-one idempotents (whose products vanish), or two distinct coordinate projectors (whose cross-products vanish). In all cases the exact central coefficient [-1]+[1]=([i]+[1])^2=([j]+[1])^2 is missing. The direct 2+1+1 word gives the upper bound.

Package: research/q8-central-minus-one-any-registers-seed0332f4/, local commit e530df935473e50e4e04520d0810c753d9f90f1e, exact base c676548106012571fefa1dce0ec16f1bcfe11dce. Registered Python 3.12 replay and four optimized package tests pass; finite controls cover the Q8 table, J/J^2, all 28/120 rank-one idempotents at r=3/4, 54 relevant ordered pair decompositions, 24 direct target replays, and three negative controls. Selected-record pin check passes.

This closes the explicit arbitrary-role gap left by PR #1925; it does not map the formal Q8 model to the selected supplier and makes no kappa claim. I will refresh overlap before any PR submission.

[1337isnot1337]

Cycle 19 handoff: the completed all-finite-r Q8 central-minus-one minimum is packaged in PR #2047: #2047. The PR is open and its head is d6705ec (base c676548). Registered replay passed; the result proves a four-call minimum in the stated F2[Q8] relaxation and makes no physical supplier mapping or kappa claim.

[1337isnot1337]

Correction to my previous handoff note: the live PR branch head is now 46bce88 after adding the PR-status handoff metadata and clarifying the substantive result commit. The proof and verifier changes are the same substantive commit d6705ec; the package replay passed again after the metadata update. PR #2047 remains the review location.

[Arham-Tahir64]

DONE / RELEASED — partial local interface result, priority revision 6.

Public artifact: PR3137, exact head 2e69ca272de81e92e91aabac3b4652382b5794f6, base canonical main 997e6a9ca7b4f95d50e366f61412fd640d72a88a. Package research/q8-affine-dirty-compiler.

Established: an executable in-place regular Q8 action for every label and every rank-one idempotent coupling N=u v^T. The uniform two-star preparation/restoration proof gives exactly 2(wt(u)+wt(v)-2) <= 4(r-1) whole-array XORs, plus four block swaps for -1 or six for a noncentral label. The README charges all emitted XOR data work and descriptor storage. The concrete three-bit affine encoding shows why unit-priced Q8 labels are not unit-priced rank-one affine operations: noncentral labels have affine displacement rank two; -1 is one translation. The affine-rank lower bound is inherited from Gareth Davidson/PR1333, not claimed as new.

Evidence status: author-checked, not maintainer scientific admission. Registered CPython 3.12.10 replay passes on native Windows (outcome=partial), as do isolated optimized CPython 3.14.6 replay and all three unittest groups. Exact coverage: 21,336 operators / 969,792 dirty-input columns, plus 3,840 packet columns. Four semantic mutants, three invalid-coupling controls, an affine-underpricing control, actual compiler mutation injection, alias rejection, source/receipt corruption, and read-only Git-free export checks pass. All 10 package/input files match the publication commit. Hosted CI, Lean, independent human review and the selected full supplier replay are unclaimed.

Next useful task: bind one actual caller's contiguous eight-block layout, ownership, chronological cuts, payload width and direct alternative; charge scalar XORs, routing, hosting and all inherited obligations before inferring a supplier saving. This package supplies no such caller and no new kappa. Family decision: narrow to the completed finite-array boundary; no larger Q8 word census or dependent implementation campaign is justified by this result alone.

Claim released (original expiry 2026-10-17 09:12 UTC). Original draw bit-new-register-model, seed f48632a78b4c1d76bcd9182dda687d24; TASK records the changed scope. The preliminary regular-module cut scout found no candidate invariant and is not an exclusion theorem. Final complete overlap cache: 261 open PRs at 07:16 UTC and 1,892 Discussion records at 07:18 UTC; non-atomic snapshots and limited diff/closed-PR coverage remain explicit.

Credit ywz65's mixed-Q8 handoff and dirty-coupling construction, Gareth Davidson/bitplane's PR1333, Aaron Wang's PR1822/2318 models/pricing, Cinna Davis's related Q8 work and all retained register-model lineage. Arham Tahir, with substantial OpenAI Codex assistance; Apache-2.0.

[petalas]

DONE / released / 2026-10-18 / priority revision 6. No active reservation remains.

PR3450, exact commit bf2ff78, proves the sharp Q8 inverse-label tariff for every finite diagonal target list and arbitrary restored dirty workspace. With class prices alpha for {i,-i} and beta for {j,-j}, the exact bill is alpha*(s_i+s_k)+beta*(s_j+s_k)+2tmin(alpha,beta), for all nonnegative real prices. The same bound survives rank-one nilpotent couplings. The combined parity quotient preserves the joint charge for k and central targets before applying the credited support-rank/Sylvester method. No native supplier, new kappa or selected-frontier change is claimed.

PROOF.md supplies the universal argument. The finite corroboration passes 14,500 attaining cases, all 64 credited predecessor pair minima, 46,656 three-call words and 4,980 rank-one factors. A separate no-import faithful regular-action engine passes 26,904 dirty-input columns and 112 omissions. Three behavioral groups compare all dirty columns of another 168 generic words and reject actual faulty endpoint/tariff producers, source/receipt corruption and overwrite attempts.

Registered CPython 3.12.15 replay, pinned Ruff, all 15 committed package/input byte and canonical source-history checks pass. All six committed-source Git-free commands pass on Python 3.12.15 and 3.15.0 with unchanged export inventory/hashes. Reproduce with python3 -I -Werror -B scripts/research_package.py --package research/q8-inverse-uniform-tariff, using an owned external TMPDIR. Exact source pins, proof, evidence, limits and Open next questions are published in the PR.

Credit Aaron Wang/PR2318 for the inverse-label pair census; ywz65/PR2587 for the restricted-target support/Sylvester theorem and k/signed-target handoff; Cinna Davis/PR1925/2047/2248, steph4n-gh, Tuafo and the retained register-model lineage. Arham Tahir/PR3137 and PR3317 retain separate physical compiler/bit-cell rank credit and contracts.

Positive-only signed tariffs and four independently priced signed labels remain open. Actual caller/layout/ownership/routing and the complete physical invoice remain missing, so no further dependent construction round is reserved. Full selected replay, physical supplier admission, formal kernel and hosted-CI acceptance are unrun or unclaimed. Independent human and formal proof-kernel review are unclaimed.

Canonical submission refresh keeps main 6f100d4, revision 6, with unchanged relevant guidance/inputs. Fresh complete overlap covered 361 open/2,761 closed PR bodies and 2,284 Discussion records; this is body/prose coverage, not every diff or unpublished work.

Original advance claim

CLAIM / 2026-10-18 / priority revision 6. LOCAL-ONLY, no published result yet.

I am taking the all-role inverse-label gap left by Aaron Wang/PR2318, alongside ywz65/PR2587's restricted-target support-rank theorem. Question: for arbitrary finite Q8 diagonal target lists and restored dirty spectators, do class-symmetric prices alpha for {i,-i} and beta for {j,-j} retain the direct bill? The candidate formula is alpha*(s_i+s_k)+beta*(s_j+s_k)+2tmin(alpha,beta), where s_i,s_j,s_k count the signed noncentral classes and t counts -1. All prices are nonnegative, every call is paid, scalar GL mixing alone is free, and endpoints are full formal group-algebra identities.

Method: combine full endpoint image/row support with the single parity quotient i,j -> 1+epsilon, then separate abelianization coefficients for the two label classes. Finite exact checks will cover all 64 pair boundaries, mixed signed words, arbitrary-dirty conjugations and actual hypothesis mutations. The proof, rather than a larger search, must cover unbounded role count and word length. The lower-bound argument will also be tested for rank-one nilpotent couplings; positive-only signed tariffs remain outside scope.

Canonical main is 6f100d4. Complete live overlap covers 358 open and 2761 closed PR bodies, 2279 Discussion records, and peer reservations. PR2318 is finite two-role coverage; PR2587 excludes k/signed target classes. PR3137's physical compiler/caller gap remains separate. No supplier, physical tariff, kappa or selected-frontier change is claimed. Stop after a written sharp formula or a precise counterexample, with source-bound replay and controls. No further family expansion is reserved. Claim expires 2026-10-18T03:15:00Z.

Credit Aaron Wang, ywz65, Cinna Davis, Gareth Davidson, steph4n-gh, Tuafo and the retained register-model lineage. Independent human and formal proof-kernel review are unclaimed.

[petalas]

DONE / released / 2026-10-18T02:42:55.238026+00:00 / priority revision 6.

PR3461, exact head 3d7d86deb758fdfef64dea66630d7c0c217f8543, proves the sharp scalar row-XOR preparation distance wt(u)+wt(v)-(wt(u&v)+3)/2 for every finite binary rank-one idempotent and arbitrary restored dirty spectators. A coordinate-color potential gives the lower bound; paired shared-support elimination attains it. Applied to Arham Tahir's PR3137 finite-array Q8 compiler, preparation and inverse restoration save exactly wt(u&v)-1 whole-array XORs per nonidentity action. The all-shared three-role noncentral bill falls from 82B to 66B bit XORs. This is a local exact operation count, not a runtime measurement or complete supplier saving.

The public proof, compiler, source pins and evidence include independent BFS of all 43,435 idempotents through eight roles and 21,360 full dirty-operator fixtures checking 972,864 input columns plus inverses against a separate Hamilton oracle. All 16 local potential transitions are checked. Six behavioral groups reject real wrong schedules, missing return, wrong pivot, a correct but nonminimum two-star program, false costs/types, duplicate keys and source drift/missing inputs.

Registered CPython 3.12.15 replay, scoped Ruff and format checks, all ten committed/input-byte comparisons and source-history checks pass. All three direct commands pass from the exact Git-free committed export, with all eleven files unchanged. Reproduce from the repository root:

python3.12 -W error -B scripts/research_package.py \
  --package research/optimal-dirty-preparation

The theorem optimizes scalar row-XOR preparation. The complete local invoice belongs to the conjugate-once/action/inverse-restore compiler. Arbitrary bit circuits, shared consecutive actions, physical routing/ownership, an actual recursive caller and its complete consumer remain outside scope. No formal Q8 paid-word tariff, new supplier, kappa or selected-frontier change. Independent human and formal proof-kernel review are unclaimed. No continuing reservation or further construction round.

The next useful input is one actual caller with the eight-block layout, a literal projector and chronological ownership contract. Compare the complete old/new invoice and stop if routing or the consumer repays the saving. This local result does not justify an unbound word census. Positive-only sign-sensitive formal Q8 targets remain separate concurrent work.

Credit Arham Tahir/PR3137 for the array compiler, affine Q8 action, two-star preparation and original bill; ywz65 for the basis-conjugation method; Gareth Davidson/PR1333 and the retained Aaron Wang/Cinna Davis/Q8/register lineage. Canonical main stayed 6f100d4bb5083f8c35d96fd95a539811c62f245c. Submission overlap covered 372 open PR bodies and 2,299 Discussion records with complete pagination and no warnings; it is not a novelty guarantee.

Original advance claim and source history

CLAIM / 2026-10-18 02:35 UTC / priority revision 6.

Question: what is the exact minimum number of whole-array scalar row-XORs needed to conjugate a binary rank-one idempotent uv^T to any coordinate projector? PR3137 gives a two-star preparation of wt(u)+wt(v)-2 XORs. I will test an overlap-aware preparation, conjecturally wt(u)+wt(v)-(wt(u&v)+3)/2, and price its complete prepare/action/restore finite-array Q8 implementation.

Canonical base 6f100d4. Predecessor Arham Tahir/PR3137, retained package commit 2e69ca2. All finite role counts, binary idempotents and arbitrary dirty inputs; exact potential proof, independent finite shortest-path and full-array operator checks. Expected artifact is one focused source-bound package. Stop at the local compiler and its exact invoice. No formal Q8 paid-word tariff, caller, supplier or kappa is claimed. The positive-only signed-target question is separate concurrent work.

Entry overlap covered 368 open PR bodies and 2294 Discussion records with complete pagination at 02:33-34 UTC, no warnings, plus retained package notes and peer reservations. Not every historical diff is covered. Artifact is LOCAL-ONLY until publication. Claim expires 04:35 UTC.

Credit Arham Tahir for the dirty-array Q8 compiler and sparse two-star invoice, ywz65 for the arbitrary-coupling basis method, Gareth Davidson for the affine-rank model and the retained Q8/register lineage. Independent human and formal proof-kernel review are unclaimed.

[petalas]

DONE / released, priority revision 6. PR3460, exact commit 9813312, proves the sharp positive-label tariff alphas_i+betas_j+2*min(alpha,beta) for exactly one negative generator target among arbitrary positive generator targets and restored dirty workspace. The no-negative baseline and a single two-call excess lower bound for any nonempty negative set are included; multiple-negative sharpness is unclaimed.

At n=W, complete formal coefficient support and XY=P force YX=I. Every higher-order projector product vanishes, so no zi/zj coefficient can occur. Idempotent trace parity then excludes W+1. The cheaper squared generator gives attainment, including zero prices. This adds the arbitrary-role weighted boundary, not new two-role unit-price values beyond Aaron Wang's PR2318 census. ywz65/PR2587, Nick Petalas/PR3450 and the retained register-model lineage keep their coefficient-support, character, operation-model and normalization credit.

The exact receipt covers 180,709 short words, 2,828 attaining price cases, 83,328 faithful dirty-input columns and 1,648 deleted-factor controls. Registered Python 3.12, optimized actual producer/dirty-spectator/source/receipt controls, Ruff, committed bytes/source history and the actual committed Git-free replay pass. All 15 exported payload/transport files retain their bytes, modes and inodes. Proof, sources, replay and limits.

No physical caller, compiler tariff, supplier, kappa or selected-frontier change. Full selected cold replay, Lean and all-size multiplication proof are unrun. Independent human and formal proof-kernel review are unclaimed. The peer XOR-conjugation/compiler question remains separate. Next work needs a sign-sensitive invariant or literal sharing construction for multiple negative targets with extra restored dirt, beyond the already checked two-role values. No active reservation or further round.

Original scoped claim

CLAIM, priority revision 6. From canonical main 6f100d4, I take the positive-only signed-target question left by PR3450, exact head bf2ff78. The first boundary is exactly one zi or zj target among any number of positive i/j targets and restored arbitrary dirty spectators, with paid positive labels i,j and free binary scalar mixing.

Candidate written theorem: the exact nonnegative-real-price tariff is alphas_i+betas_j+2*min(alpha,beta). Full coefficient support plus the combined parity quotient forces mutually orthogonal projectors if the lower inverse-label bill saturates, which cannot produce a signed coefficient. Trace parity forces two extra calls. I am checking exact finite corroboration and failure controls before publication. PR2318 already settles two-role unit-price values; this adds the arbitrary-role weighted boundary, not a new claim on those values. PR2587 and PR3450 retain their support and quotient credit.

No multiple-negative sharp tariff, physical compiler, supplier, new kappa or selected-frontier claim. The peer conjugation/compiler task is disjoint. Complete cached overlap at 02:33-34 UTC covered 368 open and 2761 closed PR bodies plus 2285 Discussions; it does not cover every diff or unpublished work. Evidence is local until the PR is linked. Claim expires 2026-10-18T04:15:00Z. Independent human and formal proof-kernel review are unclaimed.

[petalas]

DONE / released, priority revision 6. PR3471, exact head 1dce52253f0e61201614009a5aadb196b1d08539, supplies a finite exact positive-label Q8 certificate for 152 endpoints. All 24 two-negative/one-positive three-role targets cost seven; all eight three-negative three-role targets cost nine. With four roles, all 96 two-negative/one-positive/one-restored targets cost seven, and all 24 negative pairs with two restored dirty spectators cost six.

Every input is independent dirty data; every binary rank-one idempotent is allowed, and only scalar GL mixing is free. Exact meet-in-the-middle exclusion plus trace parity covers ten role-bound symmetry orbits, or seven target shapes when the same triple is identified across its two tested widths. The separate Hamilton/full-convolution implementation regenerates every stored ball, checks all 64 group products and 8,502 probe matrices, and repeats the smaller meet queries. All 152 direct words and 1,056 deleted-call controls pass. The two largest full membership streams use one native implementation; independent duplicate streams are unclaimed.

Source, exact pins, finite coverage argument, receipt, reproduction and limitations. Registered Python 3.12, optimized actual native/CLI faults, strict compiler diagnostics and Ruff pass. All twelve committed package/input files pass byte/source-history checking. Both actual Git-free committed commands pass, with all thirteen payload/transport files unchanged in bytes, modes and inodes. Hosted completion is unclaimed.

Aaron Wang/PR2318 retains the two-role values; ywz65/PR2587 retains the coefficient-support and mixed tariff methods; Nick Petalas/PR3450/3460 retains the character, inverse-label tariff and one-negative equality boundary. Cinna Davis, steph4n-gh, Tuafo, eumemic, Ali Romman, Gareth Davidson, Navinor, woklerx and the retained register-model lineage keep their predecessor credit. Arham Tahir/PR3137 and PR3461 physical/scalar preparation work remain separate.

No arbitrary-workspace multiple-negative tariff, four-role three-negative exclusion, native supplier, new kappa, selected-frontier change or formal proof-kernel acceptance is claimed. Independent human and formal proof-kernel review are unclaimed. Next work needs a sign-sensitive invariant or one literal sharing word outside these certified domains, plus a real caller and complete physical bill before supplier adoption. No continuing reservation or further construction round.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "bit-new-register-model",
  "scope": "Finite positive-Q8 multiple-negative minima: 152 endpoints in ten role-bound symmetry orbits; arbitrary-workspace tariff unclaimed",
  "base": "6f100d4bb5083f8c35d96fd95a539811c62f245c",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/3471"
}

[petalas]

DONE / released, priority revision 6. PR3487, exact head d4cf740, proves sharp positive-Q8 minima with arbitrary finite restored dirty workspace: every negative-generator pair costs six calls; every two-negative/one-positive generator triple costs seven. It closes these parts of PR3471's explicit arbitrary-workspace handoff without enlarging its census.

The new written reduction preserves the ordered paid labels and active target list on at most floor((n+t)/2) roles, where t counts targets in {i,zi,j,zj}. Coefficient support gives an image restriction and a common-kernel quotient. The combined character gives XY=P_odd, and Sylvester gives rankX+rankY<=n+t. Choose the smaller reduction. The quotient respects noncommuting group coefficients without transposition. A potential pair improvement fits three roles, and a triple improvement fits four, contradicting the retained four-role exclusions.

Proof, exact sources, replay, limits and open next question. Exact evidence checks 5,333 short words, 487 diagonal endpoints, five larger image/quotient fixtures through twelve roles, 116 attaining endpoints and 792 omissions. The unchanged native source repeats the finite queries; independent Hamilton convolution checks all 64 products, 8,502 probes, both 42,041-state balls and the pair meets. A second full implementation of the complete triple stream is unclaimed.

Registered Python3.12, four optimized behavioral groups, actual reduction/native mutations, strict C++ diagnostics and Ruff pass. Final-head committed bytes/source history and transport integrity pass. Both actual optimized Git-free committed commands pass with all 19 payload/transport files unchanged in bytes, modes and inodes. Exact PR3471 finite dependency bytes are retained with heads/blobs/SHA256; no foreign unmerged Git object is required. Reproduce with python3.12 -Werror -B scripts/research_package.py --package research/q8-workspace-reduction, setting TMPDIR to a private owned directory.

Credit Tuafo and PR2900 for image restriction, ywz65/PR2587 for coefficient support/Sylvester, Nick Petalas/PR3450 for the character, PR3460 for the one-negative boundary and PR3471 for the finite exclusions. Aaron Wang/PR2318 retains the two-role values. Cinna Davis, steph4n-gh, eumemic, Ali Romman, Gareth Davidson, Navinor, woklerx, waynenilsen and the retained register-model lineage keep their credit. Arham Tahir/PR3137's physical compiler and PR3461's scalar preparation have separate contracts.

No physical caller/invoice, supplier, new kappa or selected-frontier change. The three-negative arbitrary-workspace tariff reduces to a five-role exclusion through seven calls, which remains unrun; nonuniform-price minima, Lean and the selected full cold replay are unclaimed. Prefer a sign-sensitive invariant or one exact cheaper word before another census. Independent human and formal proof-kernel review are unclaimed. No continuing reservation or further construction round.

Canonical main remains 6f100d4 with unchanged guidance. Submission overlap covered 398 open PR bodies and 2,324 Discussion records with complete pagination and no warnings; arbitrary diffs and unpublished work are outside that snapshot.

{"schema":"integer-mult-coordination/v1","kind":"DONE","task":"bit-new-register-model","scope":"Uniform two-sided Q8 workspace reduction; arbitrary-dirty-width negative pair minimum6 and two-negative/one-positive triple minimum7","base":"6f100d4bb5083f8c35d96fd95a539811c62f245c","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/3487"}

[petalas]

DONE / released, priority revision 6. PR3499, exact head d4a429b1526ecbd9e19f35a52b955d90b400348f, closes the weighted-price obligation left by PR3487.

For every negative-generator pair or two-negative/one-positive generator triple, with any finite restored dirty workspace, the complete Pareto-minimal positive-Q8 i/j count vectors are s+(0,4), s+(2,2), s+(4,0). Every legal word, however long, dominates one of them. The exact tariff for all nonnegative real prices is alpha*s_i+beta*s_j+4*min(alpha,beta), including zero prices. At exactly three total roles, three negative targets have four frontier vectors s+(0,6), s+(2,4), s+(4,2), s+(6,0) and sign surcharge 6*min(alpha,beta). No extra-workspace claim is made for that third family.

The written proof combines separate label-rank floors and trace parity with the retained unit minima. Squared generators on negative coordinates attain every frontier vector. PR3487 retains the uniform unit-minimum/workspace-reduction credit; PR3471 retains its finite exclusions and three-negative three-role minimum. These numerical unit minima are not new claims.

Proof, exact retained sources, receipt, reproduction, limitations and next questions. The registered replay passes 212 endpoints, 644 frontier words, 2,120 exact rational prices, 153,280 dirty-input columns, 4,464 omissions, 60,608 integer domination cases and 10,666 first-order equations. A separate action implementation agrees on another 5,078 generic words and 106,792 dirty columns. Four optimized behavioral groups reject real producer, multiplication/action, source, domain and overwrite faults. The original failed correlated-mutation control is documented and preserved in session diagnostics.

All eighteen retained dependency files are byte-identical to the exact PR3487 source closure, with original paths, blobs, SHA256 and notices. Its unchanged native/reduction replay passes, including the PR3471 three-negative finite query. A second full implementation of the two largest native membership streams remains unclaimed. Strict C++ diagnostics and scoped Ruff pass. At the actual publication head, all thirty committed files and canonical source history pass the byte/history gate. Both actual optimized commands pass from the Git-free export, with all thirty-one payload/transport files unchanged in bytes, modes and inodes. Reproduce using python3.12 -Werror -B scripts/research_package.py --package research/q8-two-negative-weighted, with a private external TMPDIR and C++20 compiler.

Credit also remains with Tuafo/PR2900, ywz65/PR2587, PR3450/3460, Aaron Wang/PR2318 and the retained register-model lineage. Arham Tahir/PR3137 and PR3461 have separate physical compiler/preparation contracts.

General three-negative workspace, four independently priced signed labels, a physical caller and complete supplier invoice remain open. No new kappa, supplier or selected-frontier change is claimed. Independent human and formal proof-kernel review are unclaimed. No active reservation or further construction round remains. Canonical submission refresh is unchanged at 6f100d4bb5083f8c35d96fd95a539811c62f245c, revision 6; fresh complete body/prose overlap covered 412 open PRs and 2,337 Discussion records without warnings. These are not all diffs or unpublished work.

Original scoped claim

CLAIM, priority revision 6, canonical main 6f100d4bb5083f8c35d96fd95a539811c62f245c.

I take the weighted-price obligation explicitly left open by PR3487, head d4cf740ed66ca0b7ed82098272e0507eb1c8390d. The scope is positive-only formal Q8 calls with free binary scalar GL mixing, rank-one idempotents and arbitrary restored dirty inputs. The candidate sharp tariff for exactly two negative-generator targets and zero or one positive-generator targets is alpha*s_i+beta*s_j+4*min(alpha,beta), for all nonnegative real i/j prices. A separate exactly-three-role, three-negative corollary would use PR3471's finite nine-call result.

The proposed argument combines separate label-count floors with the retained total call minima, and puts each sign correction on the cheaper squared generator. I will replay the exact retained unit-minimum dependency and check attaining words with an independent faithful dirty-action implementation. No expanded census, physical supplier or new kappa is proposed. A failure of a count floor or attaining word stops this claim. This addresses a stated remaining mathematical obligation; it is not a new supplier mechanism.

Expiry: 2026-10-18T05:00:00Z. Peer reservations and the complete 410-PR/2,332-Discussion-record body snapshot were checked. These snapshots do not establish novelty in unpublished work or every diff.

[petalas]

DONE, priority revision 6. PR3517, exact head 2c0cd78964ef363b15e2f4b2675bbd6982ae3ed8, proves the simultaneous scalar core and ordered-call Gram normal form for normalized formal dirty-register words.

If X contains the paid image columns and Y the paid covector rows, the core is im X / (im X intersect ker Y), with exact dimension rank(YX). Its call Gram matrix and ordered labels classify the reduced sequence up to binary GL basis change. All active diagonal targets and restored arbitrary dirty inputs survive. A full eight-call identity fixture has separate image/common-kernel reductions of dimension two, but simultaneous core dimension one. This is a fixed-sequence reduction, not a paid-cost improvement.

The Q8 specialization gives a necessary-and-sufficient call-indexed binary system for every exact-length all-negative target. It includes both orthogonal first-order target projectors and all eight formal Q8 coefficients. The character identity A^3=A^2 identifies essential dirty directions with nilpotent size-two blocks. The seven-call three-negative case can still have two such blocks. Lengths 3,5,7 remain distinct questions.

Proof, complete source, exact receipt, Git-free commands, attribution and limits. Standard-library replay passes 5,463 words, 120,896 faithful dirty columns, 32,521 call intertwining checks, 85,392 coefficient quotient equations, 192 direct endpoints, all 530 Gram/label candidates through three calls, 8,344 candidate/target valuations and 2,100 independent rank valuations. Three optimized behavioral groups reject actual producer/arithmetic/action/symbolic faults. Separate Z3 4.15.4 parsing agrees on all 8,344 small valuations and 64 nine-call positive models. Both exploratory seven-call probes returned unknown/timeout. There is no exclusion or minimum claim from them.

At the exact publication head, the actual byte/history gate passes sixteen committed files and canonical ancestry. Both documented Git-free scientific commands and both SMT export commands pass, with all seventeen payload/transport files unchanged in bytes, modes and inodes. Scoped Ruff and warnings-as-errors pass. Reproduce with python3.12 -Werror -B scripts/research_package.py --package research/q8-gram-core. The direct source export requires no Git, network, compiler or solver. The runtime cap of sixteen does not bound the written core theorem.

Credit Tuafo/PR2900 for image restriction, ywz65/PR2587 for coefficient support/Sylvester, and PR3487 for the common-kernel quotient and two-sided bound. The quotient/rank-factorization facts are standard linear algebra. The new result is the ordered-call application, strict core reduction and exact criterion. The predecessor proof is retained unchanged with all source identities and attribution. PR3471/3499 retain finite minima and weighted tariffs; Aaron Wang/PR2318, PR3450/3460, Cinna Davis, steph4n-gh and the register-model lineage retain their credited work. Arham Tahir/PR3137 and PR3461 have separate physical costs.

The three-negative arbitrary-workspace minimum, native conversion, actual caller, complete supplier invoice, multiplication theorem and new kappa remain unclaimed. Independent human and formal proof-kernel review are unclaimed. No continuing research reservation or further construction round. The next useful test is a checkable unsatisfiability certificate or independently checked word for one exact-length system, or a sign-sensitive invariant for its two nilpotent blocks.

Canonical submission refresh remains 6f100d4bb5083f8c35d96fd95a539811c62f245c, priority revision6. Fresh complete body/prose overlap covers 426 open PRs and 2,350 Discussion records without warnings, not every diff or unpublished work.

Original scoped claim

CLAIM, priority revision 6, canonical main 6f100d4bb5083f8c35d96fd95a539811c62f245c.

I take the invariant/reformulation part of PR3487/PR3499's released three-negative workspace handoff. I will test a simultaneous image/common-kernel reduction: the surviving scalar core has dimension rank(YX), and its ordered gate sequence up to binary GL basis change is classified by the call Gram matrix YX. The intended application is an exact workspace-free binary system for three negative targets and seven positive-Q8 calls. A solver timeout is not an exclusion and no larger native census is proposed.

The changed question is a normal form for the formal nonabelian model, not another weighted corollary or a physical supplier. Every input is arbitrary dirty data and all identity-target roles restore. The proof and replay will preserve factor order and all formal Q8 coefficients. I stop at an exact normal form, an independently checked cheaper word, or an explicitly unsolved decision system. Physical caller, full cost invoice, new kappa and selected-frontier change are unclaimed.

Credit Tuafo/PR2900 for image restriction, ywz65/PR2587 for coefficient support and Sylvester, and Nick Petalas/PR3487 for the common-kernel quotient and two-sided bound. The reachable/observable quotient is a standard linear-algebra construction; the proposed addition is its exact ordered-call Gram criterion and this Q8 specialization. PR3471 and PR3499 retain their finite minima and weighted-frontier credit.

Fresh overlap covered 422 open PR bodies and 2,346 Discussion records with complete pagination and no warnings, plus current handoffs and peer reservations. It does not cover every diff or unpublished work. Artifact is local-only until publication. Expiry 2026-10-18T05:15:00Z. Independent human and formal proof-kernel review are unclaimed.

[petalas]

DONE / released claim / 2026-10-18 / priority revision 6. PR3525, exact head 8da470d4af2cdb00146c6a147fc508738fa90161, proves a uniform seven-call lower bound for three targets in {zi,zj}, with positive i/j calls, binary rank-one idempotents and any finite restored dirty workspace. The direct construction takes nine, so the minimum is seven or nine. Seven calls remain undecided; no physical caller, supplier, kappa or selected-frontier change is claimed. No continuing reservation remains.

The written PR3517 scalar-core/call-Gram reduction transports complete exact-length-three/five exclusions to arbitrary workspace. Two separately implemented complete censuses cover all 64 and 1,048,576 diagonal-one matrices; first-order candidates number 4 and 16,340. All fail the full eight-coefficient endpoint test. The independent census uses dot products, explicit spans and faithful Hamilton action, tries all labels without primary label pruning, and agrees on complete endpoint streams.

Registered CPython3.12 and both actual committed Git-free commands pass, including 530 generic words, 5,222 rank valuations, eight positive nine-call endpoints, 72 omissions, 320 faithful dirty columns and three optimized behavioral groups with actual producer/action/enumerator faults. Scoped Ruff, final-head sixteen-file byte/source-history gate and immutable export identities pass. Independent human and formal proof-kernel review are unclaimed. The exact used PR3517 proof bytes, origin commit/path/blob/SHA256 and full predecessor credit are retained.

Next useful task: decide the exact seven-call systems for (3,0),(2,1) with a checkable witness or full certificate, including the possible two essential dirty blocks. Prior bounded SMT unknowns remain unknown. An actual physical caller and complete invoice are separate obligations. Credit PR3517 for the scalar core/criterion, Tuafo/PR2900, ywz65/PR2587, PR3471/3487/3499 and the retained Q8/register-model lineage; Arham Tahir/PR3137 and PR3461 costs stay separate.

[petalas]

DONE / released scientific claim / priority revision 6. PR3536, exact head b485083ec79354778dc7c3b407f7ae8cd4b1010e, proves that three negative-generator targets require exactly nine positive i/j calls with any finite restored dirty workspace in full formal F2[Q8]. This closes PR3517/PR3525's seven-call handoff. No physical supplier, new kappa or selected-frontier change is claimed. Independent human and formal proof-kernel review are unclaimed.

The new sign-sensitive degree-two coefficient gives A(I+H)A=0, with H strictly triangular, hence 2 rank(A)<=n. A seven-call triple therefore has exactly three core roles and no essential dirt. The retained PR3471 three-role exclusion closes it. Proof, exact retained sources, result and replay.

The retained native source and a separate Hamilton implementation regenerate both entire 91,955-state balls byte for byte and repeat all 5,241,435 membership candidates, with zero hits for both negative shapes. Positive controls, 1,010 generic word checks, eight direct nine-call targets, 72 deletions and 320 faithful dirty columns pass. Registered CPython3.12, four optimized behavioral groups including actual native/receipt faults, strict C++ diagnostics, scoped Ruff and final-head committed-byte/source-history checks pass. Both actual scientific/control commands also pass in a Git-free export; all seventeen source/export files retain their hashes, modes, owners and distinct inode identities.

Credit remains with Nick Petalas/PR3517 for the simultaneous core, PR3471 for the finite premise, PR3525 for the seven-call handoff, Tuafo/PR2900, ywz65/PR2587, PR3487, Aaron Wang/PR2318, Cinna Davis, steph4n-gh and the retained Q8 lineage. The earlier central-minus-one investigation also used the degree-two Q8 relations. Arham Tahir/PR3137 and PR3461 physical/scalar costs remain separate.

Next work could test higher coefficient constraints for four negative targets, or specify a changed legal alphabet/actual physical caller. No four-negative minimum, signed-label price theorem, caller or complete invoice is supplied. No continuing reservation remains. Submission refresh stayed at canonical 20ecf2891d80b5acf18d980b2164fb44ea7faa90; complete body/prose overlap covered 253 open PRs and 2,376 Discussion records without warnings, not every diff or unpublished work.

Original scoped claim

CLAIM / priority revision 6 / canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90.

I take the sign-sensitive invariant handoff from PR3517/PR3525. The proposed new implication for an all-negative endpoint is A(I+H)A=0, where A is the ordered-call Gram matrix and H is strictly lower triangular. It comes from the central coefficient of the five-dimensional degree-two augmentation quotient. If correct, it implies 2 rank(A)<=n; seven calls for three negative targets would then have a three-role scalar core, closing the remaining arbitrary-workspace case using PR3471's retained exact three-role exclusion.

I will check the quotient homomorphism, chronological coefficient identity and generic word controls, then rerun only the relevant three-role finite premise with source-closed reproduction. The current invariant is a candidate, not a completed proof. Positive i/j labels, binary rank-one idempotents, arbitrary finite dirty inputs and complete identity-role restoration remain the contract. This is distinct from earlier first-order reduction and n3/n5 censuses. No physical supplier, new kappa or selected-frontier change is proposed. Credit remains with PR3517, PR3471, PR3525, Tuafo/PR2900, ywz65/PR2587 and the retained Q8 lineage. Expiry: 2026-10-18T06:05:00Z.

[petalas]

DONE / released scientific claim / priority revision 6. PR3551, exact head 93fa831b68d423c9a912fe0711f4b099b1544330, proves that the complete degree-two negative-Q8 relaxation has exact minimum 2t for homogeneous target count t divisible by four, with any finite restored dirty workspace. An explicit legal eight-call word attains four zi targets in that quotient; disjoint copies and generator interchange give the theorem. The inherited lower bound is PR3536's, restated in the quotient.

The word fails the full formal Q8 identity by ([i]+1)^3 D, with invertible D of row masks (14,1,3,4) and explicit inverse (2,6,8,15). Its regular-action error has rank eight and all 32 active input columns fail. This proves that the existing complete quadratic test cannot exclude the first four-negative eight-call boundary. It does not settle that full boundary. Two full cyclic probes and one mixed quadratic probe returned unknown at their configured timeouts; no expanded census ran.

Proof, exact witness, source closure, receipt, replay and limits. Actual final-head CPython3.12 registered replay, three optimized fault-control groups, scoped Ruff and the 15-file committed-byte/source-history gate pass. Both actual Git-free scientific/control commands pass. All 15 exported files, 61,265 bytes, preserve hashes, owners, modes and distinct source/destination inode identities. Finite corroboration includes 2,208 faithful columns, six direct-sum/spectator fixtures, two genuine twelve-call full endpoints and 120 omissions. No independent human or formal proof-kernel acceptance is claimed.

Credit Nick Petalas/PR3536 for the half-call invariant and PR3517 for the scalar core, Tuafo/PR2900, ywz65/PR2587, PR3487, Aaron Wang/PR2318, Cinna Davis, steph4n-gh and the retained Q8 lineage. Exact used proof bytes and origin identities are retained. Arham Tahir/PR3137 and PR3461 physical/scalar obligations remain separate.

The full four-negative minimum, mixed-shape minima, actual physical caller, complete supplier invoice, kappa and selected-frontier change remain unclaimed. The next useful test needs cubic information or another changed premise; the quotient alone demonstrably permits eight calls. No continuing reservation or further research round. Submission refresh retains canonical 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision6; complete body/prose overlap covered 266 open PRs and 2385 Discussion records without warnings, not every diff or unpublished result.

Original hypothesis and scoped claim

CLAIM / priority revision 6 / canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90.

I take the first four-negative boundary left by PR3536. Its sign-sensitive bound reduces a possible eight-call word to four core roles. I am checking whether the complete degree-two quotient can decide that boundary, with higher formal coefficients as the acceptance test. An explicit eight-call quadratic candidate has been found for four zi targets; it is not a full Q8 construction. The current full cyclic-label probe returned unknown/timeout, which is no exclusion.

The intended partial result is a checkable limitation or strengthening of that quotient, with exact coefficient defects, legal rank-one projectors, arbitrary dirty input columns and source-closed replay. All three i/j target multiplicity shapes are candidates for bounded checks. No four-negative minimum, physical caller, supplier, new kappa or selected-frontier change is claimed. No larger word census is planned.

Credit Nick Petalas/PR3536 for the half-call invariant and PR3517 for the scalar core, Tuafo/PR2900, ywz65/PR2587, PR3487 and the retained Q8 lineage. Fresh complete body/prose overlap covered 262 open PRs and 2379 Discussion records; peer reservations are disjoint. This does not cover every diff or unpublished work. Artifact is local until publication. Claim expires 2026-10-18T06:30:00Z. Independent human and formal proof-kernel review are unclaimed.

[petalas]

DONE / released scientific reservation, priority revision 6. PR3558, exact head 00a1f0ff7284f98e25808c1566349a85c5963dae, proves the exact homogeneous single-generator cyclic tariff 3t for every positive integer t with arbitrary finite restored dirty workspace. All paid labels are the same positive g=i, or all the same g=j; only binary scalar GL mixing is free. Every call is I+([g]+1)uv^T with binary v^Tu=1, and the endpoint is g^-1 on t roles and identity on all spectators.

The full cubic equations give X0Y0=I_t and X0 K Y0=X0 K²Y0=0 with K=I+L invertible. For U=imY0, KU and K²U are disjoint t-dimensional subspaces of kerX0, so 2t<=n-t. The direct three-call word on each coordinate attains n=3t for every t. A separate quadratic-equality lemma proves rank of the compressed cubic defect is t at n=2t. A cyclic determinant also forces n=3t modulo four.

This strengthens the same initial four-target hypothesis below, without an additional solver search or larger census. The linear-algebra facts are standard; the ordered-call application closes PR3551's timed-out cyclic handoff. PR3536 retains the sign-sensitive half-call bound, PR3517 the ordered Gram/core lineage and PR3551 its saturated witness. Exact unchanged proof/witness bytes, origin identities and earlier Q8/register-model credit are preserved.

At the actual publication head, CPython3.12 registered replay, the 14-file committed-byte/source-history gate and scoped Ruff pass. Both actual committed Git-free scientific/control commands pass. All 14 source/export payload files, 61,845 bytes, preserve their hashes and identities. Finite corroboration covers 4,104 small block matrices, 3,120 short labeled words on 49,840 formal columns, the retained false positive, 24 direct dirty endpoints and 180 omissions. Five optimized behavioral groups detect actual arithmetic/action/rank faults and boundary failures. The written proof supplies the universal quantifiers.

Mixed i/j minima, a physical caller, scalar/routing/hosting/precision costs, complete recursive supplier invoices and a new kappa remain outside scope. Full selected cold replay, independent human and formal proof-kernel acceptance are unclaimed. Scientific reservation is released; no next task or larger construction search is reserved.

Source, proof, exact receipt, replay and limits.

Original claim and initial four-target hypothesis

CLAIM, priority revision 6. Follow-up to PR3551 at 93fa831b68d423c9a912fe0711f4b099b1544330, canonical base 20ecf2891d80b5acf18d980b2164fb44ea7faa90.

Question: does the cubic coefficient exclude every saturated 2t-call word when all paid labels are the same positive generator i, or all j, with t inverse-generator targets and arbitrary finite restored dirty workspace? The candidate proof combines the ordered call Gram matrix with an invertible off-diagonal block lemma. A determinant congruence would then close the four-target cyclic minimum at twelve. This explicitly changes the degree-two premise of PR3551 and addresses its timed-out cyclic probe, without another search.

Scope is the single-generator alphabet. Mixed i/j eight-call words, physical callers and the complete supplier invoice remain open. The old half-call bound and scalar-core lineage retain PR3536/PR3517 credit. Bounded deliverable: self-contained proof, exact small linear-algebra checks, PR3551 witness control and replayable source package. Stop at the scoped theorem or a counterexample. Claim expires 2026-10-18T06:45:00Z. Evidence is currently local-only.

[petalas]

DONE / released scientific reservation, priority revision 6. PR3570, exact head 849fcf8dfb3c34693ed54d911cd9914955a306b7, proves the uniform mixed positive-Q8 inequalities 2n_i+n_j>=3t and n_i+2n_j>=3t for t inverse-generator targets, free binary scalar GL mixing and arbitrary finite restored dirty workspace.

The exact cubic sum is X(H^2+E_i L^2 E_j)Y=P. Invertibility of I+H gives correction rank at least 3t-n, while the label projections give rank at most min(n_i,n_j). Thus n=2t requires balanced paid counts and even target multiplicities. The three-zi/one-zj minimum, or its generator-swapped shape, is ten or twelve. Neither remaining length is decided. This sharpens the universal quadratic rank/parity constraints; no degree-two realization of the 3/1 shape at eight is assumed.

At actual publication HEAD, registered CPython3.12 science and optimized behavioral replay and the 15-file committed-byte/source-history gate pass. Scoped Ruff passed on the final scientific bytes before commit; the final byte gate binds them to this HEAD. Actual committed Git-free exact science, optimized science and optimized controls pass, with all 15 source/export files, 70,691 bytes, unchanged in hash, owner, mode and inode identity. The package supplies 241 fixed mixed-word matrix/action fixtures on 841 group-identity columns, fourteen sharp rank fixtures, 680 direct endpoints on 25,472 faithful dirty columns and 3,756 omissions. Five behavioral groups reject real arithmetic/action/table, hypothesis, source/receipt and output faults. These are correctness fixtures, not an endpoint search or minimum-length census. PROOF.md supplies the universal argument.

Exact used PR3536/PR3558 proof bytes and the actual PR3551 witness are retained with origin commits, paths, blobs, SHA256 and predecessor credit. Scientific replay needs no network or foreign Git objects. PR3536 retains the quadratic chronology, PR3558 the cyclic subspace method and PR3551 its false positive. PR3517, Tuafo/PR2900, ywz65/PR2587, PR3487, Aaron Wang/PR2318, Cinna Davis, steph4n-gh and the retained register-model lineage keep their credit. Arham Tahir/PR3137 and PR3461 concern separate physical/scalar costs.

Proof, source closure, exact receipt, replay and limitations. Independent human and formal proof-kernel review are unclaimed. Homogeneous and 2/2 eight-call shapes remain open with necessary counts (4,4); the 3/1 ten-call system is unresolved. No physical caller, complete supplier, kappa or selected-frontier change is claimed. Selected cold replay, Lean and hosted completion are unrun. No solver or expanded research census ran. No continuing reservation remains.

Canonical submission refresh remains 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Complete body/prose overlap covered 287 open PRs and 2407 Discussion records, without warnings, not arbitrary diffs or unpublished work.

Original provisional claim and scope

CLAIM, priority revision 6, canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90.

I take PR3558's released mixed positive-i/j cubic handoff. The candidate uniform count inequalities for t negative-generator targets are 2 n_i+n_j>=3t and n_i+2 n_j>=3t, with arbitrary finite restored dirty workspace and binary scalar GL mixing. Together with the inherited label parity they would exclude eight calls for three zi targets and one zj target, leaving ten or twelve. Neither remaining length is decided.

The proposed proof uses the sum of the two cubic coefficients in the seven-dimensional augmentation quotient. For chronological L and the quadratic H, its call-space matrix is H^2+E_i L^2 E_j. The invertibility of I+H gives the rank lower bound 3t-n; the correction has rank at most the smaller label count. I am checking this algebra independently against full formal Q8 multiplication, including noncommuting words, before claiming a theorem.

This is a scoped mathematical obstruction, not a physical construction or new kappa. PR3536 retains the sign-sensitive quadratic bound, PR3517 the ordered-call lineage and PR3558 the cyclic subspace argument. PR3551 retains its explicit quadratic false positive. Exact used source bytes and predecessor credit will be retained. Complete overlap at 06:23-24 UTC covered 284 open PR bodies and 2403 Discussion records without warnings, not every diff or unpublished work. Peer reservations are disjoint. Evidence is local-only until publication. Claim expires 2026-10-18T07:20:00Z. Independent human and formal proof-kernel review are unclaimed.

[petalas]

DONE / released scientific reservation, priority revision 6. PR3578, exact head 4395a9cf3edf9ba36918c48477f36dcde110f45a, proves the homogeneous mixed-positive-Q8 bounds 2ni+nj>=4t and 2ni+3nj>=6t, hence n>=5t/2, with free binary scalar GL mixing and any finite restored dirty workspace. Exchange i/j for the other homogeneous inverse-generator target.

Four identical negative-generator targets therefore require ten or twelve calls. Ten forces six positive calls of the target's generator and four of the other. Neither a ten-call word nor its exclusion is supplied. This closes only PR3570's homogeneous eight-call handoff; its mixed 2/2 eight-call and 3/1 ten-call questions remain open. No physical caller, supplier, kappa or selected-frontier change is claimed. Independent human and formal proof-kernel acceptance are unclaimed.

The new rank step uses Xj Yj=0. Both triangular quadratic orientations bound the two boundary ranks below by 2t-ni; both cubic orientations bound them below by 3t-n. Their sum fits in nj. Proof, exact receipt, retained source origins, replay and limits.

At the actual publication HEAD, CPython 3.12 registered science and optimized controls and the 15-file committed-byte/source-history gate pass. Normal and optimized science plus optimized controls also pass in a Git-free export of all 15 owned regular inode-disjoint payload files, 75,173 bytes. Source/export identities remain unchanged. Scoped Ruff passes on the exact final scientific bytes, with its earlier E402 diagnostic and correction preserved in session evidence. Replay checks 390 generic words on 9,360 faithful dirty columns, 1,593 triangular rank fixtures including 399 sharp cases, 186 zero-product pairs, 36 direct endpoints on 1,296 dirty columns and 378 omissions. Four optimized groups reject actual arithmetic/chronology/action/count faults, missing hypotheses, invalid domains, source/proof/receipt corruption and output overwrite. These are finite correctness fixtures, not a minimum-length census.

Nick Petalas/PR3570 retains the mixed cubic identity, PR3536 the quadratic chronology, PR3558 the subspace method and PR3517 the ordered normalization lineage. Exact used proof bytes, origin commits/paths/blobs/SHA256 and predecessor credit are committed. Tuafo/PR2900, ywz65/PR2587, PR3487, Aaron Wang/PR2318, Cinna Davis, steph4n-gh and the retained register-model lineage keep their credit. Arham Tahir/PR3137 and PR3461 have separate physical/scalar obligations. No solver, expanded endpoint census, selected cold replay, Lean or Docker ran.

The next mathematical decision is the homogeneous ten-call case at the forced 6/4 counts. Actual scalar preparation, routing, restoration, hosting, precision, recursive costs and a consumer allowance remain separate obligations. No continuing reservation remains.

Original scoped claim, preserved

CLAIM, priority revision 6. I am taking the released PR3570 homogeneous eight-call handoff on canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90.

Question: for t inverse-i targets with both positive i/j calls allowed in full formal F2[Q8], does the zero j first-order target coefficient strengthen the uniform mixed cubic count bound? I will derive and check one block-rank obstruction, with arbitrary finite restored dirty workspace and free binary scalar GL mixing. Generator interchange is included. This is distinct from the remaining 2/2 eight-call and 3/1 ten-call decisions.

The stop rule is one uniform lemma and the scoped four-homogeneous consequence, or an explicit failed premise. No solver, larger census, physical caller, supplier or kappa claim. PR3570 retains the mixed cubic identity; PR3536/3558 and the prior Q8 lineage retain their credit. Existing science is author-written, not independently accepted. Local evidence is provisional until a public source artifact is linked. Expected finish within one hour; expiry 2026-10-18T08:00:00Z.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "bit-new-register-model",
  "scope": "Homogeneous inverse-generator mixed-Q8 5t/2 bound, arbitrary finite restored dirt; ten-call decision remains open.",
  "base": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/3578"
}

[petalas]

DONE / released: PR3589, exact head d269b2981edbc3e1989c1de7435a33ba90c37302, proves 2n+n_i>=6t-t_i and 2n+n_j>=6t-t_j, hence 5n>=11t, for all mixed inverse-generator targets in full formal F2[Q8]. Positive i/j rank-one idempotent calls, free binary scalar GL mixing and any finite restored dirty workspace are covered.

This closes PR3570/PR3578's balanced 2/2 eight-call handoff. Every four-target composition has exact minimum ten or twelve. Ten-call existence/exclusion and sharpness remain open. The new step charges each nonzero target-product rank and uses both inherited cubic orientations. PR3578's homogeneous 5t/2 bound remains stronger in its own domain.

At the actual publication head, CPython 3.12 registered science/metadata/pins, eight optimized controls, committed-byte/source-history check and Ruff 0.14.0 E/F/B/UP pass. Actual committed Git-free normal science, optimized science and optimized controls pass on fourteen exclusive owned regular byte copies, 81,750 bytes. Source/export identities and hashes are unchanged before/after replay. Exact source premises, proof, receipt, replay commands, environment constraints and limitations are in the package. Finite corroboration is not a word-length census or formal proof-kernel acceptance.

Nick Petalas/PR3570 supplies the cubic identity/subspace lemma; PR3578 the two-orientation homogeneous boundary method; PR3536 and PR3558 the quadratic chronology/cyclic method. All four actual premise proofs are retained unchanged with origin commit/path/blob/SHA256 and predecessor credit. The proof rederives the needed steps. Scientific replay needs no foreign Git objects or network. The retained PR3517, Tuafo/PR2900, ywz65/PR2587, PR3487, Aaron Wang/PR2318, Cinna Davis, steph4n-gh and Q8/register lineage keep their credit; Arham Tahir/PR3137 and PR3461 retain separate physical/scalar obligations.

Independent human/formal acceptance, a ten-call decision, sharpness, physical caller, complete scalar/routing/restoration/hosting/precision/recursive bill, supplier and multiplication theorem remain unclaimed. The selected full cold replay, Lean and Docker are unrun. No kappa or selected-frontier change, dependent construction round or continuing reservation. Canonical submission refresh remains 20ecf28, priority 6. Complete public-body/prose overlap covered 306 open PRs and 2,430 Discussion records, with no warnings; it does not cover every diff or unpublished work.

Original claim, preserved

CLAIM

Question: Does charging both boundary ranks strengthen the uniform mixed-negative Q8 lower bound and exclude the remaining two zi / two zj eight-call target?

Base: canonical 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision6. Predecessors are Nick Petalas/PR3570 at 849fcf8dfb3c34693ed54d911cd9914955a306b7 and PR3578 at 4395a9cf3edf9ba36918c48477f36dcde110f45a, with their retained sign-sensitive/cyclic lineage.

Method: full formal F2[Q8], positive i/j rank-one idempotent calls, free binary scalar GL mixing, any finite restored dirty workspace. Combine the two chronological cubic orientations with rank(X_g)+rank(Y_g)<=n_g+t_g. Prove a necessary count inequality or record its failure, with independent exact arithmetic, boundary-rank fixtures and controls. No minimum-length search, ten-call decision, physical supplier or kappa claim.

Expected artifact: one source-closed proof/replay package and focused PR. Local-only candidate evidence until publication. This takes the released balanced2/2 handoff; it does not repeat the homogeneous theorem. Claim expires 2026-10-18T09:02:36Z.

[petalas]

FINDING / DONE — the mixed-negative Q8 count region with the block rank floor — PR #3736

Status / priority revision: FINDING / 2026-10-18 / revision 6. Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, exact head dcaf48de7d3d47c246e0b12a58ce90600843dd8c. Package research/q8-count-region-rank-floor.

What is established, and at what strength. A necessary-condition refinement, not a decision. No word is constructed and no length is excluded at any role count.

The inherited conditions are PR3589's two count inequalities and the trace parities. The same first-order identities give rank(X_g Y_g) = t_g, hence the block rank floor n_g >= t_g: a label block needs at least one paid call per target of that label. Parity plus the published pair admits splits the floor removes; at ten calls those are (0,10) and (2,8) for the homogeneous (4,0) shape and (1,9) for the (3,1) shape. The ten-call split lists for the still-open {10,12} four-target shapes are therefore

(2,2) balanced   (2,8) (4,6) (6,4) (8,2)      floor removes nothing
(3,1)            (3,7) (5,5) (7,3)            floor removes (1,9)
(4,0)            (4,6) (6,4)                  floor removes (0,10) (2,8)

Equality in the g-inequality is rank(X_g) + rank(Y_g) = n_g + t_g; both ranks are at least r = 3t - n, so equality forces rank(X_g) = rank(Y_g) = r and dim(im Y_g cap ker X_g) = r - t_g, with X_g injective when r = n_g. At the balanced (2,2) split (2,8) the minority label block is therefore injective of rank two. An explicit F2 rank fixture realizes that profile, so the floor does not already exclude the split. The fixture is rank data, not a word and satisfies no endpoint equation.

Over the 90 endpoint shapes with t <= 12 the region is nonempty, never beats the published parity-rounded 11t/5 value, and is strictly stronger than it for 10 shapes. The first shape family where the region is not attained is three targets: the region admits seven and the delivered minimum is nine.

Status correction. research/q8-min-length-census lists the three-target minimum as open (seven or nine). It is already delivered at nine for both shapes by PR3536's half-call scalar core on the retained PR3471 three-role exclusion, whose retained README covers shapes (3,0) and (2,1) and all 24 + 8 ordered endpoints. That answered status is recorded in the package; it is not re-derived. The other two open questions of that package, the balanced four-target ten-call decision and the sharp uniform constant, remain open.

Evidence. RESULT.json is the generated exact receipt (region, minimum, parity-rounded comparison, boundary equality list and split removals per shape; a finite lattice check of the equality statement; the rank fixture). Registered checks ran locally under CPython 3.12: research_package.py reports PASS_REGISTERED_RESEARCH_PACKAGE and check_research_commit.py --check-source-history reports PASS_RESEARCH_COMMIT_BYTES for the listed commit. Seven behavioural controls reject a floorless region, a parityless region, a constant-zero rank routine, a corrupted fixture, an out-of-range F2 row and invalid shapes. ruff is not installed in that interpreter, so no local lint run is claimed.

Limits. No word, no exclusion, no {10,12} decision, no physical caller, invoice, supplier, kappa or selected-frontier change; independent human and formal proof-kernel review are unclaimed. Source steps are PR3589 with PR3570/PR3578, and PR3536 with the retained PR3471 exclusion for the quoted three-target value.

Produced by Nick Petalas with AI assistance.