A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications

Alexandre Sedoglavic (CRIStAL); Cl\'ement Pernet (CASC); Jean-Guillaume Dumas (CASC)

REVIEW 3 major objections 6 minor 4 cited by

A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper presents an explicit 48-multiplication algorithm for 4x4 matrix multiplication using only rational coefficients, valid over any ring containing an inverse of 2.

desk verdict A genuine rational 48-multiplication 4x4 algorithm with a checkable certificate; the abstract overstates ring generality and the complexity constant has a typo. read the letter →

arxiv 2506.13242 v7 pith:O6DKOB5W submitted 2025-06-16 cs.SC

Jean-Guillaume Dumas (CASC) , Clément Pernet (CASC) , Alexandre Sedoglavic (CRIStAL) This is my paper · ORCID

classification cs.SC MSC 15A6968Q1768Q2568W30

keywords non-commutativematrixmultiplicationtensordecompositionrank4x4isotropyactionrationalcoefficientsstraight-lineprogramfast

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that two $4\times 4$ matrices can be multiplied with 48 scalar multiplications using only rational coefficients, eliminating the complex-number arithmetic of the previous 48-multiplication algorithm. The claim is backed by an explicit tensor decomposition of type $16X^{2}Y^{2}Z^{2}+32XYZ$, i.e., a sum of 48 rank-one tensors with rational coefficients that reproduces the ordinary matrix product. Since the only denominators are powers of 2, the algorithm is valid over any ring containing an inverse of 2, including finite fields and non-commutative rings. The paper also contributes optimized straight-line programs (a leading constant of about 11.65 in the recursive complexity bound, and a variant whose leading constant is 7) and a companion rational 63-multiplication algorithm for $3\times 4$ by $4\times 7$ products.

What carries the argument

The central object is the matrix multiplication tensor and its decomposition into rank-one tensors; the mechanism is the isotropy action that preserves tensor rank. Concretely, for a decomposition whose terms are $M_i\otimes N_i\otimes O_i$, an isotropy $(U,V,W)$ sends each term to $(U^{-\intercal}M_iV^{\intercal})\otimes(V^{-\intercal}N_iW^{\intercal})\otimes(W^{-\intercal}O_iU^{\intercal})$, keeping the total number of multiplications unchanged. The paper picks one specific isotropy that maps the known complex decomposition to a rational one. The explicit runtime certificate is the list of 48 rank-one terms in Equations (25)--(72), encoded as the LRP matrices $L,R,P$ in Appendix B: the $i$-th product is $(L\,\mathrm{vec}(A))_i\cdot(R\,\mathrm{vec}(B))_i$ and the output is $C=P\cdot(\text{products})$.

What would settle it

A direct verification settles the claim: take $4\times 4$ matrices $A$ and $B$ with 32 independent indeterminates, run the straight-line program of Listings 1--4, and check that each of the 16 output expressions equals the corresponding entry of $A\cdot B$. Equivalently, feed the 64 standard basis matrix pairs $(e_{ij},e_{kl})$ into the LRP representation and confirm that the 16 outputs are the right unit matrices.

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Extended reading notes

Core claim

On its own terms the paper shows that the best known 48-multiplication algorithm for $4\times 4$ matrix multiplication, originally expressed over $\mathbb{C}$, admits a rational point in its isotropy orbit. Using the stabilizer of the complex decomposition, the authors exhibit an isotropy---a triple of $4\times 4$ matrices with determinant $\pm1$---whose action, given by Equation (15), sends every coefficient to $\mathbb{Q}$. The resulting certificate is the trilinear representation $m_1,\dots,m_{48}$ in Equations (25)--(72), with type $16X^{2}Y^{2}Z^{2}+32XYZ$; contracting these rank-one terms with the input entries gives exactly the product matrix. Because all coefficients are rational, the algorithm is a genuine non-commutative matrix-multiplication algorithm over any ring with $2$ invertible.

Load-bearing premise

The load-bearing premise is that the explicit 48-term decomposition of Equations (25)--(72) sums exactly to the $4\times 4$ matrix multiplication tensor; any sign, index, or transcription error in a single term would invalidate the algorithm.

Editorial extensions

If this is right

The 48-multiplication bound for $4\times 4$ matrix product now holds over every ring with an inverse of 2, so characteristic 2 is the only ring characteristic where this bound is not obtained.
Recursive use of the optimized straight-line program gives complexity $\left(1+\frac{341}{32}\right)n^{2+\log_4 3}-\frac{341}{32}n^2\approx 11.65625\,n^{2.792481250}-10.65625\,n^2$.
The alternative-basis variant runs in $7n^{2+\log_4 3}+o(n^{2+\log_4 3})$ operations over any ring containing an inverse of 2.
The same isotropy technique yields a rational 63-multiplication algorithm for $3\times 4$ by $4\times 7$ matrices, matching the complex-coefficient record.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A systematic consequence: for any tensor decomposition with a nontrivial stabilizer, searching the stabilizer's rational points may yield rational (or integral) representatives; the same orbit method could strip complex coefficients from other machine-discovered algorithms.
Valid over rings with $1/2$, the rational algorithm is immediately usable in exact-arithmetic and finite-field settings where complex arithmetic would be inappropriate or slower.
The paper leaves a numerical-stability study for future work; whether the rational form improves conditioning compared with the complex form is a concrete empirical question raised by this construction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core result is real: an explicit rational-coefficient tensor decomposition for 4x4 matrix multiplication with 48 scalar multiplications. That matches the best known complex-coefficient rank and improves on the 49-multiplication Strassen-based bound. The authors derive it by projecting AlphaEvolve's complex decomposition via an isotropy, and they provide the full LRP matrices plus straight-line programs. This is solid computational algebra.

What's genuinely new is the rational decomposition itself and the technique of using the stabilizer to find a rational point in the orbit. The paper is honest about the derivation: it exhibits a certificate rather than claiming to prove a lower bound. The certificate is small (48 terms, 4x4 matrices) and the data files are linked. Any referee with a CAS can verify it in minutes. That's real evidence.

Soft spots, in order of importance:

1. The abstract says the algorithm is valid over "any ring except those of characteristic 2." That is wrong as stated. The algorithm divides by 2, so it needs an inverse of 2; Z has characteristic 0 but no inverse of 2. The body corrects this in the alternative-basis section, but the abstract and the opening sentence overstate. Fix that.

2. The complexity constant in the abstract (347/32) does not match the body (373/32 leading, 341/32 lower-order). Clearly a typo, but for a note whose contribution is partly about the running-time constant, it should be corrected.

3. The 3x4x7:63 rational algorithm is mentioned in the abstract and conclusion, but the decomposition is in a twin-note. That is fine if it is clearly a separate result, but the abstract should either include it or state that it is in a companion paper.

The stress-test worry about the unverified tensor identity is, in my view, too strong. The LRP matrices are the identity; the paper asserts it and points to machine-readable source. It would be better practice to include a verification script, but the omission is a presentation issue, not a correctness hole.

Who should read it: people working on fast matrix multiplication, both theoretical and exact computation. It is a useful data point and a nice example of isotropy projecting a complex algorithm onto Q. I would send it to a serious referee; the fixes are small.

Referee Report

3 major / 6 minor

Summary. The paper claims a rational-coefficient tensor decomposition for non-commutative 4x4 matrix multiplication with 48 rank-one terms, obtained by applying an isotropy to the complex-valued AlphaEvolve decomposition. The core content is the explicit decomposition in Equations (25)-(72), its LRP representation in Appendix B, and two straight-line programs with claimed complexity constants. The paper also mentions a companion rational 63-multiplication algorithm for 3x4 by 4x7 matrices. The central assertion is existential: the 48 displayed terms should sum to the 4x4 matrix-multiplication tensor over any ring containing an inverse of 2, a condition that the abstract incorrectly weakens to 'any ring except those of characteristic 2'.

Significance. If the decomposition is correct, this is a significant and useful result: it matches the best known rank of 48 for 4x4 matrix multiplication over the complex numbers while extending to rings with an inverse of 2, and it provides explicit straight-line programs and rational coefficients suitable for exact computation. The paper's strengths are its concrete, finite certificate (trilinear forms and LRP matrices), the provision of machine-readable source files in PLinOpt, and the potential for direct verification. However, the certificate is not formally verified in the manuscript, and there is an internal inconsistency in the claimed leading constant, so the significance is conditional on a correct verification and a corrected presentation.

major comments (3)

[§3, Eqs. (25)-(72) and Appendix B] The central claim that the 48 rank-one terms in Equations (25)-(72), equivalently the L, R, P matrices in Appendix B, sum to the 4x4 matrix-multiplication tensor is only asserted, not verified in the text. Because a single sign or coefficient error in any row would invalidate the algorithm, this identity is the load-bearing point of the paper. The reference to source files in Appendix A is helpful but does not by itself constitute a check. Please include a direct verification: either a short symbolic proof that the identity holds, a machine-checked certificate, or a minimal script that the reader can run to confirm the identity exactly.
[Abstract and §1] The abstract states that the algorithm is 'valid over any ring except those of characteristic 2', which is false. Equations (25)-(72) contain coefficients 1/2, 1/4, and 1/8, so the algorithm requires an inverse of 2 in the coefficient ring. For example, the ring Z of integers has characteristic 0 but does not contain an inverse of 2. The body's statement 'over any ring containing an inverse of 2' is the correct generality; the abstract and any theorem statements should be aligned with it.
[Abstract versus §4, Eq. (73)] The abstract claims a running time of (347/32)n^{2+log_4 3}, while §4 reports 341 total operations and Equation (73) gives the leading coefficient 1 + 341/32 = 373/32 ≈ 11.65625. The count in Section 4 (104 + 84 + 1 + 119 + 33 = 341) matches 373/32, not 347/32. Please reconcile the operation count and the constant; as written, the claimed leading constant is internally inconsistent.

minor comments (6)

[§4, listings and counts] Please clarify that the 34 'multiplications (binary shifts)' counted in the straight-line programs are scalar multiplications by constants and are distinct from the 48 bilinear products p0,...,p47; otherwise the title's '48 multiplications' and the count of 341 operations may appear contradictory.
[§4, line before Eq. (73)] The phrase '84 additions and 1 multiplications (binary shifts)' should read '84 additions and 1 multiplication (binary shift)'.
[§5 and Abstract] The abstract claims a rational 63-multiplication algorithm for 3x4 by 4x7 matrices, but this manuscript does not provide the corresponding decomposition or certificate; it only refers to the twin note [3]. Please either state the claim as a reference to the companion note or include the certificate.
[Appendix B, Definition B.1] The notation P_{*,i} in Definition B.1 should be explicitly defined as the i-th column of P; the current text 'P∗,i is Vect(Oi⊺)' is easy to misread.
[Appendix B] The typeset L, R, and P matrices are extremely dense and hard to audit. Adding a machine-readable checksum or a short verification command alongside the source-file references would substantially improve reproducibility.
[Eq. (73)] Writing the denominator as 48-16 is unnecessarily opaque; since 48-16 = 32, please write 341/32 directly to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rational 48-multiplication tensor decomposition is an explicit, independently checkable certificate, not a fitted or self-cited prediction.

full rationale

The paper's central claim is an explicit tensor decomposition (Equations 25-72 and Appendix B) of the 4x4 matrix multiplication tensor into 48 rank-one terms with rational coefficients. The derivation applies an explicitly stated isotropy (Equation 24) to the AlphaEvolve complex decomposition, which is an external benchmark by different authors. Correctness of the rational decomposition does not depend on this provenance: once the L, R, P matrices are stated, the identity sum(M_i ⊗ N_i ⊗ O_i) = matrix-multiplication tensor can be checked directly in any ring containing an inverse of 2, and the paper provides machine-readable source files as additional evidence. There is no fitted parameter and no quantity that is predicted from the data it is then evaluated against. The only self-citations, such as the stabilizer reference [12] and the PLinOpt tool [1], are used to guide or mechanize the construction; they do not constitute the proof of the final identity, which stands as a self-contained certificate. The abstract's phrase 'any ring except those of characteristic 2' overstates the body's correct condition 'any ring containing an inverse of 2', but this is a statement-precision issue, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The algorithm itself contains no fitted parameters; all coefficients are fixed rational numbers. The derivation uses standard results about isotropy groups, the tensor decomposition framework, the PLinOpt tool, and the prior AlphaEvolve complex decomposition as a starting point. No new entities are posited.

assumptions (4)

standard math Isotropy action (Definition 2.2) preserves the tensor decomposition of the matrix multiplication tensor.
Section 2.2 and Lemma B.1; relies on Groote [6] and the trace cyclicity in Equation (13).
standard math The tensor decomposition framework (Section 2.1) correctly represents bilinear algorithms, with rank equal to the number of scalar multiplications.
Standard algebraic complexity, cited to Landsberg [10].
domain assumption The PLinOpt software correctly produces the straight-line programs from the LRP matrices.
Section 4 and Appendix A rely on PLinOpt; source is available on GitHub but is not formally verified.
domain assumption The AlphaEvolve complex 4x4x4:48 decomposition is correct as the starting point of the isotropy projection.
Prior result [11]; the paper does not re-prove it. The output rational decomposition is independently checkable, so a flaw in [11] would not necessarily break the final algorithm if the certificate verifies.

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Cite this review

Pith. "Pith review of A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications." pith.science (2026). https://pith.science/paper/O6DKOB5W

@misc{pith2026250613242,
  author       = {Pith},
  title        = {Pith review of: A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6DKOB5W}},
  note         = {Machine review of arXiv:2506.13242}
}

abstract

The quest for non-commutative matrix multiplication algorithms over non-commutative rings in small dimensions has recently seen significant progress. Specifically, the number of scalar multiplications required to multiply two 4x4 matrices was reduced in \cite{Fawzi:2022aa} from 49 (using two recursion levels of Strassen's algorithm) to 47 in characteristic 2, and more recently to 48 in \cite{alphaevolve} over the complex numbers. We propose an algorithm requiring 48 multiplications that uses only rational coefficients, thereby removing the requirement for complex-number arithmetic, and making this algorithm valid over any ring except those of characteristic 2. We also produce a straight-line program of this algorithm reducing the number of additions and scalar multiplications, reaching a running time of $\frac{347}{32}n^{2+\log_4{3}}+o(n^{2+\log_4{3}})$ operations, as well as an alternative basis variant of it, leading to an algorithm running in $7n^{2+\log_4{3}} +o(n^{2+\log_4{3}})$ operations over any ring containing an inverse of 2. Similarly, the number of scalar multiplications required to multiply a 3x4 matrix by a 4x7 matrix was reduced from 66 in \cite{Smirnov:2021aa} to 63 in \cite{alphaevolve} by an algorithm over complex numbers. Using the same techniques, we propose an equivalent algorithm in 63 multiplications using only rational coefficients. In both cases the rational algorithm is obtained by identifying an isotropy that projects the previously known complex-valued decomposition onto the field of rational numbers.

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Reference graph

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