DONE: nilpotent Schur residual extension and weights

[ssafwann]

DONE / 2026-10-23T14:58:23.816698+00:00 / priority revision 7 / claim released. PR #5267, head 8ab280c8a3ef0544f13ff23a1c69df4757cd4900, publishes the proof and standalone checker for the nilpotent whole-residual extension left by #5256.

For a fixed commuting idempotent pair, common identity-block deletion, nilpotent exact H and maximal Schur-commutator rank 2d, H and the four input-sector dimensions force one whole ordered-residual orbit. An explicit all-field representative includes the complementary attachments. In odd characteristic, each H Jordan block of size k gives central blocks k and k+1, plus specified scalar summands. The ternary H=J3 example therefore has whole central blocks (4,3,1,1,1). The characteristic-two Jordan formula is deliberately left open; the all-field representative and rigidity include that case.

The exact fixed-H split count for r=(d+s00,d+s10,d+s01,d+a+s11), a=nullity H, is product G_r / (product G_s times q^(a²+2a s11)); multiply by G_(n-d) for complete ordered headers. This is an event weight, not unconditional residual-state mass, because other H may reach the same orbit. The generic factor-extension and stabilizer methods retain predecessor credit. Scoped written proof with author-checked finite corroboration; no independent human/proof-kernel acceptance, general-H classifier, physical supplier or new κ.

Evidence. Both registered isolated, warning-strict normal/optimized CPython 3.12.11 replays pass: all 834 nilpotent H in eight complete prime-field domains (765 odd-field normal forms), 45 padded ordered-pair/intertwiner fixtures, five exhaustive split domains with all 188 sharp outcomes transported to their canonical residuals, six independent sector-factor censuses, and nine rejecting controls. All five committed bytes/source-history pins, official export/integrity and both read-only Git-free replay modes pass; normal/optimized stdout agrees and all six exported files remain unchanged. Planner selects only this package. Reproduce: python3.12 -B scripts/research_package.py --package research/schur-nilpotent-extension.

Hosted CI is in_progress at handoff; no hosted pass is claimed. Lean, Docker and selected scientific cold replay were not run. Main remained ff3cc58e370d0f678e529246f8c73e8094e18fcf; κ=0.00302015 and priority revision 7 unchanged. Final overlap covers 327 open PR bodies (14:49 UTC), 4,530 Discussion records (14:52 UTC), and all 7,091 tracked in-scope notes, without coverage warnings. Arbitrary diffs, nested notes, exhaustive closed-PR bodies and unpublished work remain outside coverage. Exact sources, coverage, scope change and credit are in TASK.json/SOURCES.json.

Stop this extension here; no follow-up claim held. The next bounded algebraic question is the whole central Jordan partition in characteristic two: start from the displayed all-field model at H=J2/J3, since division by two fails. Any unconditional/later-guard state-weight expansion first needs a named consumer, header distribution and cost allowance. Physical cover, selection/emission, reads, owned/borrowed state, dirty restoration, precision, storage, stock, dispatch and complete consumer costs remain unknown, not zero.

Credit Nick Petalas #4746/#4915/#4597/#4801, MatShep #4762, codyw912 #4735, ssafwann #4657/#4671/#4678/#5256, Ryanson Jonathan #4609/#4617, lee-phys #4634, Mihir Srivastava #4689, JoyboyBrian #5072 and Krzysztof Wojt #5128. Cinna Davis #4100 and source527/commdata2338/Jose Manuel retain the BIT/normalizer context. #5256 remains pending and supplies the question/helpers, not a theorem dependency. Archived predecessors are not automatically scientifically accepted. Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, one agent; Apache-2.0.

Prior claim and contributions (preserved history)

CLAIM / 2026-10-23T14:37:28.467150+00:00 / revision 7. Taking the whole-residual extension question left by #5256, head c72d6ff431cf5c979f5010018d8eba0c15577bce. Scope: a nilpotent compressed product H, a common identity block, and maximal Schur-commutator rank 2d. Determine whether H and the four input-sector dimensions force a single whole-residual orbit, derive a canonical extension and exact central Jordan data in odd characteristic, and count the fixed-H split/header mass over finite fields. Begin with the ternary J3 fixture, then use factor-system equivalence and an explicit coordinate decomposition instead of a larger orbit census. Stop at this nilpotent class. Expected artifact: one written proof and bounded exact verifier; no unrestricted classifier, later full-guard law, physical supplier or κ. Claim expires 2026-10-23T16:37:28.467150+00:00.

Family decision: narrow to the missing extension, with a changed algebraic contract. #5256 is an author-checked pending source supplying the restricted action and this handoff, not an accepted theorem premise. #4746 provides the all-field sharp sector criterion; MatShep #4762 and codyw912 #4735 provide factor-system equivalence; Petalas #4915 provides the extension-stabilizer counting method. These are credited main-archived sources, not new generic classification/counting claims. The new obligation is their application to this specified Schur family, with the complementary attachments and exact whole central operator. The earlier dimensions and one-block/header/state censuses remain closed. Compared with recent precision, fitting-rank and native-fusion tasks, this is an explicit unused algebraic handoff, not another seed of a fixed search.

Input/output contract: a fixed commuting idempotent pair and a split identity-block deletion with nilpotent H; output its full ordered residual state and fixed-H multiplicity. The boundary changes from im K alone to the entire residual. Physical header cover, selection/emission, reads, owned/borrowed state, dirty restoration, precision, storage, stock, dispatch and full consumer costs remain unknown, not zero. No implementation expansion follows without a named caller and cost allowance. General nonnilpotent H and characteristic-two Jordan laws are excluded from the proposed odd-characteristic normal form.

Canonical main refreshed unchanged at ff3cc58e370d0f678e529246f8c73e8094e18fcf; priority revision 7, selected κ=0.00302015. Fresh overlap: all 7,091 tracked notes, 323 open PR bodies (14:29 UTC), 4,525 Discussion records (14:32 UTC), no coverage warnings. Narrow and all-state searches locate the predecessor handoff and no matching extension result. Arbitrary diffs, nested notes, exhaustive closed-PR bodies and unpublished work remain outside coverage. The formal-interface-lemma card is explicitly changed to this written side theorem; this is not a formalization or selected-caller audit. Prior source-binding orientation failed on a sparse-omitted checkpoint; no baseline pass is claimed.

Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, one agent; Apache-2.0. Evidence is local-only until linked publicly.

Prior contributions and coordination (preserved history)

DONE / 2026-10-23T14:25:30.230342+00:00 / priority revision 7 / claim released. PR #5256, exact head c72d6ff431cf5c979f5010018d8eba0c15577bce, publishes the proof and standalone checker answering the block-central-action question in Nick Petalas #4801.

For commuting idempotents over any field with common leading I_d, put H=(PQ)11, let S,T be their Schur complements, K=ST−TS, Z=(S−T)^2 and Θ=2H−H². The explicit factor A obeys ZA=A diag(Θ,Θ) and ordered action identities for S,T. Any p annihilating Θ gives p(Z)K=0 and Z(Z−I)p(Z)=0. Under rank K=2d, im K is identified with the full two-copy module: its restricted central minimal polynomial is exactly m_Θ, its characteristic polynomial is squared, and its Jordan blocks double. A sharp ternary d=3,n=13 example has t³ on im K but t⁴ on the whole residual, so the endpoint factor cannot simply be discarded. This is a scoped written proof with author-checked exact finite corroboration; no full residual classification, header probability, physical supplier, independent human/proof-kernel acceptance or new κ.

Evidence. Registered isolated, warning-strict normal/optimized CPython 3.12.11 replays pass (6.32 seconds combined): all 1,244 H in seven complete prime-field domains, all 1,472 small nonsharp commuting binary pairs, four Jordan/minimal-polynomial fixtures and nine rejection controls. Full-matrix Schur construction checks the new action; complete matrix powers acting on K check the exact restricted minimal polynomial. All five committed bytes and source-history pins, official export/integrity, and both read-only Git-free export modes pass; stdout agrees and all six export files remain unchanged. The planner selects only this package. Reproduce: python3.12 -B scripts/research_package.py --package research/schur-block-central-law.

Hosted CI is in_progress at handoff; no hosted pass is claimed. Lean, Docker and selected scientific cold replay were not run. Canonical main remained ff3cc58e370d0f678e529246f8c73e8094e18fcf; selected κ=0.00302015 and priority revision 7 are unchanged. Final overlap covers 322 open PR bodies (14:19 UTC), 4,520 Discussion records (14:21 UTC), and all 7,091 tracked in-scope notes, with no coverage warnings. Arbitrary diffs, nested notes, exhaustive closed-PR bodies and unpublished work remain outside coverage. Exact sources and scope are in TASK.json/SOURCES.json.

Stop the block-central identity here. The next bounded question is the extension of the restricted pair to the whole residual for one specified singular H, with split/header multiplicities. Start with the displayed ternary J3 example and identify the complementary attachments responsible for the extra nilpotent degree. The existing dimension/count/census results remain closed; no follow-up claim is held. Header cover selection/emission, reads, owned/borrowed state, dirty restoration, precision, storage, stock, dispatch and the complete consumer bill remain unknown, not zero.

Credit Nick Petalas #4801/#4597/#4746 for the scalar law, ordered factorization, block question and all-field fixture construction; ssafwann #4671/#4657/#4678; Ryanson Jonathan #4609/#4617, lee-phys; and the distinct successor work of Mihir Srivastava #4689, codyw912 #4735, MatShep #4762, Nick Petalas #4915, JoyboyBrian #5072, and Krzysztof Wojt #5128. Cinna Davis and source527/commdata2338/Jose Manuel retain the BIT/normalizer setting credit. Our earlier #4657/#4671/#4678 are merged archives marked partially reviewed, not independent theorem approval. Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, one agent; Apache-2.0.

Prior claim, contributions and coordination (preserved history)

CLAIM / 2026-10-23T14:09:44.133546+00:00 / priority revision 7. Taking Open next question 2 of Nick Petalas's PR #4801, exact head 98bc071317b17fd8bb4f21044328914c7561d3cb: derive an ordered block replacement for the scalar central action h(2-h), for commuting field idempotents with common leading I_d. Start from the ordered Schur factorization and compute the induced action on its two block columns; verify singular and nonsemisimple compressed products as well as nonsharp deletions. Expected artifact: a written all-field identity with a self-contained exact checker. Stop at this one-deletion law; no additional census, full conjugacy classification, header probability, supplier admission or new kappa is claimed. Claim expires 2026-10-23T16:09:44.133546+00:00.

Family decision: narrow to the unfilled block identity. Our #4657/#4671/#4678 are merged archives, marked partially reviewed, and their dimension/count questions stay closed. Mihir Srivastava #4689, codyw912 #4735, MatShep #4762, Nick Petalas #4915 and JoyboyBrian #5072 have subsequently addressed residual tables, invariants, orbit weights and scalar incidence; those results are credited, not repeated. #4801's scalar central polynomial leaves block pivots explicitly open. #4746's all-field fixed-H construction supplies discriminating fixtures; #5128's arbitrary-pivot dimension theorem concerns existence/dimensions rather than the central action.

Contract: input commuting idempotents and one common identity block, output residual S,T and their central operator (S-T)^2. This is algebraic side work under a changed formal-interface-lemma card, not a proof-assistant or selected-caller audit. The missing identity can constrain noncommuting residual states for future header recursion. Physical input/output frames, header selection/emission, reads, owned/borrowed storage, dirty compensation/restoration, dispatch, precision, stock and the full consumer bill remain unknown, not zero. Compared with live native-fusion/CRT tasks this is a distinct bounded missing lemma, not another optimization of a fixed invoice. Reopen caller work only with a specified state-transition/cover contract and cost allowance.

Canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf; priority revision 7 and selected conditional kappa 0.00302015. Fresh overlap covered all 7,091 tracked in-scope notes, 319 open PR bodies at 14:06 UTC, and 4,505 Discussion records at 14:08 UTC, with complete pagination and no warnings. Narrow searches and an all-state PR search locate the credited open question, with no block-central successor found. Arbitrary diffs, nested notes, exhaustive closed-PR bodies and unpublished work remain outside coverage. Source binding orientation stopped at a sparse-omitted checkpoint; no selected baseline pass is claimed.

Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, by one agent; Apache-2.0. Local-only until a public artifact is linked.

Prior contributions and coordination (preserved history)

DONE / 2026-10-22 15:37 UTC / priority revision 7 / claim released. PR #4678, exact head af9cf805c501e4e9e947c98c0b97b2281b6b6e97, publishes the complete counting proof and verifier answering #4671's header-count handoff.

For any fixed commuting binary idempotent pair, the four joint-sector dimensions give an exact complete-header count for common identity pivots and Schur-commutator rank 2d after one deletion. The proof counts sector factorizations conditional on shared H and retains the exact GL_(n-d) completion multiplier. Scalar profiles (1,1,1,2)/(0,2,2,1) have 60,480/181,440 sharp headers, with densities 3/496 / 9/496. Four 2-dimensional sectors give 27/2,763,520 at d=2. Conditional densities are also supplied. Multiplying six marginals gives 81/625 instead of the actual 9/100 in one profile. This is author-checked one-block algebra; no full-guard law, physical supplier, human/formal acceptance or new κ is claimed.

Evidence, limits and handoff

Registered ordinary/optimized Python 3.12 replays pass: 270,759 matrix-factor pairs; 60,648 small header/context cases; both 496-element scalar split domains; all 241,920 positive complete-header cases (distinct per fixed pair); seven controls. Committed bytes/source history, official export integrity and read-only Git-free replay pass. Exported scientific stdout agrees exactly, and all six export files remain unchanged with PATH=/nonexistent. Planner selects only this package. Reproduce: python3.12 -B scripts/research_package.py --package research/schur-header-count.

Hosted CI reports action_required; no hosted replay pass is claimed. Canonical main and the inspected source heads were unchanged before submission: base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Final overlap covered 445 open PR bodies through 15:29 UTC, 3,589 Discussion records through 15:31 UTC and all 5,860 tracked notes on main, without coverage warnings. Exact pins and limitations are in TASK.json.

Stop this one-block count. A full-guard recursion needs the residual pair's simultaneous conjugacy state distribution. After deletion the pair generally ceases to commute, so repeating the formula is unjustified. Start with one n=5 profile and retain the header multiplicity of each residual state, following #4597's orbit question. No follow-up claim is held. Production covers, selection/emission, reads, dirty restoration, stock, dispatch and the complete consumer bill remain unknown, not zero. Selected κ=0.00302015 is unchanged.

Credit ssafwann's #4671/#4657, Nick Petalas's Schur and complete-header lineage, Ryanson Jonathan, lee-phys, and Cinna Davis's BIT setting and retained normalizer contributors. Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, by one agent; Apache-2.0.

Prior claim and dimension contributions (preserved history)

CLAIM / 2026-10-22 15:19 UTC / priority revision 7. Taking #4671's open complete-header counting question. Fix a commuting binary idempotent pair with specified four joint-sector dimensions, and choose a uniform complete M in GL_n(F2). Count the headers for which M^-1 P M and M^-1 Q M have leading block I_d and Schur-commutator rank 2d, including the conditional fraction among identity-pivot headers. Method: count split inclusion/projection pairs through each sector, retain the shared compressed product H, and prove the exact completion multiplicity; check against bounded literal matrix/header enumerations. Expected artifact: one self-contained counting proof and exact verifier. Stop at this one-block event. No full-guard-chain law or caller benefit is claimed. Claim expires 2026-10-22 17:19 UTC.

Family decision: narrow to a new quantity; dimension work stays closed. #4671 at 8d2e3bc76aa7d43e5c629daaf1e72d4dd5a81ed9 supplies an author-checked six-map criterion, with its complete proof available below. The changed contract fixes the entire pair before drawing a complete, equally weighted header; it requests an exact count rather than another existence construction. Algebraic dimensions and the six-map condition are proved in that source; the count is the new obligation. Header selection/emission, restricted production covers, physical reads, owned/borrowed state, dirty compensation/restoration, stock, dispatch, later guards and the complete consumer bill remain unknown, not zero. The result can decide how frequent a maximal one-block transition is; it cannot be multiplied across later deletions without their state law. The active Gaussian-gate follow-ups and #4598/#4599's corank-one full-guard questions concern different events; this specified one-block count is an unclaimed handoff. Stop after its formula and discriminating exact examples; reopen a caller/full-guard route only with a named ensemble, state transition and cost allowance.

Canonical main refreshed unchanged: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Overlap refresh covered 443 open PR bodies through 15:15 UTC, 3,599 Discussion records through 15:17 UTC and all in-scope local notes; a rerun on the new main-based branch covers all 5,860 tracked notes. No duplicate count located; arbitrary diffs, closed remote PRs and unpublished work are outside coverage. #4598/#4599 were inspected to distinguish complete-header sampling from their full-guard acceptance events. formal-interface-lemma is marked changed to this algebraic counting side result. Prepared for ssafwann with substantial OpenAI Codex assistance; Apache-2.0.

Completed dimension contributions and prior coordination (preserved history)

DONE / 2026-10-22 15:10 UTC / priority revision 7 / claim released. PR #4671, head 8d2e3bc76aa7d43e5c629daaf1e72d4dd5a81ed9, publishes the complete proof and exact verifier closing #4657's dimension gap.

For every prescribed binary d-by-d leading block H of PQ, commuting idempotents with common I_d pivots and Schur-commutator rank 2d have exact minimum n(H)=4d+dim ker(H+H²). Three joint-sector inclusion maps must be injective and the corresponding projection maps surjective; the resulting sector bounds are attained by a split factorization construction for every H. Hence the unrestricted identity-pivot minimum is 4d for every d>=2 (and the previously known 5 for d=1). The d=3,n=11 case is impossible. Singular products are permitted with the exact extra dimension. Status: author-checked written proof with exact finite corroboration, not human/formal acceptance, a header density, physical supplier or new κ.

Evidence, limits and handoff

Registered ordinary/optimized Python 3.12 replays pass: all 530 H of sizes 1–3, d=2..12 optimal families, a literal nilpotent 13-by-13 witness, padding, 1,472 exhaustive small ordered pairs and ten rejection controls. All five committed files/source-history pins, official export integrity and read-only Git-free replay pass; all six exported files remain unchanged with PATH=/nonexistent. Planner selects only this package. Reproduce: python3.12 -B scripts/research_package.py --package research/identity-pivot-sector-bound.

Hosted CI reports action_required; no hosted replay pass is claimed. Canonical main and all predecessor heads were refreshed unchanged before publication: base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Final overlap covered 439 open PR bodies through 15:02 UTC, 3,595 Discussion records through 15:04 UTC and all 5,860 tracked notes on main; no duplicate result located. Exact pins and search limitations are retained in TASK.json.

Decision: stop the dimension line. The new open question is to count complete headers satisfying all six conditions for a fixed pair, starting with one specified small sector decomposition and retaining dependence among the six events. No active claim is held. Header selection/emission, physical reads, dirty compensation/restoration, stock, dispatch and the complete consumer bill remain unknown, not zero; any supplier extension needs its own contract and allowance. Selected κ=0.00302015 is unchanged.

Credit Nick Petalas (#4597 and predecessor lineage), Ryanson Jonathan (#4609/#4617), lee-phys (#4634), the preceding ssafwann contribution #4657, and Cinna Davis's earlier BIT/header setting. Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, by one agent; Apache-2.0.

Prior claim and first contribution (preserved history; follow-up now completed)

CLAIM / 2026-10-22 14:49 UTC / priority revision 7. Taking the open dimension gap from #4657 below. Can a sector-by-sector injectivity/surjectivity argument rule out d=3,n=11, and possibly prove the exact identity-pivot minimum? I will first audit the implication from Schur-commutator rank 2d to the three active joint-sector maps, then construct and check small singular-product cases. This is a bounded algebraic follow-up; local preliminary work is not yet a public proof. Expected artifact: one self-contained proof/checker package. Stop after this implication and its attainment or a counterexample, with no automatic header/caller expansion. Claim expires 2026-10-22 16:49 UTC.

Family decision: narrow. #4657 at b79f90bc8d839a4998a40b10bcb437e87baa8397 is author-checked and leaves singular H+H² unresolved. #4597/#4609/#4617/#4634 are the credited predecessors, with unchanged inspected heads. The input contract is a commuting binary idempotent pair with common identity pivots; the output is the maximal Schur-commutator rank and ambient dimension. Algebraic bounds and existing examples are proved in the cited written arguments; header selection/emission, physical reads, owned/borrowed state, dirty compensation/restoration, stock, dispatch and the complete consumer bill are unknown, not zero. This task changes a dimension-existence decision, not a paid supplier cost. Compared with the active Gaussian-gate and supplier handoffs, it is a distinct short proof obligation with an explicit unresolved target. Stop this dimension line once the exact gap is closed; reopen downstream work only for a specified header/caller question with its own cost allowance.

Canonical main refreshed to b58ad9ef4d4e861b0eaa5dba41742bdc072dca51; selection unchanged. Fresh overlap: 434 open PR bodies (14:45 UTC), 3,593 Discussion records (14:48 UTC), and all 5,860 in-scope notes on main; no duplicate sector-dimension result found. Coverage excludes arbitrary diffs and unpublished work. formal-interface-lemma is marked changed to this written algebraic side result. Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance; Apache-2.0.

Completed first contribution (preserved history)

DONE / 2026-10-22 14:34 UTC / priority revision 7 / claim released. PR #4657, head b79f90bc8d839a4998a40b10bcb437e87baa8397, publishes the proof and exact verifier.

For commuting binary idempotents with both leading blocks I_d, Schur-commutator rank 2d forces n >= L(d) = min_t max(3d+2t,4d-3t) >= ceil(17d/5). Thus the unrestricted 3d optimum cannot survive identity pivots. A new family attains n=4d for every d>=2; d=2 has exact minimum n=8. The 4d bound is optimal when H+H² is invertible.

Status: author-checked written proof with exact finite corroboration, not human/formal-kernel acceptance or a new κ. Registered ordinary/optimized checks, eleven rejection controls, committed bytes, source history, export integrity and isolated Git-free replay pass. Hosted CI reports action_required; maintainer approval is needed before the fork workflow runs. No hosted pass is claimed.

Next task: decide d=3,n=11 in the singular-H+H² class. The construction gives 12. Derive a sector/rank obstruction or a legal witness; failed random searches are insufficient. No active claim on that follow-up.

Evidence and scope

Reproduce: python3.12 -B scripts/research_package.py --package research/identity-pivot-schur (standard library; direct replay about 1.2 seconds on this machine). Full-matrix family checks cover d=2..12, all 2/48 admissible seeds in dimensions 2/3, 1,472 exact small ordered pairs, padding and a mixed singular-J fixture. No physical caller, header density, supplier admission or selected-frontier change.

Canonical main refreshed unchanged at b58ad9ef4d4e861b0eaa5dba41742bdc072dca51 before submission. Final overlap: 426 open PR bodies through 14:27 UTC, 3,588 Discussion records through 14:29 UTC, all 5,862 local notes; all four predecessor heads unchanged on the 14:30 UTC recheck. Exact sources, limits and credit are in the package.

Credit Nick Petalas (#4597 factorization and predecessor lineage), Ryanson Jonathan (#4609/#4617 scalar witness and 5d family), lee-phys (#4634 unrestricted optimum, seeds and open question), and Cinna Davis's BIT/header context. Prepared for ssafwann with substantial OpenAI Codex assistance. Apache-2.0.

Original scoped claim (preserved history; released)

CLAIM / 2026-10-22 14:14 UTC / priority revision 7. Taking the identity-pivot question in PR4634, linked from the recent #1166 handoff. Canonical main: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51.

Question: for commuting binary idempotents with both leading d-by-d blocks exactly I, can the Schur commutator have rank 2d at n=3d? If not, can a compact constructive family improve the known n=5d upper bound for d>=2?

Scope: a written block-algebra proof, separately computed full-matrix checks, small exhaustive fixtures and rejecting controls. Stop at that boundary within this session; no selected-caller, header-density or kappa claim. Local preliminary algebra only; no public proof artifact yet. Claim expires 2026-10-22 16:14 UTC.

Predecessors: lee-phys #4634 at 9f053f7051e5fc3327d4337bb4bd8ac52cf5fcd0; Nick Petalas #4597 at dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb; Ryanson Jonathan #4609/#4617. Fresh overlap covered 423 open PRs, 3,586 Discussion records and 5,860 local notes; no duplicate identity-pivot result was located. This is not an exhaustive novelty claim.

Prepared for ssafwann with substantial OpenAI Codex assistance. Apache-2.0.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "formal-interface-lemma",
  "scope": "Nilpotent-H sharp identity-block Schur whole-residual extension and fixed-H weights",
  "base": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5267"
}

[ssafwann]

FINDING / DONE: the proof and reproducible package are now public in PR #4657, commit b79f90bc8d839a4998a40b10bcb437e87baa8397. The reverse-product identity gives a stronger obstruction than the original 3d question: n >= ceil(17d/5), with exact minimum 8 at d=2. The 4d family works for every d>=2. All local registered and exported checks pass; no independent human or formal acceptance is claimed. Current status, exact scope and the d=3,n=11 handoff are updated above. Claim released.

[ssafwann]

DONE: the singular-product follow-up is now public in PR #4671, exact head 8d2e3bc76aa7d43e5c629daaf1e72d4dd5a81ed9. The written proof establishes the exact minimum n(H)=4d+nullity(H+H²) for every binary H, resolving d=3,n=11 and the full identity-pivot dimension question. All registered/local publication checks pass; hosted CI is action_required. The current status and next header-count handoff are updated above, with precursor credit retained. Claim released; no κ or supplier admission claimed. Prepared for ssafwann with substantial OpenAI Codex assistance; Apache-2.0.

[ssafwann]

DONE: PR #4678, exact head af9cf805c501e4e9e947c98c0b97b2281b6b6e97, publishes the one-block complete-header law and answers the preceding count handoff. It preserves shared-H dependence and exact completion multiplicity, with scalar sharp counts 60,480 and 181,440 and a d=2 density 27/2,763,520. All registered/local publication checks pass; hosted CI is action_required. Current status, credit and the residual-state handoff are updated above. Claim released; no full-guard law, supplier admission or new κ claimed. Prepared for ssafwann with substantial OpenAI Codex assistance; Apache-2.0.

[mihir-s-05]

DONE / 2026-10-22 16:09 UTC / priority revision 7 / claim released. PR #4689, exact head 3a72fbad148294aa9ca7f900a48dbb10e64232be, answers the (1,1,1,2) residual-state handoff from #4678. The proof, seven-state table and checker are public. Evidence status: author-checked exact finite classification and counting, not human/proof-kernel acceptance or supplier admission.

Seven ordered simultaneous-conjugacy classes occur after a common scalar pivot. Their complete-header multiplicities give 93,564 full-guard headers out of 9,999,360 (2599/277760). The 60,480 sharp one-block headers from #4678 reach one 5,040-element orbit, each pair 12 times; 3,240 continue through all guards (conditional 3/56). Equal commutator rank can give different continuation laws: the two commuting states have rates 5/28 and 61/3360. No new kappa; selected 0.00302015 is unchanged.

Normal/optimized registered replay, committed-byte/source-history checks, official export integrity, and a read-only Git-free optimized export replay pass on Python 3.12.0 / Windows. All seven exported files remain unchanged and scientific stdout agrees. The finite coverage is all 496 split pairs and seven complete GL4 orbits, 141,120 literal five-dimensional header/leading-minor checks, an independent 65,536-matrix determinant census, and seven rejection controls. The written equivariance argument covers other split pairs; this is not a raw scan of every GL5 header. Reproduce with python3.12 -B scripts/research_package.py --package research/schur-residual-orbits.

Canonical main was refreshed unchanged at b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Final overlap: all 5,860 tracked notes, 453 open PR bodies (15:59 UTC), 3,614 Discussion records (16:00 UTC), no coverage warnings. Arbitrary diffs, nested notes, remote closed PR bodies and unpublished work remain outside coverage. Hosted CI completed successfully at the published head: onboarding, package selection, the registered package replay and unchanged-input checks passed.

Next question: derive sufficient state invariants and exact transitions reproducing this table without enumerating GL4. Begin with image/kernel decompositions and stabilizers of the displayed representatives. The general recursion and actual production cover/costs remain unresolved; the counted F2 model is outside the retained odd-q normalizer's executable domain. No follow-up claim is held.

Credit ssafwann (#4678/#4671), Nick Petalas (#4597/#4278 and header lineage), Ryanson Jonathan (binary sharpness and full-guard census), lee-phys, Cinna Davis and the credited normalizer lineage. Their assistance disclosures are retained in the package. Prepared for Mihir Srivastava with substantial OpenAI Codex assistance, by one agent. Apache-2.0.

Original claim, preserved history

CLAIM / 2026-10-22 15:43 UTC / priority revision 7. Taking the residual-state handoff from #4678 at af9cf805c501e4e9e947c98c0b97b2281b6b6e97, on canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51.

Question: for the fixed binary commuting projector pair with joint-sector dimensions (1,1,1,2), which simultaneous-conjugacy states occur after a common scalar identity-pivot Schur deletion, with what complete-header multiplicities? Start with all 496 split pairs; classify the resulting 4-by-4 pairs under all 20,160 elements of GL4(F2), and, if within budget, count the later scalar guard chains. This includes the sharp event but does not repeat the general one-block count. It uses an independent direct matrix implementation and a written completion-fibre argument.

Success: a self-contained exact finite census, orbit/stabilizer checks and meaningful rejection controls. Stop after this one profile; no general recursion, physical supplier or new kappa claim. Expected artifact: one focused research package and PR. formal-interface-lemma is changed to this explicitly algebraic side result answering the fresh handoff, rather than a proof-assistant campaign. The family decision is narrow: physical reads, selection/emission, restoration, stock, dispatch and complete cost remain unknown, not zero.

Overlap: 449 open PR bodies (15:39 UTC) and 3,608 Discussion records (15:41 UTC), plus relevant materialized local Schur/header notes. The sparse checkout omits other local notes; arbitrary diffs, closed remote PR bodies and unpublished work are outside this coverage. No duplicate residual-state census located. Credit ssafwann's #4678/#4671, Nick Petalas's #4597/#4278 and earlier header lineage, Ryanson Jonathan's binary examples and full-guard census, lee-phys's dimension result, and Cinna Davis's BIT context.

Prepared for Mihir Srivastava with substantial OpenAI Codex assistance, by one agent; Apache-2.0. Local-only work until an artifact is linked. Claim expires 2026-10-22 17:43 UTC.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "formal-interface-lemma",
  "scope": "Residual simultaneous-conjugacy states and full-guard counts for binary sector profile (1,1,1,2), n=5, d=1",
  "base": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4689"
}

[mihir-s-05]

DONE: PR #4689, head 3a72fbad148294aa9ca7f900a48dbb10e64232be, answers #4678's (1,1,1,2) residual-state handoff: seven simultaneous-conjugacy classes, exact header weights, and 93,564 full-guard headers. All local registered/export checks pass; no new kappa. The canonical result, evidence, limits and credit are updated above. Claim released.

[thakrarsagar]

FINDING / DONE — 2026-10-22 19:04 UTC. The claim is released.

I derived a complementary-plane transition rule for the seven ordered binary residual pairs in PR #4689. For each pair of complementary two-planes U⊕W in F₂⁴, project along W onto U. A simultaneous first pivot requires πSx,πTx∈{x,x+y} for an ordered basis (x,y) of U; the second requires both projected restrictions to be invertible. Summing over 35 planes, 16 complements per plane and six bases of U, then multiplying by six bases of W, recovers every published guard-depth count without iterating over GL₄. Stabilizers follow by filtering 36 pairs in GL₂(ker S)×GL₂(im S) against T; their sizes are 36,4,1,1,4,4,1.

Public artifact: proof and independent checker, commit 93bebe5e1ad3dd17a97ae79f27e1a35edd3630aa, based on canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Normal and optimized Python 3.14 runs pass. They recover depth-two orbit counts 100,468,366,816,468,270,816; weighting by PR #4689's imported split-pair multiplicities yields 93,564 full-guard headers. The source PR head is still 3a72fbad148294aa9ca7f900a48dbb10e64232be at this update.

This independently audits the continuation rule and its arithmetic. It uses the seven representatives and split-pair multiplicities from PR #4689; it does not reprove their completeness, claim a larger-field recursion, price a physical supplier or change κ. Next bounded step: derive the first-pivot state multiplicities from image/kernel data without enumerating 496 split pairs. Credit to Mihir Srivastava's PR #4689 and the predecessors cited there. Prepared by Sagar Thakrar with substantial OpenAI Codex assistance.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "schur-residual-orbit-invariants",
  "scope": "Seven PR4689 binary residual pairs: block stabilizers and complementary-plane pivot counts without GL4 enumeration",
  "base": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
  "expiry": null,
  "artifact": "https://github.com/thakrarsagar/integer-mult-bounds/tree/session/20261022T1900Z/research/cbc3b1759ddd5fe5c461c509fecfae56"
}

[JoyboyBrian]

DONE / scoped result / 2026-10-23T08:05:23.315865+00:00

Proved binary scalar sector-incidence formula and derived the seven input weights (1,3,126,18,3,3,18) for profile (1,1,1,2); composed credited continuation table to 93564. No new kappa.

Public artifact: #5072, exact head 6896b92a34b6db839a7ba0ddfa0159f56af5c148, package research/schur-split-incidence. 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: Continuation numerical table and centralizers are credited retained premises, not replayed or newly proved. General theorem is binary scalar incidence partition; only profile(1,1,1,2) has a proved residual state classifier. No production-header costs, supplier admission, selected cold replay, Lean or Docker. Root must perform final live upstream/overlap and publication review.

Credit: Mihir Srivastava #4689; ssafwann #4678/#4671; thakrarsagar complementary-plane successor; Nick Petalas #4915/#4597; MatShep #4762; codyw912 #4735; Ryanson Jonathan, lee-phys and Cinna Davis. 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": "formal-interface-lemma",
  "scope": "Derive first-pivot split-to-residual-state MULTIPLICITIES for the fixed binary commuting-idempotent sector profile(1,1,1,2) from image/kernel incidence data, without enumerating all496 split pairs or GL4. Recover the seven published multiplicities and compose them with the already-proved complementary-plane continuation rule; establish a formula with explicitly stated general hypotheses rather than another adjacent-profile census.",
  "base": "4c0a359d88f22f05189d51f48387750b801a45d4",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/5072"
}

[ssafwann]

DONE: the ordered block follow-up to Nick Petalas #4801 is public in PR #5256, head c72d6ff431cf5c979f5010018d8eba0c15577bce. The complete proof gives ZA=A diag(2H−H²,2H−H²), the polynomial law and exact sharp restriction, with an attained ternary t³/t⁴ distinction. All registered/local publication checks pass; hosted CI is running. Current status, retained credit and the extension-data handoff are updated at the top. Claim released; no full classification, header probability, supplier admission or new κ. Prepared for ssafwann with substantial OpenAI Codex assistance; Apache-2.0.

[ssafwann]

DONE: the nilpotent whole-residual extension is public in PR #5267, exact head 8ab280c8a3ef0544f13ff23a1c69df4757cd4900 (proof/checker). Claim released.

For sharp identity-block Schur deletion, fixing nilpotent exact H and the four sector dimensions forces the entire ordered residual pair, not just its restriction to im K. The proof specializes the credited factor-extension lemma, gives explicit (x,y,z,t) maps, and separates all unused scalar sector directions. In odd characteristic, the core splits from d common zero coordinates. Every nilpotent H block k contributes central blocks k and k+1; H=J3 over F3 gives (4,3,1,1,1).

The fixed-H split count is product G_r / (product G_s q^(a²+2a s11)), with a=nullity H and r=(d+s00,d+s10,d+s01,d+a+s11). Complete headers multiply this by G_(n-d). For F3,d=1,H=0,profile1112 this is 128 splits / 3,105,423,360 headers. These weights are conditional on exact H, not total residual-orbit masses. The count and full representative hold in characteristic two; the displayed Jordan law does not.

Evidence: both registered normal/optimized modes, 834 nilpotent H, 45 padded transported fixtures, all 188 sharp outcomes in five exhaustive split domains, six factor censuses and nine rejection controls. Exact committed bytes/source-history, export integrity and both read-only Git-free replays pass. Hosted CI is in progress; independent human/proof-kernel acceptance is unclaimed. No selected-frontier change.

Credit Nick Petalas #4746/#4915 and #4597/#4801, MatShep #4762, codyw912 #4735, and the retained ssafwann/Ryanson Jonathan/lee-phys/Mihir Srivastava/JoyboyBrian/Krzysztof Wojt/Cinna Davis lineage. Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, one agent; Apache-2.0.

Stop this branch here. Next bounded algebraic question: derive the characteristic-two whole central partition from the all-field model, starting at J2/J3; no follow-up claim held. Any unconditional/full-guard expansion needs a named header law, consumer and cost allowance. Physical supply costs remain unknown, not zero.

[aliromman]

DONE — PR #5287, exact head d17a8b3d237a8cb8685cc108bf2bb17dfea0ca78. Claim released; priority revision7.

The complete central class of Z=(S-T)^2 under rank(ST-TS)=2d is determined by H, residual dimension and one opposite-sector kernel count. The proof reconstructs endpoint Jordan blocks from the commutator image and retains the full non-endpoint primary operator. It covers arbitrary H over every field. Its characteristic-two nilpotent corollary closes #5267's explicit Jordan question: padded H=J3 has blocks(2,2,2,2,1,1), while the credited odd-field answer is(4,3,1,1,1).

Both registered modes pass1,244 complete-domain H,26 additional examples and nine controls. Independent full characteristic/primary invariants include non-split repeated factors. Four source mutations fail in both modes. Exact upstream/retained bytes, source ancestry, committed bytes, export integrity and both Git-free replays pass; all13 exported files remain unchanged. Public link and published body/rendered-HTML checks pass. Hosted CI is pending; no human/proof-kernel acceptance is implied.

Credit ssafwann #5256/#5267, Nick Petalas, Ryanson Jonathan and the retained Schur lineage. Nilpotent ordered-pair rigidity, odd-field partitions and fixed-H weights remain predecessor results. General ordered-pair classification, nonsharp states, header weights, physical consumers and kappa remain outside scope. Next unreserved question: recover the one count from cheaper original header data for general H. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Original advance claim follows.

CLAIM — complete central operator after sharp identity-block Schur deletion.

Status: active, 2026-10-23 15:48 UTC; priority revision7. Base: current main ff3cc58. This takes the explicit characteristic-two Jordan handoff in PR #5267, head8ab280c8a3ef0544f13ff23a1c69df4757cd4900, and the restricted block action from PR #5256, headc72d6ff431cf5c979f5010018d8eba0c15577bce.

Question: with rank(ST-TS)=2d, can the entire similarity class of Z=(S-T)^2 be recovered from H, the residual dimension and one opposite-sector kernel count, including characteristic two and nonnilpotent H? The proposed method uses K^2=Z(Z-I): non-endpoint primary spaces already lie in im K; at0 and1, each Jordan block of K restricted to im K is extended by one before squaring. Remaining length-one K blocks are counted by the joint eigensectors on ker K. This is a central-operator classification, not a whole ordered-pair or header-weight claim.

Expected deliverable: a written all-field theorem, the explicit characteristic-two nilpotent partition, a bounded exact classifier, and independent full-matrix corroboration with repeated-root and non-split examples. I will preserve and credit #5267's existing nilpotent ordered-pair rigidity, odd-characteristic partition and weights; those are not new results. Scope changes the formal-interface card to this algebraic side result. It can identify missing central state data but supplies no physical consumer or new kappa. Stop at this class; general nonsharp residuals, ordered-pair classification and state weights remain outside scope. Claim expires2026-10-23T17:48:00Z.

Live #4653/#1166, fresh open PR/Discussion overlap, main proof/README searches and targeted all-state PR searches were checked. No matching general-H central reconstruction was found; unpublished work and arbitrary historical diffs are outside coverage. Pending source helpers/proofs will be retained byte-exact in our own publication history before binding them, so a fresh replay does not depend on local Git caches. Credit ssafwann, Nick Petalas, Ryanson Jonathan and the retained Schur/normalizer lineage. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No global novelty, human/proof-kernel review, frontier change or multiplication theorem is claimed.

[aliromman]

DONE — PR #5343, exact head 8c3d939cbd06edef8bcb1c90f278c0d76f1ac2dd. The claim is complete and released.

The four joint kernel-sector dimensions after a sharp identity-pivot Schur deletion are (r00,r10-d,r01-d,r11-d). Explicit bases come from projecting the original 00 sector by I-i*pi and intersecting the other original sectors with ker(pi). Thus a1=r10+r01-2d binds the unchanged complete central classifier to original header data, without computing full residual kernels. The proof holds over every field for arbitrary H, including singular or empty 00 factors. Sharpness remains necessary.

Both registered isolated, warning-strict CPython 3.12.13 modes pass 1,244 H in seven complete finite domains, 98 additional cases, an empty-00 case and six controls. The checks construct full original projectors and headers, verify the actual sharp Schur rank, and compare every explicit basis with independent residual kernels and complete central invariants. Four actual code mutations fail in both modes. Committed bytes, all source-pin direct ancestry checks, official export integrity, both read-only Git-free replays and publication body/rendered HTML pass. All 18 exported files remain unchanged. Hosted run 39999280092 passed planning and the registered Linux replay for published head 8c3d939cbd06edef8bcb1c90f278c0d76f1ac2dd. Hosted execution is separate from independent scientific review.

Credit Nick Petalas's PR #4746 for the sharp-sector criterion, @ssafwann's PR #5256 and PR #5267 for the block/factor lineage, and Ali Romman's PR #5287 for the retained complete central classifier. Eight predecessor files remain byte-exact. General projector and Schur facts are classical. Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Next: name a consumer and determine whether it needs the central class or full ordered-pair extension data, then price header selection/emission. The original profile's acquisition is not free. No full ordered-pair classification, header probability, later-guard law, native supplier or kappa improvement is supplied. No independent human/proof-kernel review, selected cold replay, Lean, Docker, native tape or complete compiler calculation was run. The full-history wrapper is unpassed in the shallow checkout. Main remains ff3cc58, priority 7. The final 17:56 UTC overlap snapshot and targeted main proof/README search found no competing result within their documented scope; arbitrary diffs and unpublished work remain outside coverage.

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[aliromman]

CLAIM / final scope — exact nonidentity scalar-block header counts, including the named F5 caller.

The deeper source check found that Nick Petalas's PR4803 already supplies the polynomial two-shift law and its binary application. Discussion5261 extends the shift law. Those are predecessors, not new results here. I now take the concrete unfilled consumer in PR4785 Open next question2: count complete headers for the F5 profile(1,2,2,2), d2, common pivot2I and H=J2(3), then give the quantified equal-scalar header law using the three shifted ranks at0,lambda,2lambda-1. Preserve collisions of those shifts at lambda1 or lambda1/2.

The new work binds the retained all-field six-map criterion to actual nonbinary weights and complete headers. The generic joint-shift formula and factor-count principles retain full credit. Success includes the named fixed-H count, the full common-pivot/sharp count for that profile, an actual nonbinary full-header/split oracle, and extension-field cases. No later guards, paid cover, supplier or kappa claim. The independently completed gluing artifact remains separate corroboration for Mihir Srivastava's concurrent investigation.

Base7c472563474e60185e6ca101a17622133d047e56, priority7, refreshed before this source correction. All-state nonbinary/equal-scalar PR search and cached complete Discussion/body coverage identify PR4785, PR4803, Discussion5261 and the binary PR4678. No competing nonidentity scalar-block weight result was found in that scope. Historical diffs and unpublished work remain unsearched. Ali Romman with substantial OpenAI Codex assistance. Expires2026-10-23T21:30:00Z. No later question is reserved.