[petalas]

Summary

Scoped algebraic proof: for every central field K, associative local K-algebra R, role count n>=1 and invertible target M, the paid right-vector cost is exactly the column lower bound, even with arbitrary free GL_n(K) maps anywhere.

E = {v in K^n : Mv belongs to K^n}
cost(M) = delta(M) = n - dim_K(E) <= n
delta(M) = min_{T in GL_n(K)} dim_K((MT-I)K^n)

Fix E by free charts.
Permute complementary rows to obtain unit pivots.
Replace each remaining column with one legal I+u e_j^T factor.

The residue division algebra need not equal K or be commutative. Finite dimension and a nilpotent radical are not assumed. PROOF.md gives the construction and multiplication order. This extends #4276's two-role local theorem to arbitrary roles and settles the invertible local case left by #4265. It is a formal-alphabet result, with no new kappa, supplier admission or complete multiplication theorem.

This contribution was produced by Nick Petalas. Credit Nick Petalas #4265/#4276/#4243 for the question, two-role theorem and free-map bound; Wentao Li #4099 following #2513 for the alphabet and column expansion; and ToikY #4334 and Nick Petalas #4601/#4749 for the distinct two-role product/semilocal comparisons. Basis extension, Gaussian elimination, Schur complements and local-ring unit lifting are classical. The new contribution is their exact arbitrary-role restricted-alphabet consequence. No predecessor verifier is imported or audited.

Evidence

• Before: Prove the exact two-factor right-vector criterion at n = 2; the M2(F2) failure lifts to F2[S3] (no new κ) #4276 proves the local theorem only at two roles; Refute the right-vector general-target formula: M2(F2) targets need 3 factors, not 2 (no new κ) #4265 records finite three-role binary agreement without a universal local proof. After: PROOF.md establishes the arbitrary-field, arbitrary-role theorem and an attaining word with no auxiliary roles.
• The self-contained standard-library verifier checks 2,380 distinct sampled targets at n=3,4,5 across five local algebras: binary dual numbers, a binary two-generator square-zero algebra, ternary cube-zero coefficients, F4 over F2, and noncommutative F4[e;Frobenius]/e^2 over F2. These are bounded samples, not complete target censuses.
• Every literal word and 27,616 separate basis-data executions pass. A calibration evaluates all 168 GL3(F2) charts on each of 16 targets, totaling 2,688 direct delta-chart evaluations. Eight controls distinguish factor order, omitted/changed factors, free endpoints, singular inputs, unit pivots, nilpotents and free targets.
• Both warning-strict registered commands pass locally, separately timed at 0.805 seconds ordinary and 0.806 seconds optimized. Complete generated receipts agree and bind all six package files. The package runner passes.
• Eight-file committed-byte/source-history preflight passes at 4b58e8cd5690bd35cf01b0018eefa2c39a298920. Official committed export integrity and optimized replay from its read-only Git-free export with PATH=/nonexistent pass with the identical receipt. In-memory compilation and staged whitespace checks pass. No receipt is handwritten.

From the repository root, with CPython 3.12 and its standard library:

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

Any CPython 3.12 installation may replace the local interpreter path. Expected markers are PASS_RIGHT_VECTOR_LOCAL_ALL_ROLES targets=2380 controls=8 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical main is pinned at 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, refreshed before submission. TASK.json explicitly changes the connector card to this algebraic side question. Its 945-pairing positive control passed. The optional selected metadata baseline stopped at sparse input research/fresh-helper-thinned/PROOF.md; no baseline pass is claimed. Live claims and open PR bodies, distinctive main README terms and merged searches were refreshed before committing. The overlap helper reported 299 open PRs and 3,713 Discussion records with complete remote pagination, but incomplete sparse local-note coverage. Those searches do not certify novelty.

Advance claim.

Limitations and unrun obligations

The universal claim rests on the written scoped proof. Executed corroboration uses F2 and F3 as base fields and the fixed samples above. Singular targets, products of local rings and nonlocal coefficient algebras are outside this theorem. Native coefficient preparation, frames, routing, restoration, precision, stock and the complete recursive invoice remain unknown. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected operator replay were not run; every registered package command ran locally.

Open next questions

• Characterize simultaneous pivot flags for invertible targets at n>=3 over products of local algebras. Begin with one three-role pair over F2 x F2; the present chart and pivot order may conflict across components.
• Specify one literal native producer/consumer and its algebra-valued coefficient preparation, access, frame conversion and restoration before using a column replacement in a complete supplier bill.

Related open work

• Count rank-one lifts over residue-field extensions (no new κ) #4781, head b838121c65e1bfd17a65466c95940eb00f12a225, counts two-role singular rank-one residue lifts over field extensions. Our targets are invertible, so no target-cost count overlaps; its 567 one-factor lifts are not compared to this sample count.
• Prove the sharp singular rank-one product threshold (no new κ) #4780, head a4436fab4164eccc498d4d9885eec68211c869de, studies singular rank-one tuples over nonlocal field products. Both its endpoint and coefficient-algebra hypotheses differ; no threshold or target count overlaps.
• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776, head 0764ab259a8ce496a4c5d4debeed4c4e4d9f7ca4, classifies singular square-zero radical pencils at arbitrary role counts. The binary square-zero coefficient algebra also occurs in our samples, but its zero-residue endpoints and our invertible endpoints are disjoint. No numerical census or cost value is asserted on a shared target.
• Prove a binomial law for polynomial right-vector cost (no new κ) #4784, head 4d0a6363fe4f45dd9a769f5b74d02fc5d9061ae3, counts invertible polynomial-target costs over F_q[t]. That ring is nonlocal and outside this theorem; no target count or cost distribution overlaps.
• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789, head 7fdbf8040ea9a957af0576067df2d9cb6fcf0e51, classifies singular ordinary rank-two targets over M_m(K), m>=3. Both its nonlocal matrix algebra and singular endpoints are outside this theorem; no target count or exact-cost value is asserted on a shared target.

The cited merged two-role and binary three-role predecessors retain their credit. Their finite censuses are not the deliverable here. No open PR claiming this arbitrary-role local theorem was found in the submission searches.

Related open work refresh before DONE

This account's open PRs created since the CLAIM at 2026-10-22T19:47:59Z were listed before DONE. The five related same-alphabet PRs are #4776, #4780, #4781, #4784 and #4789, with the exact heads and domain comparisons recorded above. Their bodies and changes since publication were read. Each main target/census domain is disjoint from this invertible local-algebra theorem, so no overlapping numerical count requires comparison. No conflicting overlapping claim was found within this coverage.

The other newly open account PRs, #4774/#4777/#4778/#4785, concern Schur dimensions or full-guard probabilities. They do not compute this right-vector factor cost.

CI notes

Both hosted checks passed on contribution head 4b58e8cd5690bd35cf01b0018eefa2c39a298920 in Registered research packages run 39844664836: plan in 2m57s and the registered research/right-vector-local-all-roles replay in 1m31s. Run metadata confirms that exact head and conclusion success. The replay executes both warning-strict ordinary and optimized verification commands. All registered local checks, committed-byte/source-history preflight, export integrity and read-only Git-free replay also passed. Hosted CI certifies execution of these bounded package checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only six files in one new research package are added. Shared code, frozen evidence, predecessor credit and the selected record are unchanged.