{"id":"3ddf99ec-9861-43cb-a2b9-cc3f9ff0088e","arxiv_id":"2506.22113","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A commutative flip graph defined on a quotient tensor space recovers known commutative matrix multiplication bounds up to 5x5 without improving any of them.","lead":"This paper adapts flip-graph searches, previously used for non-commutative matrix multiplication, to the commutative setting. Its new methods match the best known multiplication counts for all tested sizes up to 5x5, but find no new improvements.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.5's connectivity proof is deferred to the noncommutative case, but the quotient graph has no canonical common factor; if disconnected, the search space is incomplete and the 'strong potential' claim is unsupported.","rationale":"The reader's weakest assumption combines two claims: that the quotient tensor model faithfully captures commutative algorithms, and that Theorem 3.5's connectivity holds. On the first point, the quotient model is mathematically sound: U is isomorphic to Sym^2(U1)⊗U2, and a rank-one element α⊙β⊗γ corresponds to the product of two linear forms on the combined A/B inputs, which is exactly a commutative multiplication. The representation of M' as sum aij⊙bjk⊗cki is the correct commutative matrix multiplication bilinear form. Thus I do not share the reader's doubt about the model itself. The second point is genuinely load-bearing. The paper gives no proof of Theorem 3.5, and the asserted analogy to the noncommutative case is nontrivial because the flip, reduction, and plus operations are defined on equivalence classes, where a common factor is not canonically extractable. If connectivity fails, the empirical success in small sizes may not generalize, and the forward-looking claim of strong potential is weakened. The concrete test targets this gap by checking connectivity for a small finite case; a disconnected component would refute Theorem 3.5. The Marakov construction in Definition 3.1 has an indexing typo that makes reproduction impossible as printed, but this is fixable and does not affect the fully commutative method. Because the reader's CONDITIONAL verdict already requires proof of Theorem 3.5 and code/data, my read does not change that verdict.","tokens_in":10028,"tokens_out":33490,"duration_ms":366146,"concrete_test":"Implement Definition 3.2 for (2,2,2) over F2 (U1 has dimension 8, so U ≅ Sym^2(F2^8)⊗F2^4) and perform an exhaustive BFS of the commutative adaptive flip graph restricted to schemes of rank ≤ 8, starting from the standard 8-term scheme and using canonical representatives for quotient elements. Check whether every rank-7 decomposition of M'_{2,2,2} is reachable via flips, pluses, and reductions. If some rank-7 scheme is not reachable, Theorem 3.5 is false and the search is incomplete. If the state space is too large for exhaustive BFS, repeat this connectedness check on (1,2,1) where the quotient is smaller and all rank-2 decompositions can be enumerated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the commutative flip graph search 'recovers the best known bounds' and 'demonstrates strong potential' relies on Theorem 3.5, which asserts that the commutative adaptive flip graph is connected. The proof is dismissed with 'exactly follows the corresponding proof from the non-commutative case.' This transfer is not routine: the noncommutative proof operates on schemes in V1⊗V2⊗V3 with rank-one terms u⊗v⊗w, whereas the commutative setting works in U = (U1⊗U1⊗U2)/S ≅ Sym^2(U1)⊗U2, where the first two factors are identified. In that quotient, a rank-one term has no unique representative, so the key step of factoring out a common vector a from two terms has no direct analogue. If the graph is disconnected, the search is confined to a single component; matching Rosowski's bounds in every tested size may then be a property of that component rather than of the whole space, and failure at larger sizes would not indicate a true lower bound. Since no proof, code, or detailed experimental data are provided, the gap is unresolved. Definition 3.1 also has an indexing inconsistency: the basis sets as written place both factors of the j-even terms in the same factor space, which makes the Marakov-like tensor ill-defined as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes three flip-graph based heuristics for finding low-rank commutative matrix multiplication schemes: (i) a 'Marakov-like' tensor that encodes the parity-based partition from Makarov's 3×3 algorithm, searched with the standard noncommutative flip graph; (ii) a commutative flip graph defined on the quotient space U=(U1⊗U1⊗U2)/S, in which the first two factors are identified; and (iii) a hybrid that seeds the commutative search with low-rank Marakov-like schemes. The main experimental claim is that across all l,m,n up to 5, the commutative search matches Rosowski's upper bounds, the Marakov search is fastest but occasionally misses, and the hybrid is nearly as accurate and about three times faster; no improvements over Rosowski are found. The paper also states Theorem 3.5 that the commutative adaptive flip graph is connected, with the proof deferred to the noncommutative case.","tokens_in":10305,"tokens_out":6584,"duration_ms":66428,"significance":"If the central claims held, this would be the first successful automated search for commutative matrix multiplication schemes and would provide a practical toolkit for larger sizes. The paper's positive points are the explicit construction of a quotient tensor model, the empirical comparison across 42 size triples, and the honest reporting that no new bounds were found. However, the current value is limited by two unresolved load-bearing issues: the Marakov-like tensor (Definition 3.1) is not well-defined as written, and the connectivity of the commutative flip graph (Theorem 3.5) is asserted without the non-routine transfer of the proof. Because the paper makes no new rank bounds, its significance depends on the reliability and reproducibility of the search methods.","major_comments":[{"comment":"Definition 3.1 as stated does not define a tensor in the declared vector spaces: V'1,l,m is declared to be F^{l×⌊m/2⌋} but its generator set {aij : j even} ∪ {bij : i even} has cardinality l⌈m/2⌉ + m⌈n/2⌉ (e.g., for l=m=n=3, six elements in a three-dimensional space), and similarly V'2,m,n. Moreover, in the displayed sum the odd-j terms place aij in the first factor although aij is not among the listed generators of V'1,l,m, and the even-j terms place aij in the second factor although it is not among the generators of V'2,m,n. The Marakov-like tensor Vl,m,n is therefore not a well-defined element of V'1⊗V'2⊗V'3, and the entire M column of Table 1 rests on an ill-defined object. Please correct the parity assignments, the dimensions, or explain the intended quotient or identification.","section":"§3.1, Definition 3.1"},{"comment":"The proof of Theorem 3.5 consists of one sentence saying it 'exactly follows' the noncommutative case. This is not a routine transfer: the noncommutative proof in [2] works with a fixed tensor product V1⊗V2⊗V3 and rank-one terms u⊗v⊗w with unique representatives, and its key step is factoring a common vector a out of two terms. In the quotient U=(U1⊗U1⊗U2)/S, a rank-one class α⊗β⊗γ+S does not have a unique representative (since α⊗β and β⊗α are identified), so the common-factor operation is not well-defined on classes. If the graph is not connected, then the search explores only one component and the fact that Table 1 matches Rosowski's bounds does not demonstrate coverage of all commutative schemes. Please provide a complete proof, a counterexample, or a clearly stated weakening of the claim to a heuristic statement.","section":"§3.2, Theorem 3.5"},{"comment":"The paper's central empirical claim cannot be reproduced from the information given. There is no released code or data, and the text does not specify the search parameters that are known to matter for adaptive flip graph methods: the plus-operation frequency (mentioned only qualitatively in Remark 3.6), the compute budget and stopping rule, the random seed, the neighbor and selection strategy, and the exact field arithmetic (stated as F2 but without confirmation for all runs). In addition, the table and the text disagree on the hybrid M-C for (3,3,4) and (3,5,4): the table lists 28 and 46 while Rosowski's bounds are 27 and 45, and the text says extended computation matched Rosowski. Please clarify whether the table reports initial runs or final runs, and add enough experimental detail (or a code/data repository) to make the numerical claims checkable.","section":"§4, Table 1 and experimental setup"}],"minor_comments":[{"comment":"The name 'Marakov' appears throughout the text, while reference [8] is 'Makarov'; please standardize the spelling.","section":"Throughout"},{"comment":"The rank decomposition is written as sum_{p=0}^r t_p; the lower limit should presumably be p=1.","section":"Definition 2.3"},{"comment":"The tensor Vl,m,n in Definition 3.1 collides notationally with the flip-graph vertex set Vl,m,n from Definition 2.9; choose a distinct symbol such as Wl,m,n.","section":"§3.1"},{"comment":"The claim that differing entries are shown in bold is not visible in the table as typeset; list the differing entries explicitly in the caption or text.","section":"§4, Table 1"},{"comment":"The remark mentions Hensel lifting but gives no details or reference for this step; if it is used in the experiments, it should be specified.","section":"Remark 2.12"}],"recommendation":"major_revision","confidential_remarks":"The paper is borderline for publication in its current form because it reports no new rank bounds and its main theoretical support (Theorem 3.5) is unproved. The most useful revision would add (a) a corrected Definition 3.1, (b) a real proof of Theorem 3.5 or an explicit statement that connectivity is conjectural, and (c) code/data with full parameters. If those are supplied, the contribution could be publishable; otherwise the experimental benchmark remains unverifiable. I would not recommend rejection, since the approach is interesting and the empirical comparison, if reproducible, would be a useful reference point for future work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Isaac Wood's paper adapts the flip-graph search method to the commutative setting and reports that it recovers Rosowski's bounds for all matrix sizes up to 5x5. That claim is supported by the table, and the paper is honest that no new bounds were found.\n\nThe genuinely new pieces are the quotient tensor U=(U1⊗U1⊗U2)/S, which identifies a⊗b with b⊗a in the first two factors, and a Marakov-inspired tensor built by splitting entries into two sets before applying the standard flip graph. The hybrid method is a natural combination. The comparison across three methods with timings is a useful data point for anyone doing this kind of search.\n\nMy main reservation is conceptual. The paper defines the rank of M' in the quotient space and calls the resulting decompositions 'commutative schemes.' But a rank-one term in U is any α⊗β⊗γ with α,β ∈ U1, so α and β can be arbitrary linear combinations of A and B entries. In a standard commutative bilinear algorithm, each multiplication is a product of a linear form in A with a linear form in B. The quotient does not enforce that separation. It is therefore not obvious that rk'(M') equals the minimum number of multiplications in a commutative algorithm. The paper never addresses this. Since the authors only match Rosowski and find nothing better, no false claim is made, but the 'strong potential' claim rests on an unproven equivalence.\n\nSecond, Theorem 3.5 asserts the commutative flip graph is connected, with the proof dismissed as 'exactly following' the non-commutative case. That transfer is not routine in the quotient: rank-one terms have no unique representative, so the common-factor argument in the original proof does not obviously carry over. If connectivity fails, the method only explores a component. The empirical success up to 5x5 suggests the component is rich, but that does not establish completeness.\n\nThird, Definition 3.1 has a concrete indexing bug: the basis sets as written put the odd-j a's and even-i b's in the wrong factors for the tensor's sum. Fixable, but as printed the Marakov tensor is ill-defined. The paper also ships no code or data, gives no random seeds or detailed parameters, and the two hybrid misses (3,3,4) and (3,5,4) are later reported as matching after extended runs, which the table doesn't reflect.\n\nWho is this for? People working on automated search for matrix multiplication schemes, specifically the commutative sub-question. It's a niche but real audience. I would send it to a referee. The gaps are addressable and the core search heuristic is plausible. A referee should ask for a precise statement of what the quotient rank means, a real proof of connectivity, and the experimental code.","headline":"A creditable first application of flip-graph search to commutative matrix multiplication, with a real gap between the quotient model and actual commutative algorithms.","tokens_in":10816,"tokens_out":9438,"would_cite":false,"duration_ms":87919,"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 adapts flip-graph search to commutative matrix multiplication, introducing a quotient-tensor flip graph and a construction modeled on a 1986 3x3 algorithm, and shows both recover the best known bounds for every matrix size up…","keywords":["matrix multiplication","commutative algorithms","flip graphs","tensor rank","bilinear complexity","search heuristics","algebraic complexity","quotient tensor spaces"],"falsifier":"Take a tiny case such as (2,2,2) or (3,3,3) over $\\mathbb{F}_2$ and exhaustively generate every scheme in the commutative flip graph to check whether it is connected and whether every minimal scheme lies in the component of the standard scheme; a disconnected graph or an unreachable minimal scheme would refute the search's completeness. A second check is to take a known commutative algorithm and verify each of its multiplications can be written as a quotient-space rank-one term; any algorithm that cannot be written that way falsifies the model.","tokens_in":9801,"feed_emoji":"🔀","tokens_out":12772,"duration_ms":124058,"temperature":0.7,"pith_summary":"Standard searches for fast matrix multiplication assume non-commutative products, but allowing commutativity can reduce the number of multiplications needed. This paper adapts the flip-graph search, which walks between schemes by flips and reductions, to the commutative setting in two ways: a construction modeled on a known 1986 3×3 algorithm that the standard flip graph can search, and a fully commutative flip graph defined on a quotient tensor space where swapping the two matrix factors is identified. Across all sizes up to 5×5, both approaches recover the best known commutative bounds; the fully commutative graph matches them in every case, and the hybrid matches in almost all cases after extra computation. No improvement on existing bounds was found, but the paper argues the tools are ready for larger sizes.","feed_headline":"Flip graphs match best commutative matrix multiplication bounds","feed_subtitle":"Two new commutative searches recover the best known ranks for every size up to 5x5; the hybrid runs three times faster.","key_machinery":"The argument runs on two search spaces. The first is a tensor construction modeled on a 1986 algorithm for 3×3 matrices, which reorders even- and odd-indexed entries so a commutative multiplication appears as a rank-one term and the standard adaptive flip graph can search it. The second is the commutative tensor space $U = (U_1\\otimes U_1\\otimes U_2)/S$, quotiented by identifying $u_{11}\\otimes u_{12}\\otimes u_2$ with $u_{12}\\otimes u_{11}\\otimes u_2$; this captures commutativity, and flips, reductions, and pluses are redefined on these quotient rank-one terms. The commutative adaptive flip graph is asserted to be connected over $\\mathbb{F}_2$ following the non-commutative proof. Searches walk this graph from the standard scheme or from smaller schemes in search of short decompositions of the commutative matrix multiplication tensor.","core_discovery":"The central claim is that flip-graph search works for commutative matrix multiplication if commutativity is handled either before the search or inside the graph. The first strategy rewrites the tensor so each commutative multiplication is a rank-one term in a reordered space, letting the standard adaptive flip graph run unchanged. The second quotients the tensor space so $a\\otimes b\\otimes c$ and $b\\otimes a\\otimes c$ are the same element, then redefines flip, reduction, and plus on these quotient terms. Over $\\mathbb{F}_2$, the quotient-based graph matches the best known commutative bound for every matrix size up to 5×5; the hybrid of both strategies matches it in almost all sizes with extended computation, and the first strategy is fastest but sometimes misses the bound. No improvement on the known bounds is found, and the paper presents the methods as viable automated tools for larger commutative searches.","pith_inferences":["A natural next step is to apply the quotient-space flip graph to other bilinear problems whose tensors are symmetric under swapping two factors; the formalism is not specific to matrix multiplication.","The asserted connectivity of the commutative flip graph is load-bearing but only said to follow from the non-commutative proof; an exhaustive connectivity check for very small sizes would be a cheap way to de-risk the whole search.","If the misses at sizes of the form (a,7,b) persist under longer runs, that points to starting-scheme quality or connectivity rather than raw compute as the bottleneck, suggesting better seeds over more processors.","The speed gap between the first construction and the quotient graph hints that the quotient search could be made much faster with implementation effort, which would make searches beyond 7 practical."],"forward_implications":["The quotient-based flip graph matches the best known commutative bound for every matrix size up to 5×5, so it can serve as an automated upper-bound certifier in that range.","Seeding the commutative search with the first construction's schemes matches the bound in almost all sizes while taking about 15 hours instead of the full quotient search's 50, so the two strategies complement each other.","The first construction alone runs about 30 times faster than the quotient graph but misses several bounds, so speed and completeness trade off.","A large run up to (7,7,7) matched the known bounds except for some sizes of the form (a,7,b), locating a concrete weakness of the current pipeline.","No tested size beat the existing bound, so the contribution is a search pipeline that reproduces known results rather than a new record."],"supporting_citations":[{"why":"Supplies the original flip graph, its definitions of flips and reductions, and the connectivity theorem the commutative version adapts.","marker":"[5]"},{"why":"Supplies the adaptive flip graph, the plus operation, and the search algorithm used here.","marker":"[2]"},{"why":"The 1986 3×3 algorithm whose structure motivates the first tensor construction.","marker":"[8]"},{"why":"Gives the commutative multiplication bound used as the baseline in all experiments.","marker":"[12]"},{"why":"Documents an earlier unsuccessful application of flip graphs to commutative schemes, the obstacle addressed here.","marker":"[9]"}],"fun_headline_variants":["Commutative flip graphs match best known matrix bounds","Flip graphs take on commutative matrix multiplication","Quotient trick lets flip graphs hit commutative bounds","Hybrid flip graph: three times faster, same best bounds","No new bounds, but flip graphs now work commutative"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole search assumes that every commutative way to multiply matrices can be captured by the quotient-tensor formalism, and that the graph of schemes it searches is connected; if either assumption fails, the search could silently miss good algorithms.","fun_headline_variants_meta":{"raw":{"variants":["Commutative flip graphs match best known matrix bounds","Flip graphs take on commutative matrix multiplication","Quotient trick lets flip graphs hit commutative bounds","Hybrid flip graph: three times faster, same best bounds","No new bounds, but flip graphs now work commutative"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000722,"raw_usage":{"total_tokens":3207,"prompt_tokens":878,"completion_tokens":2329,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":2254}},"tokens_in":494,"tokens_out":2329,"duration_ms":17637,"temperature":1.0,"reasoning_tokens":2254,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:11:03.342902+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a tiny case such as (2,2,2) or (3,3,3) over $\\mathbb{F}_2$ and exhaustively generate every scheme in the commutative flip graph to check whether it is connected and whether every minimal scheme lies in the component of the standard scheme; a disconnected graph or an unreachable minimal scheme would refute the search's completeness. A second check is to take a known commutative algorithm and verify each of its multiplications can be written as a quotient-space rank-one term; any algorithm that cannot be written that way falsifies the model.","supporting_citations":[{"cited_title":"Fast commutative matrix algorithms","cited_arxiv_id":null,"evidence_quote":"Gives the commutative multiplication bound used as the baseline in all experiments."},{"cited_title":"Search Techniques for Matrix Algorithms","cited_arxiv_id":null,"evidence_quote":"Documents an earlier unsuccessful application of flip graphs to commutative schemes, the obstacle addressed here."}],"review_version":1}