{"id":"05b3313a-f766-4f82-8a88-9957ad9cfc06","arxiv_id":"2508.01748","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims a 44x44 base-case matrix multiplication algorithm with exponent 2.773203, beating Pan's 2.773372, but the proof of the key lemma is flawed.","lead":"Two algorithm families built on Pan's 1982 trilinear aggregation claim the best asymptotic exponents for feasible matrix multiplication, reaching O(n^2.773203) at base case 44. The proof of the key existence lemma has a concrete gap, so the headline improvement is not currently established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix A's proof of Theorem 2.22 is flawed as the reader says, but the theorem is true and repairable; the claimed rank bound and exponent are not falsified by this gap.","rationale":"The reader's REJECT is driven by the claim that Theorem 2.22 is unsupported and that the central kin-row reduction therefore fails. The first half of this is correct: the Appendix A proof as written applies Lemma A.2 twice in a way that does not preserve the first row fixed in the first application. However, this is a proof bug, not a mathematical obstruction. A correct proof is a small modification: choose the first target to be K_U K_V rather than K_U, rotate, and then set the current U row to K_U; the other row automatically becomes K_V. Alternatively, the substitution A' = K_U^T A, B' = B K_V^T, C' = K_V^{-T} C K_U^{-1} maps Strassen's first term to the desired term and preserves Trace(ABC). Therefore the claimed ⟨2,2,2;7⟩ algorithms with prescribed first rows exist, and Corollary 3.2's kin-row pairing is valid. The rest of the counting in Theorem 3.4, the rank bounds, and the exponent 2.773203 are conditional on exactly this lemma and are consistent once the lemma is fixed. I would not reject the paper outright: the defect is localized and easily verified. However, the manuscript as submitted does not contain a correct proof of the key existence lemma, and the supplemental matrices are not independently machine-checked in the text, so acceptance should be conditional on supplying a corrected proof or an explicit Brent-equation certificate.","tokens_in":23712,"tokens_out":24149,"duration_ms":266926,"concrete_test":"Construct the ⟨2,2,2;7⟩ algorithm explicitly for the matrices K_U = [[-1/γ, 1], [1, 0]] and K_V = [[1, γ], [1, 0]] of Corollary 3.3, using the corrected two-step Lemma A.2 construction (first target K_U K_V, then target K_U) starting from the Strassen triple in Claim A.3, with γ = 1 - 9/23 for n0 = 44. Then evaluate Brent's equations (Claim 2.9) for the resulting U, V, W and check that [U]_0 = [-1/γ, 1, 1, 0] and [V]_0 = [1, γ, 1, 0]. If this certificate passes, Theorem 2.22 is true and the proof gap is repairable.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing point is the existence lemma Theorem 2.22. The reader is right that the Appendix A proof is invalid: Lemma A.2 re-encodes both U and V, so after rotating and applying it a second time the previously fixed first row of U is not preserved in the way the proof's variable naming suggests. That is a real defect in the submitted proof. But it is not a false theorem. A two-step repair works: start from Strassen (first rows I, I); apply Lemma A.2 with target K1 = K_U K_V, rotate, then apply Lemma A.2 with target K_U; the second encoding row becomes K_U^{-1} K1 = K_V, so the final algorithm has rows (K_U, K_V). Even more directly, the change of variables A' = K_U^T A, B' = B K_V^T, C' = K_V^{-T} C K_U^{-1} preserves Trace(ABC) and sends the first Strassen term (I, I, I) to a term whose A- and B-coefficients are K_U and K_V. Thus the special ⟨2,2,2;7⟩ algorithms needed for Corollary 3.2 exist. The n0/2+1 kin-row reduction and the resulting rank bounds and exponents are not invalidated; the paper needs a corrected proof or an explicit certificate, not rejection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two families of matrix multiplication algorithms over the same feasible base-case sizes as Pan's 1982 algorithms. The first family uses de Groote-equivalent variants of Strassen's 2x2 algorithm inside the trilinear aggregation framework to create kin rows, reducing the number of multiplications by (n0/2+1) for even n0>=28, n0 != 16; for n0=44 this yields a ⟨44,44,44;36110⟩-algorithm with exponent log44(36110) ≈ 2.773203. The second family applies two recursive steps and replaces ⟨4,4,4;49⟩ sub-algorithms by ⟨4,4,4;48⟩-algorithms, yielding a ⟨1936,1936,1936;1303676064⟩-algorithm with exponent ≈ 2.773177. The paper also reports leading-coefficient reduction via sparse decomposition, with leading coefficient about 8. All exponents are claimed to improve over Pan's 1982 baselines.","tokens_in":23840,"tokens_out":18462,"duration_ms":189465,"significance":"If the construction is valid, these are the best known exponents among algorithms with base cases below 1000, improving Pan's 1982 algorithm for many feasible sizes. The contribution is a mathematical derivation rather than a parameter fit: the rank bounds are explicit formulas, and the authors provide supplemental encoding/decoding matrices. The counting arguments, including the h^2 count of embedded ⟨4,4,4;49⟩ tensors in Section 4, are internally consistent. The main caveat is that the proof of the key existence lemma (Theorem 2.22) is defective as written, so the central rank bounds are not yet established by the submitted manuscript.","major_comments":[{"comment":"The proof of Theorem 2.22 does not work. After the first application of Lemma A.2, we have M([U1]_0)=KU and M([V1]_0) invertible. The proof then cyclically rotates to ⟨V1,U1,W1⟩ and applies Lemma A.2 again with target KV. In that second application the 'U' matrix is V1 and the 'V' matrix is U1, so the lemma transforms the first row of the second matrix (i.e., U1) to R^{-T}M([U1]_0), where R is defined from M([V1]_0); explicitly M([U2]_0) = M([V1]_0)^{-1} KV KU, which need not equal KU. Rotating back gives a final triplet with [U]_0 = vec(M([V1]_0)^{-1} KV KU), not vec(KU). Thus the asserted simultaneous prescription of [U]_0 and [V]_0 is not established. This is load-bearing: Claim 3.3 uses Theorem 2.22 to obtain the required Strassen variant, and Theorem 3.4's reduction by (n0/2+1) multiplications, the rank bounds in Table 1, and Corollary 3.6's exponent all depend on it. The lemma itself is plausible (a two-step construction different from the one in the appendix can achieve the goal, or an explicit ⟨2,2,2;7⟩-algorithm can be certified), so I am not recommending rejection; rather, the proof needs to be corrected, or the claimed existence should be supported by an explicit machine-checkable certificate.","section":"Appendix A, Theorem 2.22"}],"minor_comments":[{"comment":"There is a typographical error: the text reads 'M([U'']_0) = KU' where it should read 'M([U']_0) = KU'.","section":"Appendix A, proof of Theorem 2.22"},{"comment":"The last line says 'the minimum is indeed obtained for n0 = 44', but the variable in that proof is m0 (with n0=m0^2); it should say m0=44.","section":"Corollary 4.3 proof"},{"comment":"The displayed formula has '244n3_0 1.5', which appears to be missing a denominator; the proof and Corollary 4.3 use 244 m0^3/3 = 244 n0^{1.5}/3. Please correct the typesetting.","section":"Theorem 4.2 statement"},{"comment":"The word 'wereas' should be 'whereas' in the sentence 'However, wereas Pan utilizes a single ⟨2,2,2;7⟩-algorithm...'.","section":"Section 3"},{"comment":"Definition 4.1 states informally that each occurrence of a ⟨4,4,4;49⟩-algorithm in the composed algorithm can be replaced by a ⟨4,4,4;48⟩-algorithm. This is a standard tensor-product substitution, but the paper should spell out the embedding: after two recursive steps, the h^2 pairs of intact 2x2 algorithms produce independent sub-tensors, and replacing each decomposition reduces the total rank by one per pair. The arithmetic in the proof of Theorem 4.2 is consistent with this reading, but the formal justification is only one sentence and should be expanded.","section":"Section 4, Definition 4.1 and Theorem 4.2"},{"comment":"The claim 'Checking all even values ... we see' is a finite computation; please include the data or a script in the supplemental material to make the check reproducible.","section":"Corollary 3.6 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper's main weakness is the dependence on the unproved Theorem 2.22. If the authors can supply a corrected proof or a machine-checked certificate for the special Strassen variants, the result would be a solid contribution. Please ensure the supplemental zip file is accessible to reviewers; the external link may not be stable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the reader is right that Appendix A's proof of Theorem 2.22 is broken, but the theorem itself is true, and the paper's main claims look correct once that proof is repaired. This is a repair-grade flaw, not a rejection-grade one.\n\nWhat's genuinely new: picking ⟨2,2,2;7⟩-algorithms from de Groote's equivalence class so that the first multiplication in the cancellation step is kin to an aggregation term. That is a real trick. It yields n0/2+1 fewer multiplications than Pan's 1982 algorithm for all even n0 ≥ 28, n0 ≠ 16. The resulting rank bounds, e.g. Rank(⟨44,44,44⟩) ≤ 36110 and exponent 2.773203, are new, and the second family with base 1936 improves the exponent to 2.773177. The leading coefficient analysis via sparse decomposition gives constants around 8, comparable to [26]. The paper also ships explicit encoding/decoding matrices as supplemental material — good practice, though I have not independently verified them.\n\nThe soft spot: the proof of Theorem 2.22. The reader's objection lands. Lemma A.2 changes both U and V rows. Applying it, rotating, and applying it again does not leave the earlier U row fixed; in the final algorithm the first row of U is K_V^{-1} rather than K_U. So the proof as written does not establish the existence of the required special Strassen variant. That matters because Corollary 3.2 and the whole n0/2+1 reduction depend on it.\n\nHowever, the lemma is repairable. The stress-test note gives two structurally simple repairs: either use Lemma A.2 with target K_U K_V, rotate, then use it again with target K_U (which makes the other row K_U^{-1} K1 = K_V), or note that the change of variables A' = K_U^T A, B' = B K_V^T, C' = K_V^{-T} C K_U^{-1} preserves Trace(ABC) and maps Strassen's first term to a term with the desired U and V rows. I checked the trace identity; it works. So the special ⟨2,2,2;7⟩-algorithms exist, and the claimed rank bounds and exponents are not falsified.\n\nOther soft spots are minor. The improvement over Pan is tiny (~0.00017), but it breaks a forty-year-old record in the feasible regime, so it is meaningful for this subfield. The abstract's phrase about a 'fundamental step towards outperforming in practice' is overpromising; the paper itself concedes the algorithm is impractical until the decomposition. That is fine.\n\nWho this is for: people who care about feasible matrix multiplication algorithms and rank bounds for small tensors. It is not a breakthrough for the general ω problem. It deserves a serious referee; the referee should require a corrected proof of Theorem 2.22, ideally with an explicit certificate for the Strassen variant.","headline":"The reader's proof objection is correct, but the theorem is repairable; the new feasible matrix multiplication exponents likely stand, and the paper deserves peer review with a mandatory corrected Appendix A.","tokens_in":24516,"tokens_out":3347,"would_cite":true,"duration_ms":35891,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q17","68Q25","15A69"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims a 44-by-44-by-44 matrix multiplication algorithm using 36,110 multiplications, with complexity $O(n^{2.773203})$, the fastest among feasible base cases under 1000.","keywords":["fast matrix multiplication","trilinear aggregation","feasible algorithms","tensor rank bound","kin rows","leading coefficient","matrix multiplication exponent","implicit canceling"],"falsifier":"Verify Appendix A on a concrete instance: take $K_U = \\begin{pmatrix}-1/\\gamma & 1\\\\1 & 0\\end{pmatrix}$ and $K_V = \\begin{pmatrix}1 & \\gamma\\\\1 & 0\\end{pmatrix}$ with $\\gamma = 1 - 9/(n_0/2+1)$, apply the two row-rewriting steps, and check whether the final algorithm has $M([U]_0)=K_U$ and $M([V]_0)=K_V$. If the second step overwrites the first, the kin pairing of Corollary 3.2 fails and the $\\langle44,44,44;36110\\rangle$ bound must be re-examined. Independently, the supplemental $\\langle44,44,44;36110\\rangle$ encoding/decoding matrices can be tested directly against the trace equations of matrix multiplication: if the equations hold with 36,110 rows, the rank bound and exponent stand regardless of the intermediate proof.","tokens_in":23373,"feed_emoji":"⚡","tokens_out":15601,"duration_ms":149065,"temperature":0.7,"pith_summary":"The paper aims to make fast matrix multiplication work at 'feasible' input sizes, where the recursive base case is small enough to run on real hardware. It constructs two families of trilinear-aggregation algorithms; the first contains a $44\\times44\\times44$ algorithm using 36,110 multiplications, with complexity $O(n^{2.773203})$, and the second, at base case 1936, achieves $O(n^{2.773177})$. These are claimed to be the best exponents among known algorithms whose base cases are smaller than 1000, and for the second family among all known feasible algorithms. The construction chooses variants of the 7-multiplication $2\\times2$ algorithm with prescribed first rows, creates duplicate 'kin' rows shared between the aggregation and cancellation steps, and unites each pair to save one multiplication. The paper also reports a sparse decomposition that reduces the leading coefficient from about 736 to about 8, a step toward practical applicability.","feed_headline":"A 44x44 base case computes matrix products with exponent 2.773203","feed_subtitle":"It is the fastest known matrix multiplication algorithm with base case size under 1000, beating the 1982 exponent.","key_machinery":"The carrying object is the trilinear aggregation scheme of the 1982 algorithm, written as encoding/decoding matrices $\\langle U,V,W\\rangle$. In the cancellation step, the scheme computes a sum of traces, each a $2\\times2$ matrix product that any 7-multiplication algorithm can compute. The paper's move is to pick, for the diagonal traces, an equivalent 7-multiplication algorithm whose first $U$-row is $(-1/\\gamma_{i,i}, 1, 1, 0)$ and whose first $V$-row is $(1, \\gamma_{i,i}, 1, 0)$, with $\\gamma_{i,i} = 1 - 9/(n_0/2 + 1)$; these rows reproduce the aggregation-step rows, creating $n_0/2 + 1$ disjoint kin pairs. Uniting each pair removes one multiplication per pair, cutting the count from $t_{\\mathrm{Pan}}$ to $t_{\\mathrm{New}}$. The second family exploits the observation that two recursive calls of the first family contain $\\langle4,4,4;49\\rangle$ subtensors and replaces each by a $\\langle4,4,4;48\\rangle$ algorithm to trim additional multiplications.","core_discovery":"On its own terms, the central discovery is that the trilinear aggregation method can be sharpened by replacing the generic 7-multiplication $2\\times2$ base algorithm inside the cancellation step with an equivalent algorithm whose first encoding and decoding rows match rows already present in the aggregation step. Because the paper argues all such $2\\times2$ algorithms lie in one equivalence class, it claims the first rows can be prescribed at will (Theorem 2.22). The matched rows become kin rows, meaning two rows of the encoding/decoding triple that coincide in two of the three matrices, and Lemma 2.20 removes one multiplication per disjoint kin pair. This yields the family TA-New25 with $t = n_0^3/3 + 15n_0^2/4 + 61n_0/6 + 8$ multiplications for even $n_0 \\neq 16$, whose exponent is minimized at $n_0 = 44$: a $\\langle 44,44,44; 36110\\rangle$-algorithm with complexity $O(n^{2.773203})$. Squaring the base case and replacing the resulting $\\langle 4,4,4;49\\rangle$ subtensors by $\\langle 4,4,4;48\\rangle$ algorithms gives a second family whose best exponent at base case 1936 is $O(n^{2.773177})$.","pith_inferences":["If the prescribed-first-row theorem can be repaired, the same kin-row trick could apply to rectangular or odd-dimension algorithms through the symmetrization and padding techniques the paper discusses, possibly improving exponents for base cases below 28 as well.","Iterating the subtensor replacement on higher powers of the base case may squeeze the exponent further, at the price of larger base cases; nothing in the paper rules out or quantifies such a limit.","Because the exponent gain over the 1982 algorithm is only about 0.00017 while the sparse decomposition raises communication costs, the practical payoff is an empirical question about constants and I/O behavior, which the paper leaves open.","A direct computer search over the equivalence class of 7-multiplication $2\\times2$ algorithms could verify the prescribed-first-row property independently of the proof's bookkeeping."],"forward_implications":["It establishes $\\mathrm{Rank}(\\langle 44,44,44\\rangle) \\le 36110$, so multiplying $n\\times n$ matrices by recursion on this base case runs in $O(n^{2.773203})$.","Among algorithms with base cases smaller than 1000, this is the fastest claimed asymptotic complexity, improving on the previous 2.773372 exponent from 1982.","The same construction yields the best known exponents for many even base cases $n_0 \\ge 28$, not only $n_0 = 44$.","The second family, with base case $44^2 = 1936$, reaches $O(n^{2.773177})$, claimed to be the best exponent of any known feasible algorithm.","A sparse decomposition brings the leading coefficient of the new algorithms from about 736 down to about 8, comparable to the best prior coefficient reductions."],"supporting_citations":[{"why":"Baseline 1982 trilinear-aggregation algorithm with implicit canceling whose multiplication count and exponent the first family improves.","marker":"[48]"},{"why":"Supplies the aggregation tables, transformations, and sparse-decomposition construction the new algorithms adapt.","marker":"[26]"},{"why":"Shows all 7-multiplication $2\\times2$ algorithms fall into one equivalence class, enabling the prescribed first rows.","marker":"[19]"},{"why":"Introduces the equivalence operator used to transform base-algorithm rows in proving the prescribed-first-row claim.","marker":"[18]"},{"why":"The classical 7-multiplication $2\\times2$ algorithm that serves as the starting point of the equivalence-class search.","marker":"[54]"},{"why":"Provides the $\\langle4,4,4;48\\rangle$-algorithm used to replace $\\langle4,4,4;49\\rangle$ subtensors in the second family.","marker":"[32]"},{"why":"Supplies a non-complex-coefficient $\\langle4,4,4;48\\rangle$-algorithm, keeping the second family valid over the same fields as the first.","marker":"[22]"}],"fun_headline_variants":["44x44 base case yields exponent 2.773203, beating 1982 Pan","Trilinear aggregation lowers matrix multiply exponent to 2.773203","Sub-1000 base case beats Pan's exponent: 2.773203","Fastest feasible matrix multiplication: exponent 2.773203"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that any two invertible $2\\times2$ matrices $K_U$ and $K_V$ can be realized as the first encoding and decoding rows of a 7-multiplication $2\\times2$ algorithm (Theorem 2.22). The proof in Appendix A applies the row-rewriting step twice, and the second application appears to overwrite the row set by the first, so the resulting algorithm may have $K_V^{-1}$ in the first encoding row instead of $K_U$. The claimed saving of $n_0/2 + 1$ multiplications, and therefore the exponents $2.773203$ and $2.773177$, depend on this premise.","fun_headline_variants_meta":{"raw":{"variants":["44x44 base case yields exponent 2.773203, beating 1982 Pan","Trilinear aggregation lowers matrix multiply exponent to 2.773203","Sub-1000 base case beats Pan's exponent: 2.773203","Fastest feasible matrix multiplication: exponent 2.773203"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000974,"raw_usage":{"total_tokens":4186,"prompt_tokens":1041,"completion_tokens":3145,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":657,"completion_tokens_details":{"reasoning_tokens":3063}},"tokens_in":657,"tokens_out":3145,"duration_ms":27966,"temperature":1.0,"reasoning_tokens":3063,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:29:51.637829+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify Appendix A on a concrete instance: take $K_U = \\begin{pmatrix}-1/\\gamma & 1\\\\1 & 0\\end{pmatrix}$ and $K_V = \\begin{pmatrix}1 & \\gamma\\\\1 & 0\\end{pmatrix}$ with $\\gamma = 1 - 9/(n_0/2+1)$, apply the two row-rewriting steps, and check whether the final algorithm has $M([U]_0)=K_U$ and $M([V]_0)=K_V$. If the second step overwrites the first, the kin pairing of Corollary 3.2 fails and the $\\langle44,44,44;36110\\rangle$ bound must be re-examined. Independently, the supplemental $\\langle44,44,44;36110\\rangle$ encoding/decoding matrices can be tested directly against the trace equations of matrix multiplication: if the equations hold with 36,110 rows, the rank bound and exponent stand regardless of the intermediate proof.","supporting_citations":[{"cited_title":"Towards practical fast matrix multiplication based on trilinear aggregation","cited_arxiv_id":null,"evidence_quote":"Supplies the aggregation tables, transformations, and sparse-decomposition construction the new algorithms adapt."},{"cited_title":"On varieties of optimal algorithms for the computation of bilinear mappings II. optimal algorithms for 2× 2-matrix multiplication","cited_arxiv_id":null,"evidence_quote":"Shows all 7-multiplication $2\\times2$ algorithms fall into one equivalence class, enabling the prescribed first rows."},{"cited_title":"Gaussian elimination is not optimal","cited_arxiv_id":null,"evidence_quote":"The classical 7-multiplication $2\\times2$ algorithm that serves as the starting point of the equivalence-class search."},{"cited_title":"Finding complex-valued solutions of Brent equations using nonlinear least squares","cited_arxiv_id":null,"evidence_quote":"Provides the $\\langle4,4,4;48\\rangle$-algorithm used to replace $\\langle4,4,4;49\\rangle$ subtensors in the second family."}],"review_version":1}