{"id":"2752c577-e404-42db-8236-551fb9582e3e","arxiv_id":"2607.27407","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every complete graph, every blow-up of an r-vertex graph, and every tree admits an ε-matrix-product factorization with ε = O(1/n), despite many having no exact factorization.","lead":"This paper introduces an approximate (\"ε\") version of matrix-product factorization of graphs, where two factor graphs may fail to reproduce the target graph on a small fraction of ordered vertex pairs. It shows that exact obstructions—like complete graphs needing n ≡ 1 mod 4—vanish under this Hamming-type relaxation, with complete graphs, blow-ups, and trees all admitting near-factorizations with error O(1/n).","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest-assumption analysis correctly identifies the cited exact cyclic factorization as the main external dependency. However, that dependency is true and can be verified directly from the difference-set structure; the deletion argument and the error counts in Theorem 3.3 are correct. The other main results (blow-ups, trees) do not depend on this external theorem and their constructions check out. Since the central argument holds under scrutiny and no internal flaw was found, the reader's ACCEPT verdict should remain unchanged.","tokens_in":11680,"tokens_out":19641,"duration_ms":158289,"concrete_test":"Verify the difference-set identity underlying Lemma 3.1: for N=4t+1, S={±1}, T={4q+2,4q+3} (q=0..t−1), check that T−S = {1,...,N−1} with no repetitions. Equivalently, compute A(P_N)A(Q_N) for a small case such as N=9 and confirm it equals A(K_N). This settles the base case on which the complete-graph theorem depends.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I traced the main constructions and found no load-bearing flaw. The complete-graph theorem (Theorem 3.3) rests on the exact cyclic factorization of K_{4t+1} (Lemma 3.1, cited from [10]). This is the only external dependency, and it is valid: for S={±1}, T={4q+2,4q+3}, the difference set T−S equals Z_N\\{0} with each residue appearing exactly once, so A(P_N)A(Q_N)=A(K_N). The subsequent deletion argument is sound: after deleting D, errors are exactly the ordered pairs whose unique witness belonged to D, and the casewise counts for n≡0,2,3 (mod 4) are correct (I spot-checked n=4, which gives exactly 4 errors). The blow-up construction (Theorem 4.2) correctly transfers adjacency through the matching inside parts, with errors only in unmatched columns; the r/n bound follows. The bipartite/tree construction (Lemma 5.1, Corollary 5.2, Theorem 5.3) realizes one orientation exactly and pays exactly m or m+d_G(y0) for the reverse orientation; the counts are correct. I found no internal inconsistency, circular reasoning, or missing proof affecting the central claims.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a Hamming-type relaxation of matrix product factorization: a graph G on n vertices admits an ε-factorization if there exist simple graphs H,K on V(G) and a set E of at most εn² ordered pairs outside which A(H)A(K) agrees with A(G). After recording elementary equivalences and quantitative constraints (witness sets S_w=N_H(w)×N_K(w), walk-count and degree inequalities), the authors give three construction families. Theorem 3.3 shows every K_n has an ε-factorization with ε=O(1/n), with case-by-case bounds for n mod 4, using a cited exact cyclic factorization of K_N for N≡1 mod 4 and deleting vertices. Theorem 4.2 gives an ε≤r/n factorization for any blow-up of an r-vertex graph, exact when every non-isolated part has even size. Section 5 gives one-sided factorizations for bipartite graphs, yielding ε≤1/n for every tree. The paper concludes that exact congruence obstructions are unstable under a vanishing proportion of entrywise errors.","tokens_in":11929,"tokens_out":15627,"duration_ms":150243,"significance":"The main contribution is conceptual: it defines a natural stability parameter δ_MPF and demonstrates that the rigid exact theory becomes flexible under a small number of entrywise errors. The constructions are explicit and the error counts are exact and parameter-free. I checked the arithmetic in Theorem 3.3, the unmatched-column count in Theorem 4.2, and the matching-witness construction in Lemma 5.1/Corollary 5.2; all are consistent. The tree bound ε≤1/n improves on the trivial zero-product bound 2(n-1)/n² for n>2. The sole external dependency is the exact cyclic factorization of K_{4t+1} cited from [10]; the underlying difference-set identity is valid, so this dependency is not a correctness risk. If published, this paper should be useful to researchers studying matrix products of graphs and stability of factorization.","major_comments":[],"minor_comments":[{"comment":"The paragraph beginning 'Here we have an example' defines A(H)=J_2⊗R_{2n}, A(K)=R_2⊗J_{2n} and computes A(H)A(K)=J_{4n}. Since J_{4n} is not the adjacency matrix of a simple graph, this is not an example of an exact factorization; as written it is disconnected from the surrounding argument. Please either remove it or state explicitly that it illustrates that products of 0-1 adjacency matrices need not be graphical.","section":"Section 3, after Lemma 3.1"},{"comment":"In Case 2, the text 'Also 1<N_QN(0)' presumably means '1∉N_QN(0)'. The current symbol is wrong and confusing.","section":"Theorem 3.3, proof"},{"comment":"The notation 'A(G)B{(i,j)∈[n]×[n] : ...}' contains a stray 'B' and appears to define the ordered adjacency set of G. It should read 'A(G):={(i,j) ...}' or use a different symbol, to avoid collision with the matrix A(G).","section":"Section 2"},{"comment":"The condition 'x,y, x,y, x^{-1}y∈S' is garbled; it should be simply 'x^{-1}y∈S'.","section":"Definition 3.2"},{"comment":"Typo: 'non-factrizable' should be 'non-factorizable'.","section":"Section 5, closing question"},{"comment":"The same symbol E is used both for a set of ordered pairs and for a 0-1 matrix. This is standard but should be noted explicitly for readability. Also, the phrase 'disagree in at most εn² entries' in the introduction is strictly equivalent only if E is chosen as the exact disagreement set; a one-sentence clarification would help.","section":"Definition 1.1 / Lemma 2.1"},{"comment":"'equivalent matrix, and witness formulations' should be 'equivalent matrix and witness formulations' (stray comma).","section":"Section 1/Abstract"}],"recommendation":"minor_revision","confidential_remarks":"I see no novelty or scope problem. The manuscript fits a combinatorics journal. The external dependence on [10] is legitimate; I verified the difference-set identity behind Lemma 3.1. My recommendation of minor revision is purely for exposition: the misplaced example in Section 3 and a few corrected notation/typos should be addressed before final publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You can read this one. It defines ε-matrix-product factorization for graphs—where A(H)A(K) and A(G) may differ on at most εn^2 ordered entries—and proves that the rigid exact obstructions are unstable under vanishing entrywise error. The main results: every K_n admits an ε-factorization with ε = O(1/n) even when n ≠ 1 mod 4; every blow-up of an r-vertex graph admits ε ≤ r/n; every tree admits ε ≤ 1/n.\n\nWhat is genuinely new is the stability parameter itself, which no one in the exact theory seems to have written down, and the constructions are not just routine scans of known cases. The papers is strongest where it counts: short elementary proofs, explicit error counts, and no fitted constants. I spot-checked the deletion arguments for K_4, the unmatched-column count in the blow-up, and the leaf case in the tree theorem; all are consistent. The paper also openly says it makes no optimality claim and ends by asking for lower bounds.\n\nSoft spots, in proportion: the complete-graph theorem leans entirely on the cited exact cyclic factorization from [10], so if that reference had a gap, the O(1/n) bounds would fall. But I checked the difference set—for S={±1}, T={4q+2,4q+3}, every nonzero residue appears exactly once as t-s—so the base factorization is solid. The bipartite construction is genuinely one-sided: it realizes only the X→U orientation exactly and pays m or m+d_G(y0) for the reverse orientation. That is a limitation of the technique, not a hidden flaw. Also, because the zero-product bound gives 2m/n^2 automatically, the ε ≤ 1/n bound for trees is not impressive on its own; the authors acknowledge this and keep the unnormalized error count as the real claim. Minor typos and a few malformed displays, including the closing question's 'factrizable', do not affect the math.\n\nThis is a solid contribution to a small, active literature. It deserves a serious referee, and I would send it to peer review. I would also cite it if I were working on algebraic graph factorization.","headline":"A clean, small paper that defines a Hamming relaxation of matrix-product factorization and shows the exact modulo-4 obstruction for K_n collapses under O(1/n) error; the constructions check out.","tokens_in":12406,"tokens_out":1600,"would_cite":true,"duration_ms":18518,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","15A23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Approximate matrix-product factorization lets every complete graph, blow-up, and tree factor with error O(1/n), even where exact factorization is impossible.","keywords":["matrix product factorization","approximate factorization","adjacency matrix product","complete graphs","trees","blow-up graphs","Hamming distance","Cayley graphs"],"falsifier":"Compute the exact minimal error δ_MPF(K_6) by exhaustive enumeration of all simple graphs H,K on six vertices; if the minimal number of error entries is not 6 or fewer, the claimed O(1/n) rate for n ≡ 2 (mod 4) would be falsified.","tokens_in":11567,"feed_emoji":"🧮","tokens_out":5287,"duration_ms":50707,"temperature":0.7,"pith_summary":"This paper introduces an approximate version of matrix-product factorization of graphs, in which two factor graphs on the same vertex set are required to reproduce the target adjacency matrix except on a small, explicitly marked set of ordered entries. The main results are constructions: every complete graph K_n admits such a factorization with error at most O(1/n), every blow-up of an r-vertex graph admits one with error at most r/n, and every tree on n ≥ 2 vertices admits one with error at most 1/n. These statements hold despite exact factorization being possible for K_n only when n ≡ 1 (mod 4), and impossible for any nontrivial tree. The paper argues that rigid congruence obstructions in the exact theory are zero-error phenomena only: they vanish when a vanishing proportion of entrywise errors is allowed.","feed_headline":"Complete graphs factor to within O(1/n)","feed_subtitle":"An approximate construction shows exact congruence obstructions vanish under tiny entrywise error.","key_machinery":"The central identity is the witness decomposition: for each vertex w, the set S_w = N_H(w) × N_K(w) collects all ordered pairs whose two-step walk through w uses an H-edge then a K-edge; outside the error set these sets form an approximate disjoint decomposition of the ordered adjacency relation of the target graph. Two construction tools carry the proofs: deletion of a few vertices from exact cyclic Cayley-graph factorizations of complete graphs, and matchings placed inside the parts of a blow-up so that the K-factor mirrors the structure of the target adjacency matrix.","core_discovery":"The core discovery is a stability principle: the congruence obstructions to exact matrix-product factorization are not stable under small entrywise perturbations. Concretely, the paper constructs, for every complete graph K_n, a pair of simple graphs H,K on the same vertex set such that A(H)A(K) and A(K_n) agree on all but O(n) of the n^2 ordered entries. The construction deletes a small set D of vertices from an exact cyclic factorization of K_N for N ≡ 1 (mod 4) and counts errors as the ordered pairs whose unique two-step witness was deleted; depending on n mod 4, the error count is at most n or n+1. For a blow-up of a fixed r-vertex graph, the construction places a matching inside each pa","pith_inferences":["The deletion idea likely extends to other infinite families with exact factorizations: any exact factorization with a bounded-degree witness structure can be perturbed by deleting vertices, giving O(1/n) error for nearby sizes.","The matching-inside-parts construction suggests a general template principle: graphs that are quotients of a small template by independent set replacement are approximately factorizable; this might extend to graphons or limits of dense graph sequences.","The one-sided bipartite construction is inherently asymmetric; a two-sided analogue, realizing both orientations of almost all edges, would require pairing the two sides symmetrically and may yield improved constants, perhaps optimal O(m/n^2).","The open question about lower bounds on ε for non-factorizable graphs is a natural next step: determining the exact order of δ_MPF(K_n) for n not ≡ 1 mod 4 would confirm the optimality of the O(1/n) rate."],"forward_implications":["If correct, the exact congruence classification of complete graphs becomes an asymptotic irrelevance: K_n is O(1/n) close to a product for every n.","The blow-up bound ε ≤ r/n shows that any graph family that is a blow-up of a fixed r-vertex template is approximately factorizable with error shrinking linearly in the template size; complete multipartite graphs are a direct corollary.","The tree bound ε ≤ 1/n shows that sparse graphs, despite failing exact factorability, admit structured factorizations that preserve almost all two-step witness structure.","The walk-count and row-support inequalities give quantitative lower bounds on error in terms of factor degrees, usable as a diagnostic for candidate factorizations."],"fun_headline_variants":["Tiny errors erase modular obstacles to graph factorization","Exact obstructions vanish under tiny entrywise errors","Approximate graph factorization beats congruence barriers","Every tree factors to within 1/n error"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The O(1/n) bound for complete graphs rests on the cited exact cyclic factorization of K_N when N ≡ 1 (mod 4): if that base factorization were false, the deletion argument would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Tiny errors erase modular obstacles to graph factorization","Exact obstructions vanish under tiny entrywise errors","Approximate graph factorization beats congruence barriers","Every tree factors to within 1/n error"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000658,"raw_usage":{"total_tokens":2897,"prompt_tokens":846,"completion_tokens":2051,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":1993}},"tokens_in":590,"tokens_out":2051,"duration_ms":16728,"temperature":1.0,"reasoning_tokens":1993,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:51:58.253180+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact minimal error δ_MPF(K_6) by exhaustive enumeration of all simple graphs H,K on six vertices; if the minimal number of error entries is not 6 or fewer, the claimed O(1/n) rate for n ≡ 2 (mod 4) would be falsified.","supporting_citations":[],"review_version":1}