{"id":"955de3c1-ce4a-4ef1-8f73-ed5711030a89","arxiv_id":"2507.15723","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"All even subdivisions of arbitrary graphs satisfy Sidorenko's inequality in Cayley graphs over finite abelian groups.","lead":"This paper proves that every graph obtained by replacing each edge of any starting graph with a path of even length satisfies Sidorenko's inequality when the host graph is a Cayley graph over a finite abelian group. The proof uses Fourier analysis to write the homomorphism count as a sum of nonnegative terms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.1's construction of L from L1 is invalid even for bipartite H1; the proof can be repaired by defining L directly from H, so the theorem likely stands but needs rewriting.","rationale":"The reader's weakest-assumption flags the non-bipartite definition issue. Our check shows the column-replacement step fails even in the bipartite case, so the gap is more severe than stated. However, the theorem itself is not threatened: a direct circuit matrix of the standard subdivision H has the required paired-negative-column structure, so the Fourier argument can be rewritten without mentioning L1 at all. Thus the conditional verdict stands: the main claim is very likely correct, but the proof as written needs a concrete revision before it is fully rigorous.","tokens_in":5960,"tokens_out":30487,"duration_ms":344753,"concrete_test":"Take H1=C4 with the circuit-matrix row L1=(1,-1,1,-1) from Definition 2.2, form the 1x8 matrix L=(1,-1,-1,1,1,-1,-1,1) as in the proof of Proposition 3.1, and check it against Definition 2.2 for H=C8. The row has two consecutive equal entries at the subdivision vertex common to the paths for e1 and e2, so it is not a proper 2-edge-coloring of the fundamental cycle; equivalently, L times the signed incidence matrix of H is not zero. If this check succeeds, the paper must either supply a different valid L or justify the Fourier step without Proposition 2.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.1's construction of L is not just under-specified for non-bipartite H1; it is false for bipartite H1. With H1=C4 and L1=(1,-1,1,-1) as in Definition 2.2, replacing each column v by (v,-v) gives the C8 row (1,-1,-1,1,1,-1,-1,1). A valid circuit-matrix row for C8 must alternate signs around the cycle; this row has equal signs on the adjacent edges a-2 and 2-b, so it is not a proper edge coloring and does not satisfy Definition 2.2. For this L, the kernel is not the image of the signed incidence map of H, so Proposition 2.3 cannot be invoked. The proof can be repaired by taking L directly as a circuit matrix of the always-bipartite standard subdivision H; each fundamental cycle contains both edges of any subdivided original edge, and because those two edges are adjacent in the cycle they receive opposite signs, so the paired-column property needed for the Fourier argument still holds. As written, however, the central step of the proof of Proposition 3.1 is invalid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Sidorenko-type inequality for even subdivisions of arbitrary graphs when the host graph is a Cayley graph over a finite abelian group. The main theorem (Theorem 1.4) states that if H is an even subdivision of an arbitrary graph H0, then t(H, Cay(G,S)) >= t(K2, Cay(G,S))^{e(H)} for every finite abelian group G and symmetric subset S. The proof reduces arbitrary even subdivisions to standard subdivisions, represents the homomorphism count as an average over the kernel of a circuit matrix, and then applies a Fourier expansion over the dual group. The Fourier step pairs the two columns of the circuit matrix that correspond to the two edges of each subdivided original edge, obtaining a sum of nonnegative terms whose zero-character term gives the desired lower bound.","tokens_in":6116,"tokens_out":5899,"duration_ms":72544,"significance":"If the technical gap in the proof is repaired, this is a substantial contribution: it gives a broad new class of graphs satisfying Sidorenko's inequality in all abelian Cayley host graphs, extending known results for standard subdivisions and theta substitutions. The Fourier-analytic argument is clean and structural: it uses no fitted parameters, and the positivity arises directly from the pairing of conjugate Fourier coefficients. The paper also draws an explicit spectral-quasirandomness consequence for near-extremal abelian Cayley graphs, which is a useful and falsifiable byproduct. However, the current manuscript has a load-bearing gap in the construction of the circuit matrix for the subdivision, so the main theorem is not fully established as written.","major_comments":[{"comment":"The construction of L from L1 is not valid as written. Definition 2.2 defines the circuit matrix only for a bipartite graph, but H1 is arbitrary and may have odd cycles, so L1 need not exist. Even when H1 is bipartite, replacing each column v of L1 by (v,-v) does not produce a circuit matrix of the standard subdivision H. For H1=C4 with L1 row (1,-1,1,-1), the construction gives the C8 row (1,-1,-1,1,1,-1,-1,1), which has equal signs on adjacent edges and is therefore not a proper alternating edge coloring of the 8-cycle; it does not satisfy Definition 2.2, and Proposition 2.3 cannot be invoked. The kernel of this L need not equal the image of the signed incidence matrix of H, which is exactly what Proposition 2.3 requires. Since the Fourier argument depends on this equality, this is a load-bearing gap. The proof can likely be repaired by constructing a circuit matrix of H directly, since H is always bipartite and each fundamental cycle contains both edges of any subdivided original edge with opposite signs, but this construction is absent from the manuscript.","section":"Section 3, Proposition 3.1"},{"comment":"The proof assumes connectedness without stating or proving the reduction for disconnected graphs. Definition 2.2 and Proposition 2.3 are stated for a connected graph, and the dimension argument in Proposition 2.3 uses connectedness through rank(L) = k - (n-1) and dim Im(M) = n-1. However, Theorem 1.4 and Proposition 3.1 are stated for arbitrary graphs H0 and H1. If these graphs are disconnected, the homomorphism density tensorizes and the inequality for each connected component implies the full inequality, but this reduction is not written. Please add an explicit statement that the argument may be applied componentwise, or extend the definitions to disconnected graphs.","section":"Section 2.2 and Section 3"}],"minor_comments":[{"comment":"The phrase 'using 2 colors i1 and i−1' should read 'using the two colors +1 and −1'; as written it appears to introduce symbols i1 and i−1 that are not defined.","section":"Section 2.2, Definition 2.2"},{"comment":"There are several typographical errors: 'C' should be '\\mathbb{C}' in the definition of characters, 'Erd˝ os' contains a broken accent, and 'the conjecture remains still very open' is awkward. Please proofread for such issues.","section":"Abstract and Introduction"},{"comment":"The final paragraph on spectral quasirandomness is informal: inequality (3) gives a bound on |\\hat f(χ)|, but the subsequent statement about eigenvalues being o(|S|/|G|) lacks explicit quantifiers and a proof of uniformity. If this is intended as a theorem, it should be stated with precise hypotheses; otherwise it could be moved to a discussion remark.","section":"End of Section 3"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is real: every even subdivision of an arbitrary graph satisfies Sidorenko's inequality in every finite abelian Cayley host. This extends the F_q^n work of [1] to all finite abelian groups, and the Fourier core is sound. The key move—pairing the two subdivision edges of each original edge as negatives in every cycle—does make the Fourier terms nonnegative, and the zero character gives the bound. The soft spot is in Proposition 3.1 as written, and it is more than a presentation issue. Definition 2.2 defines the circuit matrix only for bipartite graphs, yet Proposition 3.1 starts with an arbitrary H1. Worse, the explicit construction “replace each column v by (v,-v)” does not yield a valid circuit matrix of the standard subdivision even when H1 is bipartite. Take H1=C4 and L1=(1,-1,1,-1). The constructed row for C8 is (1,-1,-1,1,1,-1,-1,1), which is not a proper alternating coloring of the C8 cycle. Its kernel is not the image of the signed incidence map, so Proposition 2.3 cannot be applied. This is a load-bearing flaw in the proof. That said, the flaw is repairable. Define L directly as a circuit matrix of the always-bipartite H, using a spanning tree that contains exactly one of the two subdivision edges for each original edge. Then every fundamental cycle contains both edges of any subdivided original edge, consecutively, so they receive opposite signs, and the paired-column negativity still holds. The theorem likely stands, but the proof of Proposition 3.1 needs to be rewritten. There is also a minor issue in Proposition 2.1: the identity needs |ker L| = |G|^{k-m}, i.e., the integer matrix must be surjective over the finite abelian group. This holds for totally unimodular circuit matrices, but not for every rank-m matrix with ±1 entries. And the comparison with [6] is ambiguous—if their “generalized theta” allows a single even path, then part of the novelty disappears. The author should clarify. Who is this for? People working on Sidorenko's conjecture and Fourier analytic graph inequalities. With the repair, it deserves a serious referee. I would send it out, with instructions to fix Proposition 3.1 and address the [6] overlap explicitly.","headline":"A genuine extension of Sidorenko-type inequalities to abelian Cayley hosts for all even subdivisions, with a repairable but real gap in the circuit-matrix construction.","tokens_in":676,"tokens_out":716,"would_cite":true,"duration_ms":93495,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every even subdivision of any graph satisfies Sidorenko's inequality in every Cayley host graph over a finite abelian group.","keywords":["Sidorenko's conjecture","even subdivisions","homomorphism density","Cayley graphs","finite abelian groups","Fourier analysis","circuit matrix","spectral quasirandomness"],"falsifier":"To check the proof's key step, take $H_0 = K_3$ so that the standard subdivision is $C_6$, build the $1 \\times 6$ matrix by replacing each column of an oriented cycle vector of $K_3$ with $(v, -v)$, and compare the resulting Fourier sum with the true density $t(C_6, \\mathrm{Cay}(\\mathbb{Z}_5, \\{\\pm 1\\}))$; the constructed row is $(1,-1,-1,1,1,-1)$, not an alternating circuit matrix of $C_6$, so this calculation would reveal whether Proposition 2.3 can be invoked there or whether the proof needs a different definition.","tokens_in":5669,"feed_emoji":"📐","tokens_out":12882,"duration_ms":129186,"temperature":0.7,"pith_summary":"The paper proves a restricted form of Sidorenko's conjecture: take any finite graph $H_0$, replace each edge by a path of even length (the lengths may differ per edge), and the resulting bipartite graph $H$ satisfies $t(H, \\mathrm{Cay}(G,S)) \\ge t(K_2, \\mathrm{Cay}(G,S))^{e(H)}$ for every finite abelian group $G$ and symmetric subset $S$ of $G$. Since $t(K_2, \\mathrm{Cay}(G,S)) = |S|/|G|$, this says the density of every even subdivision is at least the density one expects from a quasirandom graph with the same edge density. The proof converts the homomorphism count into an average over the kernel of a circuit matrix and then into a Fourier sum over characters of $G$, where each term is a product of Fourier coefficients of the edge indicator. The column structure of the circuit matrix for a subdivision forces each pair of factors to be complex conjugates, so every term is a nonnegative squared modulus; the trivial-character term alone yields the lower bound. A sympathetic reader should see this as a substantial widening of the class of graphs known to satisfy Sidorenko's inequality, and as evidence that the algebraic reduction of the full conjecture leaves only the non-abelian Cayley case.","feed_headline":"Even subdivisions obey Sidorenko's inequality on abelian Cayley graphs","feed_subtitle":"Even subdivisions of arbitrary graphs never dip below random-graph density in abelian Cayley hosts.","key_machinery":"The circuit matrix: a $\\{-1,0,1\\}$-matrix whose rows encode the fundamental cycles of a graph, with each cycle entry $+1$ or $-1$ according to a proper two-coloring of that cycle. For a bipartite graph, the density of homomorphisms into an abelian Cayley graph equals the uniform average, over the kernel of this matrix, of the edge-indicator product (Proposition 2.3); expanding that average by characters turns it into a sum of products of Fourier coefficients of the indicator of $S$. The crucial mechanism is column replacement: for the standard subdivision of $H_1$, each column $v$ of the circuit matrix of $H_1$ is replaced by the pair $(v, -v)$, making the two Fourier coefficients attached to a subdivided edge complex conjugates. Their product is a squared modulus, so every term in the Fourier sum is nonnegative, and the trivial-character term alone gives the Sidorenko lower bound.","core_discovery":"The central claim is Theorem 1.4: for a finite abelian group $G$, a symmetric subset $S$, an arbitrary graph $H_0$, and any even subdivision $H$ of $H_0$ (each edge replaced by a path of even length, possibly different lengths per edge), the homomorphism density obeys $t(H, \\mathrm{Cay}(G,S)) \\ge t(K_2, \\mathrm{Cay}(G,S))^{e(H)}$. Since $\\mathrm{Cay}(G,S)$ has edge density $|S|/|G|$, the right-hand side is $(|S|/|G|)^{e(H)}$, the value a quasirandom graph of the same edge density would give. The proof views $H$ as the standard subdivision of an intermediate graph $H_1$ whose edges get length-two paths, so each original edge of $H_1$ contributes a pair of columns to a circuit matrix; the Fourier expansion of the kernel average then has all terms nonnegative, with the trivial character contributing exactly the baseline.","pith_inferences":["The column-pairing positivity might extend to non-abelian Cayley hosts if conjugate characters are replaced by the appropriate representation-theoretic pairing; if so, the same mechanism could prove the inequality for even subdivisions in all Cayley graphs.","The near-equality spectral statement suggests an untested converse: abelian Cayley graphs that are not spectrally quasirandom should contain noticeably more than the random number of copies of every even subdivision.","The circuit-matrix gap for non-bipartite $H_1$ may be repairable by defining the matrix through oriented cycle vectors instead of proper two-colorings; testing this repair on $H_1 = K_3$ would determine whether the proof covers all cases exactly as stated.","One could probe robustness by replacing the abelian group with a finite group that is only locally abelian (for example, a dihedral group with a symmetric generating set) and checking whether the nonnegativity of the Fourier terms survives."],"forward_implications":["Every even subdivision of any finite graph, with independent even path lengths, satisfies the Sidorenko inequality in every abelian Cayley host graph.","Because the subdivided graph is bipartite regardless of the base graph, the theorem adds infinitely many new Sidorenko graphs, including subdivisions of dense graphs and of graphs with odd cycles.","The quantitative refinement implies that if any nontrivial character of $G$ has Fourier coefficient at least $\\varepsilon \\hat{f}(0)$, then the density exceeds the random baseline by a factor at least $1 + \\varepsilon^{e(H)}$.","Near-equality in the inequality forces the host Cayley graph to be spectrally quasirandom: every non-principal eigenvalue is $o(|S|/|G|)$, so the host behaves roughly like a random regular graph at density $|S|/|G|$.","Combined with the known reduction of Sidorenko's conjecture to Cayley graphs over finite groups, the remaining open case inside that reduction is the non-abelian one."],"supporting_citations":[{"why":"Establishes the reduction of Sidorenko's conjecture to verifying the inequality on Cayley graphs over finite groups.","marker":"[10]"},{"why":"Supplies the construction that converts vertex-transitive host graphs into Cayley graphs, completing the reduction used in Proposition 1.2.","marker":"[7]"},{"why":"Provides the Fourier-analytic identity for averages over kernels of $\\{0,1,-1\\}$-matrices that Proposition 2.1 adapts.","marker":"[1]"},{"why":"Proves earlier Sidorenko results for standard subdivisions, the baseline class this paper extends.","marker":"[3]"},{"why":"Covers the nearest prior family, graphs obtained by replacing each edge with a generalized theta graph of even paths.","marker":"[6]"},{"why":"Originates the correlation inequality and establishes the basic Sidorenko cases of trees and even cycles.","marker":"[9]"}],"fun_headline_variants":["Even subdivisions obey Sidorenko on abelian Cayley graphs","Sidorenko holds for even subdivisions on abelian Cayley hosts","Even subdivisions always meet Sidorenko on abelian Cayley","Even subdivisions keep Sidorenko on abelian Cayley","On abelian Cayley graphs, even subdivisions satisfy Sidorenko"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on applying a matrix built from a graph's cycles, a construction defined only for graphs whose cycles can be properly two-colored, to an intermediate graph that may contain odd cycles; if that application is invalid, the argument as written collapses.","fun_headline_variants_meta":{"raw":{"variants":["Even subdivisions obey Sidorenko on abelian Cayley graphs","Sidorenko holds for even subdivisions on abelian Cayley hosts","Even subdivisions always meet Sidorenko on abelian Cayley","Even subdivisions keep Sidorenko on abelian Cayley","On abelian Cayley graphs, even subdivisions satisfy Sidorenko"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000873,"raw_usage":{"total_tokens":3781,"prompt_tokens":948,"completion_tokens":2833,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":2745}},"tokens_in":564,"tokens_out":2833,"duration_ms":23169,"temperature":1.0,"reasoning_tokens":2745,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:26:42.114792+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To check the proof's key step, take $H_0 = K_3$ so that the standard subdivision is $C_6$, build the $1 \\times 6$ matrix by replacing each column of an oriented cycle vector of $K_3$ with $(v, -v)$, and compare the resulting Fourier sum with the true density $t(C_6, \\mathrm{Cay}(\\mathbb{Z}_5, \\{\\pm 1\\}))$; the constructed row is $(1,-1,-1,1,1,-1)$, not an alternating circuit matrix of $C_6$, so this calculation would reveal whether Proposition 2.3 can be invoked there or whether the proof needs a different definition.","supporting_citations":[{"cited_title":"Lov´ asz and B","cited_arxiv_id":null,"evidence_quote":"Supplies the construction that converts vertex-transitive host graphs into Cayley graphs, completing the reduction used in Proposition 1.2."},{"cited_title":"Conlon, J","cited_arxiv_id":null,"evidence_quote":"Proves earlier Sidorenko results for standard subdivisions, the baseline class this paper extends."},{"cited_title":"Sidorenko","cited_arxiv_id":null,"evidence_quote":"Originates the correlation inequality and establishes the basic Sidorenko cases of trees and even cycles."}],"review_version":1}