Research · Papers · Direct sums, wedges and the p14 frontier · ML-051
On the functional direct-sum additivity of M₂ over GF(2)
Functional direct-sum additivity for M₂ over GF(2) in unrestricted sequential Boolean XAGs is OPEN
Published 2026-08-29
For everyone
Plain summary
ML-051 asks whether running s separate 2×2 matrix multiplications over GF(2) costs s times the single-copy cost when programs can use any combination of Boolean XOR and AND gates. This property is functional direct-sum additivity: separate jobs shouldn't get cheaper just because you compute them together. The answer is known to be 7s in two restricted models: the bilinear model, where multiplications only cross the two input matrices, and the formal-quadratic model, where intermediate expressions stay quadratic. It remains open for unrestricted sequential Boolean programs. An unrestricted program can build higher-degree terms, cancel them out later, and use the fact that x = x when squared in Boolean logic. The register contains no proof ruling out a six-gate program for a single copy. Ruling out six gates is the first milestone. If six is impossible, Strassen's seven-gate construction pins the single-copy cost at MC(M₂) = 7. A 13-gate search on two copies would then check additivity directly. Alder–Strassen and Strassen settle 7s only in the two structured models; the unrestricted Boolean question is still open.
Result
Functional direct-sum additivity for M₂ over GF(2) in unrestricted sequential Boolean XAGs is OPEN. For bilinear and formal-quadratic circuits, the exact multiplicative complexity across s independent copies is 7s, established by Alder–Strassen's lower bound and Strassen's upper bound. That structured result does not resolve the unrestricted Boolean setting.
The identified next step is proving the nonexistence of an unrestricted sequential six-gate XAG for M₂. Combined with Strassen's upper bound, this would establish MC(M₂) = 7. A subsequent 13-gate SAT search over two independent copies would then test functional additivity directly.
Setting and definitions
M₂ denotes the 2×2 matrix multiplication map over GF(2). The unrestricted sequential Boolean model corresponds to general straight-line programs over the basis {XOR, AND}. Functional direct-sum additivity holds for a complexity measure C if C(f ⊕ ... ⊕ f) = s · C(f) for s disjoint copies. The bilinear and formal-quadratic models restrict intermediate expressions to bilinear or formal quadratic forms, respectively. Tensor rank measures bilinear decomposition length, distinct from functional circuit complexity.
Method
The open status and underlying SAT search formulations are documented in this paper's downloadable evidence pack. There is no UNSAT certificate for six-gate unrestricted sequential M₂, leaving the one-copy Boolean lower bound open. The bilinear and formal-quadratic baseline is established via Alder–Strassen and Strassen.
Discussion
The gap stems entirely from unrestricted sequential Boolean semantics. These circuits can generate high-degree intermediate polynomials, exploit Boolean idempotence (x² = x), and cancel terms across multi-output targets. Because M₂ has four outputs (eight for two copies), quadratic normalization does not automatically carry over to this vector-output setting. The structured 7s theorem provides a baseline, not a transfer theorem.
Ruling out six gates for single-copy M₂ is the immediate computational target needed to fix the base cost at 7. The 13-gate two-copy search would then test subadditivity across independent instances, isolating functional circuit reuse from tensor rank decomposition. No corrections are recorded for ML-051.
For everyone — the takeaway
What this means
We know 2×2 matrix multiplication takes exactly seven multiplications per copy if the program follows strict algebraic templates. We don't know if a general Boolean logic circuit can beat that number by building and canceling higher-degree terms along the way. The first question to answer is simple: can one 2×2 multiplication run in six AND gates? If not, the true one-copy cost is seven, because we already have a seven-gate design. From there, testing whether two copies can run in 13 gates will reveal whether combining separate tasks lets them share work.
Attribution and prior art
Prior art: Alder-Strassen and Strassen establish 7s only within the bilinear and formal-quadratic models. The question of unrestricted Boolean functional additivity remains open.
Register references
- Entry: ML-051.
- Receipt artifacts:
02-direct-sum-tensor/REPORT.md§§8–9;mm2_boolean_mc.py;boolean-mm-sat/. - Prior-art works named by the register: Alder–Strassen; Strassen.
- Unrestricted Boolean result: the register does not record this.
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 2 of 2 receipt files bundled (13 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.