[petalas]

Summary

Scoped algebraic proof with an exhaustive finite certificate: for every field K and block size m>=3, every ordinary rank-two target on two roles over M_m(K) has exact paid right-vector cost two or three. The paid alphabet is I+u y^T with base-field right vectors and arbitrary free GL2(K) maps.

Write M=A B, with A=[A1;A2] injective and B=[B1 B2] surjective through K^2.
cost(M)=3 iff
  B1=c1 b and B2=c2 b for a common nonzero row b and a basis c1,c2;
  every nonzero scalar combination s A1+t A2 has rank two.
Otherwise cost(M)=2.

This answers the rank-two, two-role part of Open next question 1 of #4772. The new proof classifies the deficient rank-two factor pencils and supplies a literal three-factor word for every exceptional target. The two-factor criterion and its attaining word are cited from #4276, not re-audited.

The explicit target [[E11,E21],[E21,E31]] has ordinary rank two and exact cost three over every field and every m>=3. Only research/right-vector-rank-two/ is added. No new kappa, supplier admission or selected-frontier change.

This contribution was produced by Nick Petalas. Credit Nick Petalas #4772/#4276/#4265/#4243 for the open question, rank bound, criteria and free-map normal form; Wentao Li #4099 following #2513 for the alphabet; and ToikY #4334, followed by Nick Petalas #4601/#4749, for the related cost framework. Rank factorization and the rank-one pencil lemma are classical linear algebra. Predecessor files and credit are preserved.

Evidence

• Before: Prove exact rank-one matrix-algebra cost at every role count (no new κ) #4772 proves ordinary rank-one cost n and leaves higher singular ranks open. After: PROOF.md gives the complete ordinary rank-two classification on two roles for m>=3, including both attaining cases.
• All 2,542,806 binary 6 by 6 rank-two targets are enumerated uniquely through their canonical column spaces and surjective right factors. Exactly 2,535,162 cost two and 7,644 cost three. The literal full-matrix two-factor criterion agrees with the separately computed new classification on every target. All 7,644 exceptional legal three-factor words multiply to their complete targets.
• The complete target/cost table has SHA256 37a68b33c4175157da0147c0b165f82909956a87750ff354a366f542f432c623. Separate dense chart solvers check both attaining and excluded categories. Ten explicit field/block cases cover F2,F3,F4,F5,F7 at m=3,4; all 64 binary dirty inputs and the boundary, solver, alphabet, omission and reversal controls pass.
• Both registered warning-strict commands pass locally on CPython 3.12, timed separately at 2.04 s and 2.06 s, below their 120-second limits. Normal and optimized receipts agree. The registered package runner passes. In-memory compilation, JSON parsing, function-size checks and the staged whitespace check pass.
• Eight-file committed-byte/source-history preflight and official committed export integrity pass at 7fdbf8040ea9a957af0576067df2d9cb6fcf0e51. Optimized Git-free export replay with PATH=/nonexistent emits the identical generated receipt. The receipt binds all six package files and both live predecessor proof files by SHA256; none is handwritten.

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

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

Any CPython 3.12 installation can replace the local interpreter path. Expected markers are PASS_RIGHT_VECTOR_RANK_TWO targets=2542806 cost2=2535162 cost3=7644, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical main is pinned at 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, and was refreshed before committing. TASK.json explicitly changes connector-alphabet-change to this algebraic side question. The card's 945-pairing positive control passed. Live claims, open PR bodies, distinctive merged searches and all-main README searches were checked before committing. The final overlap snapshot covered 306 open PRs and 3,724 Discussion records with complete remote pagination. Sparse local-note coverage remains incomplete; these searches do not certify novelty.

Advance claim.

Limitations and unrun obligations

The all-field statement is a written scoped proof using the cited predecessor criterion. The exhaustive target census covers F2 and m=3; the other fields have the stated example checks. The block-size restriction is essential: at m=2, ordinary rank-two diag(0,I2) costs one. More roles and other ordinary ranks remain unclassified. Independent human and formal proof-kernel review are unclaimed.

The formal factors overwrite data. Native coefficient preparation, routing, reversible dirty-state restoration, precision, stock and the complete recursive invoice remain unknown. No complete multiplication theorem follows. Lean, Docker and selected-construction replay were outside this isolated result and were not run. Every registered local command ran successfully; no package check is deferred to CI.

Open next questions

• Classify ordinary rank-three targets on two roles over M_m(K), m>=4. Start with a named deficient 3 by m matrix pencil; the rank-one pencil lemma here no longer suffices.
• Specify one literal native producer/consumer, coefficient preparation and overwrite/restoration contract, then price the complete invoice before using these formal factors in a supplier.

Related open work

• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776, head 0764ab259a8ce496a4c5d4debeed4c4e4d9f7ca4, classifies zero-residue square-zero radical pencils in the same alphabet. Its local coefficient algebra differs from M_m(K), m>=3, so no counted target or cost census overlaps.

• Prove the sharp singular rank-one product threshold (no new κ) #4780, head a4436fab4164eccc498d4d9885eec68211c869de, proves the singular rank-one component threshold over pure field products. Matrix-algebra blocks are outside its threshold theorem. Under the diagonal embedding of k nonzero rank-one components, ordinary rank is k; k=2 lies at m=2, outside our m>=3 theorem, and k>=3 lies above our rank-two domain. There is no overlapping numerical count to compare.

• Count rank-one lifts over residue-field extensions (no new κ) #4781, head b838121c65e1bfd17a65466c95940eb00f12a225, counts rank-one residue lifts over proper extension-field local algebras. Those scalar-residue local algebras differ from the noncommutative matrix blocks here, so no lift or target-count total overlaps.

• Prove a binomial law for polynomial right-vector cost (no new κ) #4784, head 4d0a6363fe4f45dd9a769f5b74d02fc5d9061ae3, proves an exact degree/cost distribution for invertible targets over F_q[t]. The polynomial coefficient algebra and invertible endpoints differ from the singular matrix-algebra targets here, so no target-cost count or distribution overlaps.

• Prove exact local right-vector costs at every role count (no new κ) #4787, head 4b58e8cd5690bd35cf01b0018eefa2c39a298920, proves arbitrary-role exact costs for invertible targets over local algebras. M_m(K), m>=3, is nonlocal and our targets are singular, so both endpoint and algebra domains are disjoint. No shared target-cost number is asserted.

All five titles and bodies were read. The account's open PRs created since the advance CLAIM were refreshed before DONE. No numerical disagreement was found on an overlapping domain; these five endpoint/algebra domains have no common census with this package. Merged #4772 is the direct predecessor; its rank-one endpoints are disjoint from this rank-two classification. Merged #4265/#4276 include binary m=2 finite examples, while this classification assumes m>=3.

CI notes

Both hosted checks passed on 7fdbf8040ea9a957af0576067df2d9cb6fcf0e51 in run 39844689687: plan in 2m44s and the registered package replay in 1m27s. The run's head SHA matches the submitted commit. This certifies the registered replay, not independent human or formal proof-kernel acceptance. All registered local checks, committed-byte/source-history preflight, export integrity and Git-free replay also passed. No CI failure or waived proof obligation remains.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Six files are added in one research package. Shared implementation, frozen evidence and selected metadata are unchanged.