[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. For a tensor isotopic to multiplication in A = product_i F_(q^d_i), its covariant stabilizer determinant image is exactly

Delta(t) = {(u,v,w) in (F_q*)^3 : (uvw)^e = 1},
e = 2 if q is odd and an even degree occurs or an odd degree repeats;
e = 1 otherwise.

The image has (q-1)^2 or 2(q-1)^2 elements. A deterministic polynomial procedure for fixed q extracts the degree multiset from the tensor's left nucleus and Frobenius-fixed spectral projectors, without an invertible slice or automorphism enumeration. This resolves the promised finite etale multiplication class of David Ganz's atlas A030 in #2849. General tensor extraction and orbit recognition remain open. No new kappa is claimed.

The contribution is produced by Nick Petalas. Credit David Ganz for the explicit A030 question, A019 normalization and A029 character-table framework; finite-field norms, normal bases and algebra decomposition retain classical credit. The new result is the exact sign criterion and its intrinsic extraction at this interface. Global literature priority is unclaimed.

Evidence

Before, A030 specified extraction but supplied no procedure for this promised tensor class. After, the written proof supplies the complete determinant image, attainment and polynomial extraction.

• Twelve literal algebra fixtures over F2/F3/F5, through dimension six, pass in original coordinates and after independent changes in all three factors.
• Every predicted determinant triple is attained by explicit unit/automorphism matrices.
• Seven complete finite stabilizer domains enumerate 521,892 invertible input-map pairs and accept 1,995 autotopisms. Eight control groups distinguish the sign cases, singularity, an outside-algebra matrix, a wrong nucleus dimension and the inverse-input convention.
• Warning-strict normal and optimized direct commands pass on CPython 3.12.15 in 5.05s and 5.00s respectively. These are observed replay times, not a performance comparison. Both registered commands pass in the package runner. Generated JSON receipts bind all five package files by SHA-256.
• Exact committed bytes and source-history availability pass at head 07e6f55d18d2d3fd6dcea19bd642d71657c5fe14. Whitespace and bounded-function inspection pass. The worktree is clean.

From the repository root:

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

Any CPython 3.12 path can replace the interpreter prefix. Expected markers are PASS_ETALE_TENSOR_DETERMINANT_IMAGE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Advance CLAIM. Base ff3cc58e370d0f678e529246f8c73e8094e18fcf, exact atlas/card/selected-record provenance and scope change are recorded in the isolated package. The current selected metadata baseline passes; historical balanced-transfer arithmetic is not a premise.

Limitations and unrun obligations

The extraction algorithm assumes orbit membership in nonzero finite etale multiplication tensors and does not recognize that promise or output a transporter. Its general-size guarantee is proved in writing; the executable is a bounded corroborator. The proof covers all finite base fields, while finite fixtures use prime fields 2, 3 and 5. Nilpotent algebras are outside scope.

Independent human and formal proof-kernel review are unclaimed. No physical program, native/dirty-state/routing/precision invoice, supplier admission or complete multiplication theorem is supplied. No selected cold replay or proof-assistant build was needed or run. All registered checks ran locally.

Open next questions

• Extend determinant extraction to finite local commutative algebras with nilpotents. Start with the unit and automorphism characters of K[z]/(z^m); the etale Frobenius splitting proof does not apply.
• Recognize the finite etale orbit promise and construct a transporter in deterministic polynomial time at fixed q. Start with regularity of the computed left-nucleus action and compatibility with the second input; this package assumes those properties.

Related open work

• Extract exact matrix-pencil stabilizer determinant images (no new κ) #5250 extracts stabilizer determinant images for n-by-n-by-two regular matrix pencils with an invertible slice. It addresses the same A030 quantity on a different promised class; this package concerns n-by-n-by-n finite etale multiplication tensors. The classes overlap in dimension two. The split and quadratic-etale F3 controls give eight determinant triples in both descriptions, consistent with this package's already completed finite censuses. Neither result depends on the other's proof or census, and no additional computations were run to compare them.

The completion refresh read this account's ten open PRs created since the CLAIM at 2026-10-23T13:55:53Z, #5245 through #5254, including this PR. The nine others concern qubit repair, Hermitian intersections, downset inertia, symplectic spread completion, diagonal-completion rank, matrix-pencil determinant images, a projective design, odd-type fitting rank and adaptive tail gates. Only #5250 studies this determinant-image quantity. No shared-number disagreement was found. Sparse historical notes and unpublished work remain coverage limits.

CI notes

Both hosted checks passed at exact contribution head 07e6f55d18d2d3fd6dcea19bd642d71657c5fe14 in Registered research packages run 39966757144: planning in 4m45s and registered package replay in 1m45s. Run metadata confirms completed status, success and the matching head SHA. The checks were read once at completion. The replay executes both warning-strict normal and optimized registered commands.

All local registered checks and the committed-byte/source-history check also passed. Hosted CI certifies execution of the bounded replay and controls, not independent human or formal proof-kernel acceptance of the all-dimension theorem. The limitations and unrun obligations above remain unchanged.

Merge danger

Door: two-way. Blast radius: package.

The change adds five files in one opt-in research package. It does not modify shared code, frozen predecessors or the selected result.