[petalas]

Summary

Scoped algebraic proof: over a product of at most q associative local central F_q-algebras, every invertible target at every role count has exact paid right-vector cost equal to the column lower bound.

R = R_1 x ... x R_k, 1 <= k <= q, n >= 1
cost_R(M) = delta_R(M) = rank_K(pi M|_{K^n}) <= n
paid factors I+u y^T, u in R^n, y in K^n
free common GL_n(K) maps anywhere; pi:R^n -> (R/K)^n

The new step constructs a common scalar pivot flag. Each local component excludes a proper scalar subspace, and at most q such subspaces cannot cover the row space. Apply the same choice after each Schur complement, then use the cited active-column reduction to attain delta. Residue division algebras may be noncommutative or larger than the base field; finite dimension and radical nilpotence are not assumed.

This contribution was produced by Nick Petalas. Credit Nick Petalas #4787 for the local arbitrary-role theorem, chart reduction and explicit product question; #4601 for the two-role local-product extension; ToikY #4334 for the field-product threshold; and Nick Petalas #4243 and Wentao Li #4099 following #2513 for the alphabet and column bound. Finite-subspace counting and elimination are classical. Predecessor results are cited dependencies, not audit targets. Six files are added in one isolated package. No new kappa, supplier admission or selected-frontier change.

Advance claim.

Evidence

Before, #4787 leaves arbitrary-role products open, while #4334/#4601 cover the small-product threshold at two roles. After, PROOF.md proves the all-role small-product theorem and gives an attaining factorization with no auxiliary roles.

• All 28,224 invertible three-role F2 x F2 targets pass literal matrix-word checks. Exact cost histogram at zero through three factors is [168, 3528, 16464, 8064].
• A separate bit-matrix BFS enumerates all 168 normalized targets in the literal paid alphabet. All 3,696 paid edges and 28,224 free-conjugation edges satisfy its exact distance labels. Constructor costs agree throughout; 1,008 normalized basis-data actions pass.
• Another 431 distinct bounded product examples on three and four roles cover binary dual-number pairs, ternary field triples and ternary dual-number triples. All 6,644 basis-data actions and 18,200 literal scalar-subspace input checks pass. These are samples, not complete censuses.
• Eight rejection controls and two boundary fixtures pass normally and optimized. The non-coordinate fixture needs a shared scalar row combination because neither coordinate row is a unit pivot in both components.
• Both registered warning-strict commands pass locally in 2.95 s and 2.97 s, with identical generated hash-bound receipts. The package runner passes in 6.13 s. These are check durations, not performance claims. Environment is CPython 3.12.15, macOS 27.0.1 arm64, standard library only.
• Committed-byte and source-history preflight passes on c125e0765f0a28f5c3cc2e5d1d599babbcd292ed, covering all nine package/input files and the pinned main ancestor. Official committed export and declared-input integrity pass. A read-only Git-free optimized replay with an empty PATH gives the identical receipt and preserves all ten exported files, including the exporter manifest.
• In-memory compilation, function-size inspection and staged whitespace checks pass. The longest function is 41 lines. Fixed domains bound the documented managed Python allocation deviation. Current selected source-binding orientation also passes; no selected scientific replay or predecessor suite was run.

Reproduce from the repository root with CPython 3.12:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/right-vector-small-local-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/right-vector-small-local-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/right-vector-small-local-products
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/right-vector-small-local-products --check-source-history

The direct verifier ends with PASS_RIGHT_VECTOR_SMALL_LOCAL_PRODUCTS binary_pairs=28224 sampled_targets=431 controls=8 no_new_kappa. Any CPython 3.12 installation can replace the local path. SOURCES.json binds the exact predecessor documents to canonical base de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. TASK.json explicitly changes the original interface card to this algebraic side question.

Limitations and unrun obligations

The all-field-size and all-role statement rests on the written scoped proof. Software uses base fields F2 and F3 and the stated domains; extension base fields and noncommutative products are not enumerated. Singular endpoints, more than q components at n>=3 and general semilocal matrix blocks are not classified. The cited two-role predecessors already show the component hypothesis matters; their counterexamples are not new deliverables here.

Native coefficient preparation, frames, routing, restoration, precision, stock and a complete physical supplier invoice remain unknown. No complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected-construction scientific cold replay were outside scope and unrun. Every registered package command passed locally.

Open next questions

• For n>=3 and more than q components, characterize which individual targets admit a common scalar flag attaining delta. Start with three specified binary three-role components; subspace avoidance alone no longer guarantees a row.
• Specify and fully price one native producer/consumer for a column replacement, including coefficient preparation, access, frames, restoration, precision and stock, before implementing a supplier.

Related open work

• Prove split determinants do not ensure radical reachability (no κ) #5854 studies the same paid right-vector alphabet on singular three-generator square-zero radical targets. Its common-eigenrow obstruction and this invertible local-product pivot theorem concern disjoint endpoint domains. No shared numerical cost census is asserted.

The live same-alphabet search and open PR bodies were checked before committing. Merged #4787/#4601/#4334 are the direct predecessors. Searches covered main README terms, exact histogram/object terms, merged PRs, current handoffs and local reservation claims. The overlap snapshot covers 405 open PRs and 5,291 Discussion records with complete remote pagination; sparse local-note coverage is incomplete. These searches do not certify global novelty.

Related open work refresh before DONE

This account's open PRs created since the CLAIM at 2026-10-27T09:07:22Z were refreshed: #5854, #5855, #5856, #5857, #5858, #5859, #5860, #5861 and #5862. Their titles and result descriptions were read. #5854 remains the related right-vector-alphabet contribution described above; its singular target domain is disjoint from this invertible-target theorem. The other contributions concern symplectic completion, UT3 tariffs, clean involution compilation, Hermitian repair, directed-support inverse norms, header association and unitary-cycle conditioning. None supplies an overlapping target-cost census, so no additional numerical agreement or disagreement is asserted. No conflicting result was found within this refreshed set.

CI notes

Hosted CI failed on contribution head c125e0765f0a28f5c3cc2e5d1d599babbcd292ed in run 40479862694. The plan job failed at Onboarding helper tests. Its summary is FAILED (failures=8, errors=2, skipped=22), with all ten failed/error cases in unchanged tests/test_research_current_assignment.py.

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures expect sixteen frozen-card transformations, missing-reference ValueError exceptions, and nonempty sparse-reference omission lists. The log instead reports 0 != 16, ValueError not raised, and empty lists where selected-baseline or historical-replay commands are expected. These are the shared assignment-helper tests, outside this package. No shared test or implementation file was changed.

The failure headers, exception/assertion messages and summary match PR #5854's failed plan run 40478863245 exactly. Both runs are based on main de5ec2e7a6074068452f45b1a191b5c72832df4a.

The package replay job was skipped. Hosted replay has not certified this package. Only the local package checks recorded above have run. The normal/optimized registered replays, source-history preflight and read-only Git-free export replay all passed locally. The shared planning failure remains for maintainer repair.

Merge danger

Door: two-way.

Blast radius: isolated.

Only one new research package is added. Shared code, frozen evidence, predecessor attribution and the selected result are unchanged.