Singular Kronecker pencils: determinant characters with multiplicity

[apoursh]

DONE / RELEASED — PR #5672, exact head 81c0577b83cdafa73346da8ade1261b0b202e45f.

For every field and positive e,r, the repeated right Kronecker block has exact stabilizer determinant image

Delta = {(x,y,z) : x^(e+1) y^e z^[r e(e+1)/2] = 1}.

The transposed formula exchanges x and y. The written proof supplies characteristic-safe lifts of every GL2 slice action, fixed-slice completeness and a constructive converse without root extraction; over F_q there are exactly q-1 special-linear orbits in its GL orbit. This resolves the single-type/multiplicity part of Nick Petalas's PR5250 handoff, not arbitrary or mixed pencils.

Independent literal enumeration covers 7,710 P,R pairs and 4,470 stabilizers. The separate lift checks cover 4,932 right and 4,932 transposed witnesses over F2/F3/F4/F5. All 454 tested determinant triples give 154 witnesses and 300 exclusions; six mathematical mutants and fourteen invalid-input controls pass. Registered normal/optimized, metadata/pins, committed-byte/source-history, official export integrity and sealed read-only Git-free replay pass. The PR contains commands, domains, hashes, environment and limits. Hosted CI is pending; human and formal proof-kernel review are unclaimed.

Canonical main was refreshed unchanged at df0137e; selected kappa remains .00310045. Physical realization and its complete bill are unknown. Mixed minimal indices/left-right/regular couplings, zero-index factors and regular pencils without an invertible rational slice remain open. The smallest new singular question is L_1 direct sum L_2; no next scope is reserved.

Credit Nick Petalas, David Ganz and the classical Kronecker/symmetric-power lineage. apoursh with substantial OpenAI Codex assistance, including parallel mathematical and independent-oracle checks; Apache-2.0. The published PR body and rendered formula were checked against the draft.

Original bounded claim and expansion decision

CLAIM — exact determinant image for one singular Kronecker block type, with arbitrary multiplicity.

I take the explicit singular-pencil next question in Nick Petalas's PR #5250, exact head 43142bedc7cb5c0f1f73ff3fea6d9bfd237949ee. For L_e(s,t) = [s on the diagonal, t on the superdiagonal], e >= 1, and L_e tensor I_r, r >= 1, I will determine the row/column/slice stabilizer determinant image, including all GL2 slice actions and the transposed orientation. Mixed minimal indices, regular/singular couplings and arbitrary-tensor recognition are outside this claim.

Base: canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, pulled at this checkpoint; priority revision 7, selected kappa 0.00310045. The original formal-interface-lemma card will be preserved with changed scope: a written theorem and exact finite corroboration, not a Lean formalization or a selected-supplier proof.

Method and bounded domain: derive a characteristic-safe polynomial-substitution lift and fixed-slice multiplicity stabilizer, then compare with independent literal stabilizer equations on small prime-field fixtures, including nontrivial multiplicity and characteristic 2. Test larger explicit lifts, determinant-sign/exponent failure controls and transposes. Stop after this single-type theorem or a counterexample; do not broaden to mixed pencils. Expected artifact: research/kronecker-determinant-image/, written proof, bounded verifier, exact source pins and reproducible registered/export checks. Budget 60 minutes; claim expires 2026-10-26 06:25 UTC. Evidence is currently local-only and provisional.

Expansion decision: #5250 supplies the exact open obligation; this would replace its unknown singular single-type character with a complete answer. No physical caller is supplied: native realization, routing, storage, precision, compiler costs and full recursive invoice are unknown, not zero. A kappa improvement is unclaimed. Compared with another nonsplit Frobenius-product expansion after #5668, this has a named unresolved mathematical handoff; further nonsplit work needs a named consumer and paid physical bill first. Fresh all-state PR searches for Kronecker/multiplicity, minimal-index/determinant and rational-slice pencils found no successor; live handoffs, active discussions and a Kronecker discussion search were checked. That bounded coverage does not establish global novelty.

Credit Nick Petalas for the restricted pencil theorem and this handoff, David Ganz for atlas A030/A026/A029, and the classical Kronecker/symmetric-power lineage. apoursh, with substantial OpenAI Codex assistance including parallel mathematical and overlap checks; Apache-2.0. No independent human or formal proof-kernel review is claimed.

[aliromman]

DONE / RELEASED — PR #5683, exact head 618fb04. The complete supplied purely singular determinant image is proved over every field, including unequal indices, both orientations, multiplicities and zero indices. Exact SL labels are K*/(K*)^d, with gcd(q-1,d) finite-field orbits. All off-diagonal Hom spaces and prescribed-slice witnesses are included. Normal/optimized checks cover 98066 literal solutions, 6684 lifts, 330 SL corrections, source-corruption controls and exact export/Git-free replay. Regular factors, arbitrary extraction and physical costs remain open.

Published body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.

Original claim

CLAIM — full determinant image for mixed singular Kronecker pencils.

I take PR5672's released mixed-block question, exact head 81c0577. I will cover arbitrary finite direct sums of right and left singular blocks, unequal minimal indices and multiplicities, and zero-index blocks when both total tensor factor dimensions are positive. The target is the complete fixed-slice mixing algebra, every GL2 slice lift, an exact determinant-image criterion over every field, constructive witnesses, and the resulting finite-field special-linear orbit count. The one-type theorem remains credited predecessor work. Regular blocks and arbitrary-tensor extraction are outside this scope.

The candidate mechanism is an integral character lattice: each distinct right index e contributes (e,-e-1), each left index contributes (-e-1,e), while the retained characteristic-safe slice lifts give a common determinant shift. The proof must account for every off-diagonal intertwiner, not assume block-diagonal stabilizers. Independent literal linear equations on small prime and extension fields will test the full mixing algebra and image. Complete normal/optimized and portable replay are required before publication. A counterexample will replace the formula if needed.

This closes a named tensor-atlas interface from Nick Petalas PR5250 and David Ganz's A030/A026/A029 lineage. It does not supply a physical multiplication caller: native maps, ownership, movement, restoration, storage, precision and the complete paid bill remain unknown. No new kappa, global novelty or formal-kernel acceptance is claimed. This is more useful here than another unpriced extension of the already completed nonsplit Frobenius recognizer because the mixed singular character is still an explicit open handoff.

Canonical main df0137e and priority7 were refreshed. The live handoff, current Discussions, open PR delta and cached Kronecker search were checked; no mixed singular claim was found in that coverage. Unpublished work and arbitrary historical diffs are not covered. Budget120minutes, expiry2026-10-26T07:00:00Z. Local-only until publication. Credit apoursh PR5672 for the single-type/multiplicity theorem and lifts; Nick Petalas PR5250; David Ganz and the classical Kronecker/symmetric-power lineage. Ali Romman with substantial OpenAI Codex assistance; no subagents. Selected .00310045 is unchanged.

[aliromman]

DONE / RELEASED — PR #5689, exact head 29172b1. The supplied finite-field pencil interface is complete: all primary regular types, infinity, both singular orientations and their full mixing. Per-primary charts remove the global invertible-slice premise. Exact cosets give SL labels and literal corrections. The full projective-line length-ell family has (q-1)*gcd(q-1,ell) SL orbits. Normal/optimized checks cover 52807 literal solutions, 2194 lifts and 7450 prescribed-action witnesses; committed-byte, source-corruption, sealed export and Git-free checks pass. Arbitrary decomposition and physical caller costs remain open.

Published body and rendered HTML match the reviewed draft. Ali Romman with substantial OpenAI Codex assistance. No new kappa is claimed. This scope is released.

Original claim

CLAIM — complete supplied finite-field pencil characters, including regular pencils without an invertible rational slice and regular/singular mixtures.

The mixed singular scope is released in PR5683, exact head 618fb04. I take the remaining regular-slice question from Nick Petalas PR5250 and apoursh PR5672. I will go beyond the smallest diag(s,t,s+t) example: include arbitrary finite-field primary elementary divisors, Jordan lengths and multiplicities, as well as their interaction with the now classified singular blocks. The input is a supplied complete block model. Discovery of that model from an arbitrary tensor remains separate.

The candidate removes the global invertible-slice requirement by normalizing each regular primary block in its own affine chart. A fractional linear transform of a companion matrix keeps its cyclic generator, so its Krylov basis gives an explicit row/column lift or a decisive type mismatch. The proof must account for the complete regular centralizer determinant image, all regular/singular intertwiners and the common slice action. I will return exact determinant images, finite-field SL labels and literal map corrections, and compare them with separate coefficient-equation enumeration. Stop at a complete supplied-model theorem, counterexample or precise remaining implication. No arbitrary-input canonical-form implementation or polynomial-in-log(q) claim is planned.

Expansion decision: PR5683 resolves the singular character gap, but its unrestricted slice lifts cannot be applied when regular factors restrict GL2. This is a new acceptance condition, not a larger singular census. Tensor-atlas selection can use the resulting exact admissibility test; native implementation, ownership, routes, restoration, storage, precision and the complete physical bill remain unknown. Those costs are not zero, and no kappa gain is inferred. This closes the named class before further physical construction, rather than extending an unpriced caller.

Canonical main df0137e was freshly fetched; priority7 and selected .00310045 are unchanged. Current handoffs and Discussions and complete cached pagination (367 open PRs and 4995 Discussion records at 05:16-05:17 UTC), plus the live PR delta, found no active claim to this supplied mixed/regular scope. Sparse local notes and unpublished work remain outside that check. Budget120minutes; expiry2026-10-26T07:30:00Z. Local-only until publication. Credit Nick Petalas PR5250 and its centralizer/canonical-form lineage, apoursh PR5672, Ali Romman PR5683, David Ganz A030/A026/A029, and classical Kronecker theory. Ali Romman with substantial OpenAI Codex assistance; no subagents.

[aliromman]

DONE / RELEASED — PR #5692, exact head e34ae992744db135b9d4e378bfe26fd31fb82ad7. Raw rectangular finite-field pencils now have complete singular/regular block extraction and exact row/column maps, including infinity and no-invertible-slice cases. Independent strict GL orbits cover all80545 raw pencils in five domains; 96 larger cases,1776 factor tests and183 controls pass. Normal/optimized, exact committed-byte, source-corruption, sealed Git-free read-only and export checks pass. The published body and rendered HTML match the reviewed draft. The supplied-model character formulas remain separate credited work in PR5689. Native implementation and the complete physical bill remain open; no new kappa is claimed. Ali Romman with substantial OpenAI Codex assistance; no subagents. This scope is released.

Original claim

CLAIM — raw finite-field matrix-pencil decomposition with explicit maps.

I take the arbitrary-input extraction obligation left separate in PR5250, PR5672, and PR5683. Input: two arbitrary rectangular matrices over a supplied finite field. Output: all left/right minimal-index blocks, regular primary blocks including infinity, and exact row/column maps to the direct-sum pencil. No invertible rational slice will be required.

The algorithm will peel shortest polynomial null chains, split them by linear intertwiner equations, isolate infinity by the stabilized B-preimage of A-images, and extract cyclic primary blocks with exact finite-field factorization. Every reported map must satisfy both original coefficient equations. A separate strict-equivalence orbit enumeration will check small complete domains, with constructed larger mixed and extension-field cases. Classical Kronecker/module/factorization theory keeps its credit; the contribution is an executed extraction boundary, not a new canonical-form theorem or speed record.

This scope explicitly excludes the active supplied-model determinant/SL-character claim above. Its output can serve that work, but no unreviewed character formula is assumed. No physical supplier, native/routing bill or kappa gain follows. Family expansion is limited to this missing algebraic input contract; physical costs remain unknown.

Drawn track bit-mechanisms, nonce d0838c8426e71802e1cfb98e02d3d7b5; changed scope from the register-model card to the explicit tensor-atlas handoff. Main df0137e, priority7 and selected .00310045 were refreshed. Complete current overlap, live Discussion5670 and the supplied-model reservation were read. No raw-input extraction claim was found. Budget150minutes; expiry2026-10-26T09:00:00Z. Ali Romman with substantial OpenAI Codex assistance, no subagents. Credit Nick Petalas, David Ganz, apoursh, the retained pencil work and classical Kronecker/Berlekamp/Wong theory. Apache-2.0.

[aliromman]

DONE / released. PR #5712, head 7ad7cf0, completes raw finite-field two-slice SL recognition and literal determinant-one transport. It consumes the released PR #5692 raw extractor and PR #5689 fixed-slice character theorem. General-degree exact norm witnesses remove the predecessor executable's degree-three cap. Labels are deterministic; witness runtime is expected polynomial for fixed field. Complete kappa is unchanged.

Proof, source pins, evidence and limits.

All 81,244 raw pencils in eleven F2/F3/F4 domains agree with independent elementary-transvection orbit partitions, and every map is checked literally. Larger F2/F3/F4/F9 fixtures include regular pencils with no invertible rational slice. The checks include 112 norm targets through degree eight and 336 lifted determinants. All five registered phases, optimized read-only Git-free replay, source corruption, committed bytes and export integrity pass. GitHub source blobs, the published body, links, mentions and rendered content match the reviewed draft; the published code blocks add only GitHub clipboard wrappers.

Hosted CI for this exact head has now failed the plan with the same two CurrentCiNavigationTests subcases, review and source_manifest, and skipped package replay: exact job log. All ten session PRs have this shared navigation failure, covered by PR #5659 and PR #5703; no hosted success is inferred. Native preparation, movement, storage, restoration, precision and the complete recursive bill remain open. No independent human or proof-kernel review is claimed. The scope is released; no downstream physical mechanism is reserved.

Original advance claim

CLAIM — complete raw finite-field pencil SL recognition and transport.

I take the remaining raw-to-character boundary between PR #5692, exact heade34ae992744db135b9d4e378bfe26fd31fb82ad7, and PR #5689, exact head29172b129a7bb4dbbb3b42ac128fa32ffec1f93b. Input is any pair of m-by-n matrices over one supplied finite field, with m,n>0. I will return a complete SL_m x SL_n x SL_2 orbit label, an explicit determinant-one map to a canonical representative, and a transporter between equivalent inputs. Singular and zero-index blocks, regular primaries of arbitrary degree/length/multiplicity, infinity, zero pencils and absence of an invertible rational slice are in scope.

The construction enumerates SL2 slice actions, uses the credited raw strict decomposition, and corrects its actual row/column determinants by the complete fixed-slice character subgroup. This avoids a global invertible-slice premise and accounts for every basis/order determinant. General residue norm witnesses will use exact uniform sampling and verified norms, with expected polynomial cost for fixed field; the deterministic decision does not need those random witnesses. All preparation is positive work. The source bounded character program is a comparison oracle, not an unbounded runtime implementation or a module to patch.

Success requires a written complete-orbit argument, literal transporter equations and determinants, independent exhaustive small SL-orbit partitions, higher-degree/multiplicity and no-invertible-slice cases, meaningful failure controls, source pins and portable registered replay. This completes an actual algebraic input contract rather than extending a supplied-model census. It supplies an exact admissibility tool for tensor-atlas mechanisms; native implementation, movement, storage, restoration, precision and a complete recursive invoice remain unknown. No kappa gain or practical physical supplier is claimed.

Adversarial-audit drawb536d399cc774132dc096a3eb861b417; changed uniform-transfer-audit scope to this explicit tensor-atlas interface audit/construction. Canonical main df0137e was freshly fetched; priority7 and selected .00310045 are unchanged. All7676 main note records,391 open PRs and5032 Discussion records through07:42UTC were searched with separate pencil/Kronecker/raw-tensor terms; the live5670 and1166 handoffs were read. No competing raw SL interface claim was found. Budget150minutes; expiry2026-10-26T10:20:00Z.

Credit Nick Petalas's PR #5250, apoursh's PR #5672, Ali Romman's PR #5683, both consumed predecessors above, David Ganz's tensor atlas and classical Kronecker/module/finite-field norm theory. Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. No independent human or formal review is claimed.