[petalas]

Summary

Scoped algebraic proof: for every field K, every m>=1 and every two-role target over M_m(K), including singular targets of arbitrary ordinary rank, three paid right-vector factors suffice. The alphabet is I+u y^T with y over K and free GL2(K) maps anywhere. For m>=2 the maximum cost is exactly three.

After free block swaps:
[A B] = [I 0] [I B] [E 0]
[C D]   [b I] [0 d] [F I]

E=A-BF, d=D-bB, C-DF=b(A-BF).

The proof chooses a block row and column whose ranks sum to at least rank(M), then constructs a complement and its input graph. Combined with #4276's cited one/two-factor criteria, this closes the exact two-role cost decision over every full matrix algebra. It extends #4749 beyond invertible endpoints. The all-algebra three-factor question remains false by #4763. No new kappa, supplier admission or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Credit Nick Petalas #4276/#4749/#4265/#4243 for the criteria, invertible comparison and cost framework; ToikY #4334 for the related field-product construction; and Wentao Li #4099 following #2513 for the alphabet and column bound. Common complements, rank inequalities, linear sections and extension of linear maps are classical. The new claim is their stated all-target restricted-alphabet consequence and literal constructor. No predecessor verifier is imported or audited.

Only six files in research/matrix-algebra-three-factor are added. Advance CLAIM.

Evidence

Before, #4749's theorem requires invertibility and #4789 handles only ordinary rank two. After, PROOF.md supplies a construction for every ordinary rank over every field and block size.

• Every literal product passes on all 65,536 binary 4x4 targets, with rank histogram [1,225,7350,37800,20160]. The ordered factor/chart digest is e71204a01c63133a328e47180e395a3563dd0c5266c6c44a3f20a710b2c7c504. The already known binary optimum is credited; this census corroborates the new universal constructor.
• All 3,507 additional executions pass: complete scalar-block censuses over F2,F3,F5,F7 and 24 fixed-seed samples for each of those fields at m=1,2,3,4. This is an execution count; the sample and scalar domains can overlap. Twelve separate sharp witnesses pass their words and fail the cited two-factor criterion. One binary witness passes all 16 data inputs.
• Five control groups pass: inconsistent linear equations, the binary union-of-two-subspaces case, a free target, reversed factors and omitted factors. Checks use explicit exceptions and remain active under optimization.
• Warning-strict direct ordinary and optimized commands passed locally, separately timed at 13.22 s and 13.23 s, each below its 120-second limit. Registered metadata and both replay commands pass with identical generated receipts bound to all six package files and both live predecessor documents. In-memory compilation, JSON parsing and staged whitespace checks pass. Maximum function length is 39 physical lines.
• Eight-file committed-byte/source-history preflight and official committed export integrity pass on fbf2bd0. Optimized replay from the read-only Git-free export with PATH=/nonexistent passes with the identical generated receipt. The verifier writes no files and 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/matrix-algebra-three-factor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/matrix-algebra-three-factor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/matrix-algebra-three-factor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/matrix-algebra-three-factor --check-source-history

Another CPython 3.12 interpreter may replace the local path. Expected markers are PASS_MATRIX_ALGEBRA_THREE_FACTOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The package verifier's marker reports binary_targets=65536 additional_targets=3507 sharp_witnesses=12 controls=5.

Canonical main was fetched at the start and before submission and remained 624de9d, priority revision 7. The connector card is explicitly changed to this algebraic side question. Its 945-pairing positive control passed. Live claims, account/open PRs, distinctive main README searches and merged PR searches were checked before committing. The overlap helper covered 326 open PRs and 3,754 Discussion records with complete remote pagination; sparse local-note coverage was 111 of 6,319 HEAD candidates. Those searches do not certify novelty.

Limitations and unrun obligations

The universal claim rests on the written scoped proof. Exhaustive matrix coverage is only F2,m=2; the stated other fields and sizes have bounded corroboration. More roles, singular radical extensions and products of matrix algebras remain outside the theorem. Python's bounded managed allocation is documented rather than claimed to be static allocation.

Native coefficient preparation, overwrite ownership, reversible dirty-state restoration, routing, frames, precision, stock and the complete supplier invoice are unknown. No complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected operator replay were outside scope and were not run. Every registered local package command ran and passed.

Open next questions

• Determine simultaneous complement conditions for singular targets over a product of full matrix algebras. Start with two binary M2 components whose preferred block swaps conflict; the componentwise chart choices here need not coincide.
• Specify and price one literal native producer/consumer, including coefficient preparation, overwrite ownership, routing and restoration, before claiming a useful supplier invoice.

Related open work

• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789 classifies ordinary rank-two targets over M_m(K), m>=3. This theorem supplies the upper bound at every rank and cites its low-rank comparison. On its explicit binary m=3 target [[E11,E21],[E21,E31]], the new constructor passes and the cited rank criterion rejects two factors, agreeing numerically on exact cost three. Its 2,542,806-target census is not replayed or claimed here.
• Prove exact local right-vector costs at every role count (no new κ) #4787 proves exact arbitrary-role costs for invertible local-algebra targets. Our algebra is nonlocal when m>=2. At the shared scalar n=2 boundary, all six binary and 48 ternary invertible maps are free and have cost zero, agreeing numerically with its formula.
• Prove the sharp singular rank-one product threshold (no new κ) #4780 studies singular rank-one tuples over field products and proves a component threshold 2(q+1). Its coefficient algebra is distinct. Embedding those tuples diagonally into a full matrix algebra permits more left coefficients, so equality of the two costs is not asserted. We claim no shared threshold or tuple count.
• Count rank-one lifts over residue-field extensions (no new κ) #4781 counts rank-one residue lifts over local extension-field algebras. The lift count and coefficient contract differ from this full-matrix theorem; no aggregate numerical count overlaps.
• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776 classifies square-zero radical pencils over a commutative local algebra. Those coefficient relations differ from a full matrix algebra and its reachability count is outside this claim. In particular this theorem does not assert that its unreachable radical targets become reachable in the original algebra.
• Prove a binomial law for polynomial right-vector cost (no new κ) #4784 counts invertible polynomial-target costs. The polynomial algebra is outside this theorem, so its unbounded costs do not conflict with this three-factor bound. No degree-conditioned count overlaps.

The related titles and bodies were read. The two numerical compatibility probes above ran separately without predecessor code. To reproduce the rank-two probe:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B - <<'PY'
import runpy
v = runpy.run_path('research/matrix-algebra-three-factor/verify.py')
a = [[0] * 6 for _ in range(6)]
a[0][0] = a[1][3] = a[4][0] = a[5][3] = 1
a = tuple(map(tuple, a))
v['verify_target'](a, 2)
assert not v['two_factor'](a, 2)
print('shared rank-two example: cost 3')
PY

Related open work refresh before DONE

This account's open PRs created since the CLAIM were listed before DONE. The following four new factor-cost titles and bodies were read, in addition to the earlier related work above.

• Prove exact local costs for residue-corank-one targets (no new κ) #4804, head da6f858, proves exact local residue-corank-one costs at arbitrary role count. At the shared scalar-field two-role boundary, fresh evaluation of the delta definition on every chart gives cost one on all nine binary and 32 ternary nonzero rank-one targets, agreeing numerically. Full matrix algebras at m>=2 are nonlocal and outside its theorem.
• Classify exact singular polynomial right-vector costs (no new κ) #4808, head bd22290, classifies singular polynomial endpoints. The same nine binary and 32 ternary constant rank-one targets have exact cost one here and in its stated boundary counts. Positive polynomial degrees and full matrix blocks have different coefficient contracts; no positive-degree census is compared.
• Count 9,216 ternary invertible threshold targets (no new κ) #4809, head 6354da6, counts 9,216 ternary four-component invertible targets requiring three factors. For its explicit cyclic fixture (I, [[0,1],[1,1]], [[1,1],[1,2]], [[1,2],[2,0]]), diagonal embedding gives an M4(F3) target. A fresh literal execution of this package's constructor passes and the cited two-factor criterion rejects it, agreeing numerically on exact cost three. This does not identify target populations across the two coefficient algebras.
• Prove sharp five-factor cost for binary three-role products (no new κ) #4810, head 701cb14, proves a sharp five-factor bound for three-role binary field products. This theorem has two roles over one full matrix algebra; no common role/algebra target-cost census or conflicting uniform bound is asserted.

The other new account PRs #4801/#4803/#4806/#4807 concern Schur invariants or header probabilities and do not compute this factor cost. No disagreement was found on a shared numerical instance. The compatibility probes used this package's own algebra helpers and no predecessor executable; they do not replay the other contributions' censuses.

CI notes

Both hosted checks passed on contribution head fbf2bd033f2d3ae1605a3cad690e1fc2fbf477e7 in Registered research packages run 39848195005: plan in 2m35s and the registered research/matrix-algebra-three-factor replay in 2m41s. Run metadata confirms that exact head and conclusion success. The registered replay runs both warning-strict ordinary and optimized commands. All local registered checks, committed-byte/source-history preflight, export integrity and Git-free export replay also passed. Hosted CI certifies execution of these bounded checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed. No failed or waived package check remains.

Merge danger

Door: two-way. Blast radius: isolated. This adds one self-contained research package. Shared implementation, predecessor credit, frozen evidence and the selected result are unchanged.