[petalas]

Summary

A scoped algebraic proof determines the ordered covariant stabilizer determinant image for multiplication in M_n(F_q), for every n>=1 and every finite field.

H_n = {a^n : a in F_q*}
Delta(mu) = {(u,v,(uv)^-1) : u,v in H_n}
SL orbits inside GL orbit = (q-1) gcd(n,q-1)^2

The proof also constructs attaining stabilizers and corrects a supplied GL transporter to an SL transporter exactly when its determinant triple belongs to the required image. Under an explicit orbit promise, dimensions and q suffice to extract the three character equations. This adds the noncommutative full-matrix-algebra case of David Ganz's atlas A030 proposal. It is an original derivation for this repository; global literature priority is unclaimed.

Only six files in research/matrix-tensor-determinant-image/ are added. This contribution was produced by Nick Petalas. Credit David Ganz #2849 for A019/A029/A030, Nick Petalas #5252 for the finite-etale extraction precedent, and the classical matrix-unit innerness and cyclic-group arguments. The atlas retains the Chen, Grochow, Qiao, Tang and Zhang tensor-action context. Predecessors are cited without executable replay or an audit claim.

Evidence

Before, the cited promise-class extraction results do not supply this noncommutative matrix-algebra image. After, PROOF.md gives a self-contained all-field argument, the exact orbit count and explicit correction witnesses.

Local CPython 3.12.15 warning-strict normal and optimized checks pass, with identical generated hash-bound receipts. Each direct check takes about 1.2 seconds; these are check durations, not performance claims. The registered runner passes both commands. Exact finite corroboration includes:

• 21 contexts at q=2,3,4,5,7,8,9 and n=1,2,3; 537 attaining stabilizers; all 142,126 literal transformed tensor coefficients.
• A separate complete 65,536-map binary domain, finding 20,160 invertible maps and exactly six matrix-algebra automorphisms, equal to the inner maps.
• All 1,171 n=2 determinant triples in those fields, with 89 explicit SL corrections and 1,082 criterion rejections. This is a determinant-domain census, not a whole-transporter or whole-orbit enumeration.
• Twelve semantic and boundary controls, including the inverse-input convention, both character restrictions, an altered tensor factor, extension fields, n=1 and q=2.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/matrix-tensor-determinant-image/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/matrix-tensor-determinant-image/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/matrix-tensor-determinant-image
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/matrix-tensor-determinant-image --check-source-history

Another CPython 3.12 installation can replace the interpreter path. Expected markers are PASS_MATRIX_TENSOR_DETERMINANT_IMAGE no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier uses the standard library and writes no files. Its generated receipt binds all six package files. SOURCES.json binds the atlas and etale predecessor as historical provenance; there are no external executable inputs. At contribution commit 571194d4ce2a1ae5473a0fe9ff6d71384d877bcd, committed-byte/source-history preflight and official export integrity pass. Both direct checks also pass in a read-only Git-free export with an empty PATH, in 1.17 and 1.20 seconds. Their receipts equal the final registered receipt, and all seven exported files retain their bytes. Syntax, bounded-function inspection and git diff --check pass; the longest function has 29 lines.

Canonical main was refreshed before submission and remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. The selected source-binding orientation passed; it is metadata orientation, not a scientific supplier replay. TASK.json explicitly records the changed card scope. Live PR bodies, handoff claims, local reservations, main README terms and merged searches were checked. Sparse local-note coverage is incomplete, and the searches do not prove global novelty.

Advance CLAIM.

Limitations and unrun obligations

The tensor action is K-linear with ordered factors. No factor permutations or semilinear action is included. Extraction assumes the matrix-multiplication orbit promise. Constructive correction requires a literal source or supplied multiplication chart and an existing GL transporter. Raw-tensor recognition and chart recovery remain unproved. Finite execution corroborates the written universal proof; it is not formal proof-kernel verification.

No physical supplier, native implementation, full compiler, precision/stock invoice, new kappa or complete multiplication theorem is supplied. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected scientific cold replay were not run. Every registered command runs locally.

Open next questions

• Derive the image for products of full matrix algebras over finite extensions. Start with one extension-field block, then include permutations of identical blocks. Extension and permutation characters remain missing.
• Recognize the promised tensor class and recover a multiplication chart and transporter. Start with the three nuclei and their simple-module actions. A complete recognition and chart-recovery proof remains missing.

Related open work

No open PR studying the ordered stabilizer determinant image of this full matrix multiplication tensor was found in the refreshed open-PR body corpus. Merged #5252 studies finite-etale multiplication; merged #5459 and #5663 study commutative Frobenius products; merged #5250 studies matrix pencils. Their objects or hypotheses differ from M_n(F_q). No numerical comparison or predecessor verification is claimed.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-27T10:02:37Z were refreshed. The list contains this PR and nine others, #5865, #5866, #5867, #5868, #5869, #5870, #5872, #5873 and #5874. Their titles and bodies were read for overlap.

• Count every finite-field connector chart repair rank (no new κ) #5867 counts residual repair ranks over finite-field connector charts. It does not study a matrix-multiplication tensor's stabilizer determinant image.
• Prove singular reachability and a Veronese obstruction (no κ) #5869 classifies singular paid-target reachability over commutative domains. Its determinant-zero condition concerns the endpoint matrix, not the three covariant stabilizer characters here.

The seven remaining PRs study cyclotomic column completion, binary phase defects, Clebsch queries, sunflower inertia, dirty-word interval bounds, binary C5 fits and A5 paid-rank caps. None studies this tensor or determinant-image quantity. No shared computed number is compared, and no additional computation was run to check another contribution.

CI notes

On contribution commit 571194d4ce2a1ae5473a0fe9ff6d71384d877bcd, Registered research packages run 40486908263 completed with plan failed; replay skipped. The only failed step is Onboarding helper tests.

The log identifies tests/test_research_current_assignment.py throughout all ten failure/error records and ends Ran 511 tests and FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures are assertions in that same test file. This matches the reported shared onboarding-test failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. No package verifier ran in hosted CI, and no shared file or test is changed by this PR.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific checks have run. Both registered normal/optimized commands, committed-byte/source-history preflight, official export integrity, and both read-only Git-free direct replays passed locally as recorded above. No scientific result, package file or contribution commit was changed after publication.

Merge danger

Door: two-way.

Blast radius: isolated.

This adds one self-contained research package. Shared implementation, frozen evidence and the selected result are unchanged.