[petalas]

Summary

A prime-field recovery certificate can now be converted explicitly into an extension-field multiplication table and factor maps. Claim type: scoped constructive proof with exact finite corroboration, produced by Nick Petalas. No new kappa or selected-frontier change.

For every finite-degree field extension K/F and K-bilinear square tensor t, general commutative unital multiplication recognition is preserved by restriction to F. Given any F-certificate t(Pa,Qb)=S(ab) and its recovered unit e:

u=P e; v=Q e
R_v(x)=t(x,v); L_u(y)=t(u,y)
mu(x,y)=t(R_v^-1 x,L_u^-1 y)
return K-table(mu), unit=t(u,v), maps (R_v^-1,L_u^-1,I)

The two slices are invertible K-linear maps even when P,Q,S are not K-linear. The written proof and bounded adapter are in research/commutative-scalar-ascent.

aliromman's #5540 supplies the prime-field recovery interface and extension-field question. #5637 already implements a direct recognizer for the local characteristic-power truncated-polynomial class. Older #6464 already proves general recognition over a supplied exact field; its cyclic-module construction is not used here. This contribution is the explicit scalar-restriction certificate bridge. Restriction of scalars and unital isotope normalization are classical. Credit aliromman, apoursh, Nick Petalas, Dugongue and the retained tensor/algebra lineage. No predecessor executable or scientific suite is imported or replayed.

Evidence

Before, the existing prime-field certificate interface supplied no scalar-restriction adapter. After, the standalone adapter returns a checked K-table and K-linear isotopy, including products, non-Gorenstein algebras, nonsplit residue fields and nilpotent extensions of nonsplit fields.

Local CPython 3.12.15 on macOS arm64 passes:

• 78 newly constructed certificates over F4, F8, F9 and F25, 684 K-basis products and 75 recovery bases that are not K-linear. The largest restricted dimension is 12.
• All 16,930 field-law triples in those four arithmetic domains, ten rejecting controls and two boundary witnesses. The latter explain why K-bilinearity and a general commutative recognizer are required.
• Both warning-strict normal/optimized registered commands, generated source-hash receipts, metadata/provenance and eight-file committed-byte/mode/source-history checks. Each registered command was timed locally below one second against its 120-second limit.
• Official export integrity and input-coverage checks, followed by normal and optimized Git-free replay with empty PATH from a read-only export. Reports agree and all nine exported files retain their hashes.

Exact reproduction from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/commutative-scalar-ascent/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/commutative-scalar-ascent/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/commutative-scalar-ascent
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/commutative-scalar-ascent --check-source-history --check-modes

Another CPython 3.12 path may replace that absolute interpreter path. Expected markers are PASS_COMMUTATIVE_SCALAR_ASCENT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical base is e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8, selected #7000. The formal-interface-lemma card is explicitly changed to this algebraic side result. The selected metadata orientation passed. Live claims, full cached open-PR bodies, merged searches and all-main README searches were checked; sparse local-note and unpublished-work coverage remains limited. Advance claim.

Limitations and unrun obligations

The universal claim rests on the written proof. The executable consumes supplied certificates in explicitly supported polynomial fields with q at most 25 and restricted dimension at most 12. It is not an end-to-end recognizer integration, a prescribed algebra-class classifier or a special-linear label transfer. Independent human and formal proof-kernel review are unclaimed.

No native coefficient implementation, paid caller, dirty restoration program, physical precision/tape proof, selected cold replay, Lean or Docker run is supplied. All five selected assumptions remain. No required local package check is deferred solely to CI.

Open next questions

• Connect one actual source-owned prime-field recovery output to this adapter with a general exact finite-field interface and explicit coordinate basis.
• Recover one prescribed algebra-class or determinant label using the restricted algebra together with its embedded scalar action.
• Specify and price a native caller for one recovered table, including coefficient maps, workspace, returns and restoration.

Related open work

• #7252 computes stabilizer determinant images for split radical-square-zero multiplication tensors. This adapter constructs general K-linear recovery certificates and computes no determinant image or orbit count.
• #7270 encodes automorphisms of local Frobenius-truncated algebras. This adapter supplies a general recovered table and isotopy; it supplies no automorphism encoding or shared numerical optimum.

The completion refresh inspected all seven other open account PRs created since the advance CLAIM: #7469, #7470, #7471, #7472, #7474, #7476 and #7477. Their one-step threshold, integral biclique, rounded-lattice, partition, rounded-shear, factor-gauge and tape-retirement objects do not study commutative-tensor scalar ascent. No new related line or competing numerical value was identified, and no additional mathematical computation checked those results.

CI notes

CI did not certify this package. At unchanged head 8cfb0297ed41284b17cfcfe3e8eef9c335929b91, run 43290306518, plan job 131434472137, failed in Onboarding helper tests. I read the checks once and downloaded the job log. Exactly one of 760 tests failed, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
FAILED (failures=1, skipped=24)

At base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify and omits the selected command. I checked both base files directly. This PR changes neither file. The package replay job was skipped because plan failed, so hosted execution covered no package checks.

I also downloaded and checked the plan logs for run 42905094314, job 130140894651, and run 42903313790, job 130134720799. Each has the same sole failing test, missing-command assertion and 760 tests / 1 failure / 24 skipped summary. No fix, retrigger commit or job rerun was made. The maintainers retain the repository-level repair.

All local checks recorded above passed at the same head. Hosted replay, independent human review and formal proof-kernel acceptance remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated. Eight new files in one opt-in package. Shared code, frozen sources and the selected record are unchanged.