{"id":"4c6a170c-01ab-44e2-a419-a9aa38fb2023","arxiv_id":"2502.04514","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"New symmetric flip-graph search yields 5x5 matrix multiplication in 93 multiplications and 6x6 in 153, improving prior bounds.","lead":"This paper finds new, faster ways to multiply 5x5 and 6x6 matrices, using 93 and 153 scalar multiplications respectively. The authors modified a computer search method to look only for algorithms with a certain symmetry, which made the search faster and produced schemes that work over any field.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed new rank bounds rest on an unverified Hensel-lifting step; the schemes themselves are not in the paper, so the 'arbitrary ground fields' claim is not yet independently checkable.","rationale":"The paper's mathematical framework, especially Theorem 4, appears internally sound: orbit flips preserve G-invariance, the search algorithm correctly maintains valid schemes, and the starting-point construction is consistent with the stated ranks. The weakest point is not the flip-graph search itself but the transition from F2 schemes to schemes valid over arbitrary fields. The paper explicitly acknowledges that Hensel lifting has no convergence guarantee and that convergence is only empirical. This is a load-bearing assumption because the headline results claim improvement over arbitrary ground fields, and the specific schemes are not presented in the manuscript for direct verification. The reader's weakest assumption identifies exactly this issue, so I agree with the conditional assessment. If the proposed direct verification over Z succeeds, the central claim would be well supported and the conditional concerns resolved; if it fails, the reported bounds would not be established. No further mathematical flaw is apparent that would justify changing the verdict more drastically.","tokens_in":11358,"tokens_out":9179,"duration_ms":104609,"concrete_test":"Pin the GitHub repository at a specific commit, extract the reported rank-93 and rank-153 schemes, and evaluate over Z the difference between the sum of the rank-one tensors and the matrix multiplication tensor M_n; verify that every entry is identically zero as a polynomial in the variables. Optionally also run an independent Hensel lift from the published F2 schemes to confirm that integer coefficients are reached. If the difference is nonzero for either scheme, the headline bounds are not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the F2-search schemes for 5×5 and 6×6 matrix multiplication, using 93 and 153 multiplications respectively, lift to integer-coefficient schemes. The only evidence offered is the statement in Section 6: 'There is no guarantee for the approximation to converge to a solution, but in our experiments all candidates converged to an integer solution within a few steps.' Convergence of Hensel lifting is not proven, and the paper does not display the lifted schemes or provide an independent certificate. Since the schemes are only in an unversioned GitHub repository, a bug in the search, in the lifting solver, or a mismatch between the published numbers and the repository contents would invalidate both new bounds. Moreover, 'arbitrary ground fields' requires an actual integer solution, or a rational solution with denominators allowed in every field, not merely a 2-adic one; the paper does not state how integer-ness was certified after 2-adic lifting. Without a direct check that the sum of the 93 (resp. 153) rank-one tensors equals the matrix multiplication tensor over Z, the claim is plausible but not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a symmetry-restricted version of the flip graph algorithm for matrix multiplication schemes, working with G-invariant schemes for G = C3 or G = C3 × Z2. It defines orbit-level flips, reductions, and plus-transitions, proves in Theorem 4 that these operations preserve G-invariance when applied to full orbits, and describes a random-walk search. The authors report new upper bounds: rank ≤ 93 for 5×5 and rank ≤ 153 for 6×6 matrix multiplication, improving on 97 and 160 respectively, and state that the F2-found schemes lift to integer coefficients, hence hold over arbitrary ground fields. The search code and concrete schemes are available at a GitHub repository rather than in the paper.","tokens_in":11651,"tokens_out":8925,"duration_ms":93725,"significance":"If the reported schemes are correct, the paper provides a meaningful advance in explicit upper bounds for small matrix multiplication and a novel algorithmic idea: restricting the flip graph to symmetric schemes both reduces the search space and appears to make lifting from F2 easier. The proof of Theorem 4 is clean, and the search procedure is described in enough detail to be reimplemented. However, the central existence claims are not yet independently checkable from the manuscript because the schemes are not displayed and the integer lifting is supported only by an empirical statement in Section 6. These issues are fixable with exact certificates, so the significance is real but conditional.","major_comments":[{"comment":"The sentence \"There is no guarantee for the approximation to converge to a solution, but in our experiments all candidates converged to an integer solution within a few steps\" is the only evidence for the lifting of the 93- and 153-multiplication schemes to Z. This is load-bearing for the abstract's claim that the schemes work over arbitrary ground fields. Please either prove convergence for the specific systems and then prove that the resulting coefficients are integers, or provide an exact integer certificate for each claimed scheme and state explicitly how equality over Z was verified.","section":"§6 (Lifting)"},{"comment":"The concrete schemes of ranks 93 and 153 are not included in the paper; the reader is referred to an unversioned GitHub repository. Because these schemes are the central object of the paper, a mismatch between the repository content and the reported numbers, or a bug in the solver, would invalidate both new bounds. Please include the schemes as supplementary material or as a machine-checkable certificate, and pin the exact version of the repository used for the reported results.","section":"§5 (Results) and §8 (Implementation)"},{"comment":"The phrase \"integer solution\" after 2-adic lifting is ambiguous: an element of Z_2 is not an integer, and a solution with coefficients in Z_2 cannot be reduced modulo primes p ≠ 2 to give a scheme over arbitrary fields. The paper should explicitly verify that the lifted coefficients are in Z and that the defining identity M_n = sum of rank-one tensors holds over Z, not merely modulo 2^m. After such a verification, reduction modulo arbitrary primes is immediate, but this verification is currently absent.","section":"§6 (Lifting) and §9 (Conclusion)"}],"minor_comments":[{"comment":"The notation \"G · A⊗B⊗C\" is used both for the orbit as a set and for the formal sum of orbit elements, which makes the proof of Theorem 4 harder to follow; a separate notation for the formal orbit sum would improve clarity.","section":"Theorem 4 and Definition 5"},{"comment":"In Algorithm 1, line 13 sets m = ℓ + M immediately after resetting ℓ = 0; the intent is clear but the line is redundant, and the reset block could be rewritten for readability.","section":"Algorithm 1"},{"comment":"The comparison for n = 4 between the symmetric rank 49 and the non-symmetric rank 47 from [11] should state the ground field for both bounds, since ranks can differ between F2 and larger fields.","section":"§5 (Results)"},{"comment":"The display \"GL ×3 n\" is typeset in a confusing way; it should be written as GL(n)^{×3} or similar.","section":"§2 (Matrix Multiplication and its Symmetry Group)"}],"recommendation":"major_revision","confidential_remarks":"The main risk is not the mathematics of the flip graph but the lack of an exact, versioned certificate for the reported schemes. If the authors add the schemes and an integer verification, the paper would be suitable for publication. I would not reject the paper on grounds of disagreement with consensus; this is a computational existence claim that needs to be made independently checkable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about this one. It reports new upper bounds for matrix multiplication: 93 multiplications for 5×5 and 153 for 6×6, improving the previous 97 (94 over F2) and 160. And the theoretical machinery—orbit flips in a symmetry-restricted flip graph—is new and mostly clean, but the actual schemes live in an unversioned GitHub repo and the lifting step is empirical. That is where the paper stands or falls.\n\nThe new idea is sound. Instead of running the flip graph algorithm on all schemes, the authors restrict to schemes invariant under C3 or C3×Z2, and define flips on orbits. Theorem 4 shows the orbit-flip operation preserves the matrix multiplication tensor, and the proof using orbit-stabilizer counts is correct. This is a genuine extension of the flip graph method, not a trivial tweak. The observation that symmetric schemes lift from F2 to Z more readily than the asymmetric ones found in earlier work is interesting and, if it holds up, useful.\n\nWhat is less solid is the verification of the claimed schemes. Section 6 says Hensel lifting \"has no guarantee to converge\" but \"in our experiments all candidates converged to an integer solution within a few steps.\" That is plausible, but the paper never displays the lifted integer coefficients or provides a certificate that the sum of the 93 (or 153) rank-one tensors equals M_n over Z. The claims are for arbitrary ground fields, so an integer solution—or at least a rational one with denominators invertible everywhere—is needed, not just a 2-adic approximation. Without the explicit schemes, the reader has to trust the GitHub repository, and an unversioned repo is not a stable artifact. If any of the reported schemes fails to lift, the 93 and 153 bounds become characteristic-2 successes, which is still something but not what the abstract promises.\n\nI don't think these issues sink the paper. The proof of the flip-with-symmetry framework is solid, the search procedure is described in detail, and the reported bounds are a meaningful advance. But a referee needs to see the actual schemes, ideally in an appendix or in a versioned archive, and a clear statement that the lifted schemes were checked over Z. That is a reasonable request, not a fatal flaw.\n\nThis is a paper for researchers in exact bilinear complexity and small matrix multiplication. It deserves a serious referee. I would encourage the editor to send it out, with the request that the certificate issue be addressed before acceptance.","headline":"New matrix multiplication bounds from a symmetry-restricted flip graph; the theory is clean, but the schemes are not in the paper and the lifting evidence is empirical, so verification is needed.","tokens_in":12135,"tokens_out":4123,"would_cite":true,"duration_ms":41243,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","68Q25","68W30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces a symmetry-constrained flip graph search and reports new matrix multiplication schemes with 93 multiplications for 5×5 matrices and 153 for 6×6 matrices over arbitrary fields.","keywords":["matrix multiplication","flip graph algorithm","tensor rank","symmetry constraints","C3 invariance","Hensel lifting","bilinear complexity","rank bounds"],"falsifier":"Substitute the lifted integer coefficients of the reported 93- and 153-multiplication schemes into the defining bilinear equations for 5×5 and 6×6 matrix multiplication and check that the difference tensor is exactly zero over the integers; if any entry is nonzero, or if the mod-$2^k$ lifting system has no solution for any reported scheme, then the arbitrary-ground-field claim is false.","tokens_in":11170,"feed_emoji":"✖️","tokens_out":9536,"duration_ms":86575,"temperature":0.7,"pith_summary":"The paper claims that forcing matrix-multiplication schemes to be invariant under a small symmetry group makes the flip-graph search much smaller while also making the final schemes easier to lift from arithmetic over $\\mathbb{F}_2$ to true integer coefficients. On that basis it reports two new records: $5\\times5$ matrices can be multiplied with 93 scalar multiplications and $6\\times6$ matrices with 153, improving the previous bounds of 97 (or 94 over $\\mathbb{F}_2$) and 160. This matters because these are concrete upper bounds on the rank of the matrix multiplication tensor, and the $6\\times6$ scheme is atomic rather than a composition of smaller algorithms. A further improvement of two multiplications in the $5\\times5$ case or one in the $6\\times6$ case would beat the asymptotic exponent of the classical seven-multiplication 2×2 algorithm.","feed_headline":"Symmetry-first search finds cheaper 5x5 and 6x6 matrix schemes","feed_subtitle":"New schemes multiply 5x5 and 6x6 matrices in 93 and 153 multiplications, beating earlier records.","key_machinery":"The load-bearing object is the orbit flip: a flip applied to two full orbits of rank-one tensors inside a $G$-invariant scheme, with integer multipliers $\\lambda,\\lambda'$ inserted so that the two replacement terms again form full orbits. Theorem 4 shows this preserves $G$-invariance, so a random walk on the symmetric flip graph can move whole orbits in one step. Around this sit two supporting constructions: starting points built from diagonal partitions (which supply the $C_3$-invariant rank-one tensors that fix the missing diagonal entries), and a symmetry-preserving Hensel lifting that solves cyclic versions of the Brent equations modulo increasing powers of 2.","core_discovery":"In the paper's own terms, the central discovery is that the flip graph algorithm remains effective when the search is restricted to $G$-invariant schemes for $G=C_3$ or $G=C_3\\times Z_2$: every flip, reduction, or plus-transition can be applied to whole orbits of rank-one tensors, so three (or six) rank-one tensors are moved at the cost of one. The authors prove that an orbit flip of a $G$-invariant scheme is again a $G$-invariant scheme (Theorem 4), and they exploit the preserved cyclic symmetry in the lifting step by solving a symmetrized version of the standard bilinear equations. Running this search over $\\mathbb{F}_2$ with carefully chosen diagonal-partition starting points produces symmetric schemes of rank 93 for $n=5$ and 153 for $n=6$, and all such schemes found in the experiments lifted to integer coefficients, giving schemes valid over arbitrary ground fields.","pith_inferences":["The paper does not test larger symmetry groups; a natural extension is that larger finite subgroups of the symmetry group would shrink the search further and could make $n\\ge7$ searches feasible.","The authors do not explore composing their atomic 6×6 scheme with small rectangular schemes; such compositions might improve bounds for larger sizes beyond the two record values.","The diagonal-partition starting points look like a transferable recipe: choose block-diagonal tensors that match the symmetry, let flips fill the rest; this recipe could be tried on rectangular formats or on other small groups."],"forward_implications":["The rank bounds become $\\operatorname{rank}(M_5)\\le 93$ and $\\operatorname{rank}(M_6)\\le 153$ over arbitrary fields, improving the previous best bounds of 97 (or 94 over $\\mathbb{F}_2$) and 160.","The 6×6 scheme is atomic, so it offers a new way to build larger schemes without composing smaller ones, as prior 6×6 bounds did.","A further drop to 91 multiplications for 5×5 or 152 for 6×6 would produce an algorithm asymptotically faster than the classical 2×2 algorithm, since $\\log_5 91\\approx 2.803$ and $\\log_6 152\\approx 2.804$.","Symmetry seems to correlate with liftability: all symmetric schemes found here lifted to $\\mathbb{Z}$, whereas several previously discovered $\\mathbb{F}_2$ schemes are provably non-liftable."],"supporting_citations":[{"why":"Introduces the flip graph algorithm and gives the previous 5×5 bound of 97 that this paper improves to 93.","marker":"[16]"},{"why":"Adds plus-transitions to escape dead ends and reports a 5×5 scheme with 94 multiplications over F2, the bound this paper improves.","marker":"[2]"},{"why":"Gives the previous 6×6 upper bound of 160 that this paper improves to 153.","marker":"[21]"},{"why":"Defines the rank-one tensor model of matrix multiplication and establishes the 2×2 seven-multiplication scheme that motivates the symmetry analysis.","marker":"[22]"},{"why":"Provides the standard bilinear equations that the lifting procedure uses to test and refine F2 schemes.","marker":"[4]"},{"why":"Supplies the Hensel-lifting technique used to lift schemes from F2 to Z.","marker":"[23]"},{"why":"Provides non-symmetric schemes, including a 4×4 scheme of rank 47, and examples of F2 schemes that provably cannot be lifted to integers.","marker":"[11]"},{"why":"Determines the symmetry group of the matrix multiplication tensor used to define G-invariant schemes.","marker":"[8]"},{"why":"Shows the 2×2 seven-multiplication scheme is unique up to symmetry, motivating the search over symmetric schemes.","marker":"[9]"}],"fun_headline_variants":["Symmetry trims flip graph search, yields 93 and 153-mult schemes","Orbit flips shrink search, land 93 and 153-mult schemes","Cyclic symmetry speeds flip graph, finds 93 and 153-mult schemes","Symmetry-flip search: 5x5 in 93 mults, 6x6 in 153","Symmetry trims search, sets record-low 5x5 and 6x6 mults"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported schemes exist only as computer output in auxiliary files and are assumed to lift from $\\mathbb{F}_2$ to integer coefficients; the paper states that the Hensel-lifting step is not guaranteed to converge, and the claim that the schemes work over arbitrary fields depends on that empirical convergence holding for every reported scheme.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry trims flip graph search, yields 93 and 153-mult schemes","Orbit flips shrink search, land 93 and 153-mult schemes","Cyclic symmetry speeds flip graph, finds 93 and 153-mult schemes","Symmetry-flip search: 5x5 in 93 mults, 6x6 in 153","Symmetry trims search, sets record-low 5x5 and 6x6 mults"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001568,"raw_usage":{"total_tokens":6209,"prompt_tokens":843,"completion_tokens":5366,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":5248}},"tokens_in":459,"tokens_out":5366,"duration_ms":39058,"temperature":1.0,"reasoning_tokens":5248,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T22:27:07.059589+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Substitute the lifted integer coefficients of the reported 93- and 153-multiplication schemes into the defining bilinear equations for 5×5 and 6×6 matrix multiplication and check that the difference tensor is exactly zero over the integers; if any entry is nonzero, or if the mod-$2^k$ lifting system has no solution for any reported scheme, then the arbitrary-ground-field claim is false.","supporting_citations":[{"cited_title":"Flip graphs for matrix multiplication","cited_arxiv_id":null,"evidence_quote":"Introduces the flip graph algorithm and gives the previous 5×5 bound of 97 that this paper improves to 93."},{"cited_title":"Adaptive flip graph algorithm for matrix multiplication","cited_arxiv_id":null,"evidence_quote":"Adds plus-transitions to escape dead ends and reports a 5×5 scheme with 94 multiplications over F2, the bound this paper improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the previous 6×6 upper bound of 160 that this paper improves to 153."},{"cited_title":"Gaussian elimination is not optimal","cited_arxiv_id":null,"evidence_quote":"Defines the rank-one tensor model of matrix multiplication and establishes the 2×2 seven-multiplication scheme that motivates the symmetry analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the standard bilinear equations that the lifting procedure uses to test and refine F2 schemes."},{"cited_title":"Cambridge Univ","cited_arxiv_id":null,"evidence_quote":"Supplies the Hensel-lifting technique used to lift schemes from F2 to Z."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides non-symmetric schemes, including a 4×4 scheme of rank 47, and examples of F2 schemes that provably cannot be lifted to integers."},{"cited_title":"de Groote","cited_arxiv_id":null,"evidence_quote":"Determines the symmetry group of the matrix multiplication tensor used to define G-invariant schemes."},{"cited_title":"de Groote","cited_arxiv_id":null,"evidence_quote":"Shows the 2×2 seven-multiplication scheme is unique up to symmetry, motivating the search over symmetric schemes."}],"review_version":1}