[petalas]

Summary

Scoped proof: for every field K, block size m>=2 and role count n>=1, every target M in M_n(M_m(K)) of ordinary K-rank one has exact paid right-vector cost n, even with free GL_n(K) maps anywhere.

Each paid factor has nullity <= m.
n-1 paid factors leave rank >= m > 1.
Write M = a b; replace its n block columns using n legal paid factors.
Therefore cost(M) = n.

This adds a singular exact-cost family at arbitrary role counts to the questions of #4265/#4276. The rank inequality, rank-one decomposition and finite-field rank-one count are classical; the scoped restricted-alphabet deduction and attaining word are the contribution. Only research/right-vector-rank-one/ is added. No new kappa or selected-frontier change.

Produced by Nick Petalas. Credit Nick Petalas #4276/#4265/#4243, Wentao Li #4099 following #2513, and ToikY #4334. The invertible matrix-block successor #4749 is complementary. Predecessor files and credit remain unchanged.

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 gives exact one/two-factor criteria at two roles and leaves general-role costs open; Refute the right-vector general-target formula: M2(F2) targets need 3 factors, not 2 (no new κ) #4265 asks for an exact noncommutative cost replacement. Those results are cited, not audited.
• After: PROOF.md proves the all-field lower bound permitting arbitrary free-map positions, and gives n attaining factors and their chronological order with no scratch roles.
• The independent standard-library verifier enumerates all 33,047 rank-one targets across six domains: F2 at (n,m)=(1,2),(2,2),(3,2),(2,3), and F3/F4 at (2,2). Counts are respectively 9,225,3969,3969,3200,21675. Every literal matrix word, factor rank bound and separate basis-data execution passes. Nine mutation/boundary controls pass, including the m=1 exception already implied by Prove free basis changes keep the right-vector diagonal count (no new κ) #4243.
• All registered warning-strict normal/optimized commands passed locally, separately timed at 1.52 s and 1.48 s on CPython 3.12. Their complete generated hash-bound receipts agree. The package runner passed. In-memory compilation and the staged whitespace check passed. Committed-byte/source-history preflight passed for eight package/input files at 116992ef72c8aa9f8994137b04395584d3b350e8.
• The official committed export and integrity check passed. Optimized direct replay from its Git-free export, with PATH=/nonexistent, agrees with the local generated receipt. No receipt was handwritten.

Exact reproduction from the repository root, using the CPython 3.12 standard library:

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

Any CPython 3.12 installation may replace the local path. Markers are PASS_RIGHT_VECTOR_RANK_ONE targets=33047 controls=9, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

The exact base is canonical main 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b, priority revision 7. TASK.json records the explicit change from connector-alphabet-change to this algebraic side result. The card's 945-pairing first positive control passed. Main READMEs, distinctive merged searches, live PR bodies and recent claims were refreshed before committing. The overlap helper reported complete pagination for 426 open PRs and 3,684 Discussion records; sparse local coverage remains incomplete. Search coverage is not a novelty proof.

Advance claim.

Limitations and unrun obligations

The universal claim is the written scoped proof, with finite corroboration. Ordinary rank one and m>=2 are essential to the stated exact family. Other singular ranks and coefficient algebras are unclassified. The formal factors overwrite and discard data; native coefficient preparation, reversible dirty-state restoration, frames, routing, precision, stock and the full recursive invoice are unknown. No native supplier or complete multiplication theorem follows.

Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or selected-construction replay was run. All registered package obligations ran locally. The optional selected-baseline orientation attempt stopped at omitted sparse input research/fresh-helper-thinned/PROOF.md; no baseline pass is claimed.

Open next questions

• Which targets of ordinary rank 2 through m-1 cost n? Start with m=3,n=2 and Prove the exact two-factor right-vector criterion at n = 2; the M2(F2) failure lifts to F2[S3] (no new κ) #4276's block-restriction criterion. The required simultaneous full-rank block is no longer automatic.
• Can a literal native producer/consumer implement one of these column replacements with a favorable full invoice? First specify coefficient preparation, overwrite, routing and restoration. Unknown costs are not zero, and no supplier construction expansion is proposed here.

Related open work

• Prove the semilocal right-vector cost formula (no new κ) #4749, head 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891, treats invertible semilocal matrix-block targets. Its invertible M2(F2) endpoints are disjoint from our singular rank-one endpoints. There is no shared target-count number to equate.
• Prove exact square-zero radical reachability and binary count (no new κ) #4745, head 3edea13af13b755932852e9306a714b317630dda, treats radical targets over split square-zero local algebras. Matrix algebras M_m(K), m>=2, are outside its coefficient-algebra hypothesis. No target-cost or census domain overlaps.
• Count square-zero radical targets over every finite field (no new κ) #4766, head 5af823b1a14b2517235925145e36581bfdfcad41, extends Prove exact square-zero radical reachability and binary count (no new κ) #4745's radical count to all finite fields. Its scalar-residue square-zero coefficient algebra differs from this matrix-algebra family; no count overlaps.
• Prove a cube-zero radical target has exact cost three (no new κ) #4764, head b1b163cd5553c75c9bd5ca3ac31a2eab4e2f3511, gives one cost-three cube-zero radical target over a commutative local algebra. That coefficient algebra is outside this theorem. Its three calls and our three-role cost three concern different endpoints; no overlapping numerical result is asserted.
• Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763, head d0158a552352c09608295a7ec47dd31327b360ae, proves unbounded costs for invertible polynomial targets over F2[t]. Its algebra and invertible endpoint domain differ; no cost or count overlaps.

The separately claimed rank-one residue-lift census concerns local algebras with scalar residue and two roles. It differs from ordinary rank-one targets over noncommutative matrix blocks here. Merged #4265/#4276 already cover the binary (n,m)=(2,2) domain in their larger finite censuses and supply the exact-cost questions and two-role criterion. The new contribution is the all-field arbitrary-role proof; the old two-role finite specialization is not claimed as new.

CI notes

CI pending for 116992ef72c8aa9f8994137b04395584d3b350e8. All registered checks passed locally. Hosted results are not yet claimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only the six files in the new package are added. Shared code, selected provenance and frozen source files are unchanged.