[apoursh]

PR #5250 leaves the determinant characters of singular minimal-index blocks and their multiplicity spaces open. This supplies a scoped written proof for L_e(s,t) tensor I_r, for every field and positive e,r, including all GL2 slice actions and the transposed orientation:

Delta = {(x,y,z) in (K*)^3 : x^(e+1) y^e z^[r e(e+1)/2] = 1}.

The proof gives characteristic-safe polynomial-substitution lifts, the complete fixed-slice multiplicity stabilizer, and a constructive converse without root extraction. Over F_q, the GL orbit splits into exactly q-1 orbits under the product of the three special linear groups. This resolves the single-type part of the documented handoff; mixed indices/block types and arbitrary-tensor extraction remain open. No new kappa is claimed.

Evidence and reproduction

• Independent literal stabilizer equations on eight fixed prime-field cases: 7,710 P,R pairs, 4,470 stabilizers, with exact determinant-image agreement. The oracle does not use symmetric powers or the predicted formula.
• 4,932 right-block lifts and 4,932 transposed lifts across 13 explicit F2/F3/F4/F5 domains, including nonscalar multiplicity mixing. All GL2 actions and all nonzero multiplicity determinants in those domains are covered.
• 454 determinant triples across both orientations: 154 explicit witnesses and 300 exclusions. Six mathematical mutation controls and fourteen malformed/domain controls pass. F4 exercises characteristic two with nontrivial determinants.
• Registered warning-strict normal/optimized commands, metadata/pins, committed-byte/source-history preflight and official export integrity pass at 81c0577b83cdafa73346da8ade1261b0b202e45f.
• Read-only Git-free exported replay with empty executable PATH passes in both modes (2.193/2.146 seconds, CPython 3.12.14, macOS arm64). Scientific results match the registered run; all nine export files, including its manifest, retain their bytes and permissions. There are no external live scientific inputs.

python3.12 -B scripts/research_package.py --package research/kronecker-determinant-image
python3.12 -B scripts/check_research_commit.py --package research/kronecker-determinant-image --check-source-history
python3.12 -I -W error -B research/kronecker-determinant-image/verify.py
python3.12 -I -O -W error -B research/kronecker-determinant-image/verify.py

Expected scientific marker: PASS_KRONECKER_DETERMINANT_IMAGE. RESULT gives full finite coverage and PACKAGE_SHA256 binds package bytes. The official export SHA256 for the recorded environment is daeb5c6faf2174eee5f0c0a030d737843b729f93f5d8e9602e73794bbaa942c1; scientific equality is determined by the exact result and source bytes, not tar encoding alone.

Contribution process and scope

Canonical main was pulled and refreshed again before publication at df0137efd35e11b4445734bf612bdd4c7ebc330c, priority revision 7; the selected record remains kappa 0.00310045. The source-binding baseline passes. Canonical investigation and advance claim, with a short handoff-board pointer, records the scope, bounded stop rule and five-point family comparison. Fresh targeted all-state PR searches and live Discussions found no competing successor in that coverage; this is not a global novelty claim.

The original formal-interface-lemma card is retained verbatim with changed scope: written mathematics and bounded corroboration, not Lean formalization or a selected-supplier implication. PR5250 is pinned at 43142bedc7cb5c0f1f73ff3fea6d9bfd237949ee; its README blob equals the retained base version. Its archival merge and partially-reviewed ledger status are not treated as theorem acceptance.

Excluded: zero minimal index, unequal indices, mixtures of left/right blocks, regular/singular coupling, and regular pencils with no invertible rational slice. The verifier has explicit small domains; arbitrary dimensions rely on the written proof. Native realization, dirty-state ownership, routing, storage, precision, compiler cost and the full recursive invoice remain unknown. No physical supplier, selected-frontier edit, complete multiplication theorem, Lean build or selected cold replay is supplied. Human and proof-kernel review are unclaimed. Hosted CI status is separate from the completed local checks.

Credit and next questions

Nick Petalas retains credit for PR5250 and this explicit handoff. David Ganz retains atlas A030/A026/A029 and its cited lineage. Kronecker blocks, symmetric powers and the regular-pencil canonical-form/centralizer work remain classical predecessors; no global literature priority is claimed. Prepared by apoursh, with substantial OpenAI Codex assistance including parallel proof, overlap and independently implemented oracle checks. Apache-2.0; exact source pins and attribution are retained.

Two concrete next questions remain in README/TASK: start mixed singular blocks with L_1 direct sum L_2, and treat regular diag(s,t,s+t) over F2 where every rational slice is singular. Neither scope is reserved.

Only eight files in research/kronecker-determinant-image/ are added. Shared code, frozen predecessor packages and selected-frontier files have no changes.