{"id":"437ef91d-3e6a-4db4-a42b-4e1cedefeb6c","arxiv_id":"2602.12551","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For nearly regular tournaments, a tournament has the Sidorenko property exactly when it is transitive or a blow-up of the cyclic triangle whose three parts are transitive.","lead":"Tournaments that appear no less often in nearly regular tournaments than in a random one are exactly the transitive tournaments and the blow-ups of a cyclically oriented triangle with transitive parts. The paper also settles two open questions about which tournaments force quasirandomness among nearly regular tournaments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified — the central classification is supported; only minor convention/limiting gaps remain.","rationale":"The reader's CONDITIONAL verdict is reasonable, but the specific concern about Lemma 3 does not land after scrutiny. Lemma 3 is indeed the structural crux of the 'only if' direction, and I examined its case analysis in detail. The partition assigns vertices consistently, the cross-edge orientation between parts is forced by forbidding W4/L4/C5, and each part is transitive. The four-case figure description is accurate (one combination yields C5, the other three yield W4 or L4). The minor technical gaps flagged by the reader (undefined D_k, infinite Hölder exponents) are easily patched and do not affect the main theorem. The paper's main claims—Corollaries 10 and 11—are supported by a detailed and apparently correct argument. Therefore no change to the reader's verdict is needed; the paper remains acceptable conditioned on minor exposition fixes.","tokens_in":18006,"tokens_out":54990,"duration_ms":410882,"concrete_test":"Brute-force verify Lemma 3 for all non-transitive tournaments on at most 7 vertices: enumerate all tournaments, check which avoid W4, L4, and C5 as induced subtournaments, and test whether each is isomorphic to T[a,b,c] for some a,b,c in N. A counterexample would falsify the classification; absence of a counterexample would further confirm the proof's completeness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After close review, the central claim of the paper is sound. The most load-bearing step is Lemma 3's structural dichotomy: every non-transitive tournament avoiding W4, L4, and C5 is T[a,b,c]. The proof's partition of vertices relative to a chosen cyclic triangle is well-defined: vertices with exactly one in-neighbor or exactly two in-neighbors are placed in the appropriate part, and each part has the required in/out-neighbor relations to the triangle vertices. The case analysis for a misoriented edge between V_i and V_{i+1} is valid: checking the four combinations of the directions u_i v and u_{i+1} w shows that three yield W4 or L4, and the remaining one yields a regular 5-vertex tournament, i.e., C5. The classification thus holds. The remaining issues are technical rather than load-bearing: D_k is undefined when N+ or N- vanish, but the product N+^a D^{ab} N-^b can consistently be set to zero outside the support; the generalized Hölder step in Theorem 7 uses infinite exponents when a=1 or a=b, but those factors have exponent zero and can be handled by a standard limiting argument; and the computational identification of three 6-vertex anti-Sidorenko tournaments is reported without code, but this is an aside. None of these affects Corollary 10 or Corollary 11.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper characterizes, in the language of tournamentons, the tournaments H whose homomorphism density among regular (equivalently, nearly regular) tournaments is minimized by the constant tournamenton. The main result, Corollary 10, states that t(H,W) ≥ 2^{-binom(|V(H)|,2)} for every regular tournamenton W if and only if H is a transitive tournament or one of the tournaments T[a,b,c], i.e., blow-ups of the cyclic triangle whose parts induce transitive tournaments. The proof is built from a structural dichotomy (Lemma 3) saying that the only non-transitive tournaments avoiding W4, L4 and C5 are the T[a,b,c]; a construction of H-free regular tournamentons for all other tournaments (Theorem 4); analytic inequalities proving the Sidorenko-type lower bound and equality case for T[a,b,c] (Lemmas 5–7, Theorems 7 and 9); and a twin-blowing inequality (Lemma 8). Corollary 11 identifies the quasirandom-forcing tournaments among these, answering two questions of Noel, Ranganathan and Simbaqueba and reducing their Problem 6.1 to a finite check via Proposition 12.","tokens_in":18250,"tokens_out":20445,"duration_ms":169297,"significance":"The result, if correct, is a complete and striking classification: unlike the graph setting, the Sidorenko property over regular tournamentons has a very small family of extremal tournaments. It resolves two open problems and provides a clean equality case. The proof is unusually transparent: the key inequalities (10), (15)–(17) are explicit and checkable, and the argument relies only on standard external facts (transitive tournament densities, the P4 quasirandomness criterion, compactness of tournamentons). The twin-blowing lemma is a nice tool. The paper is honest about where its method does not extend (the entropy-method sketch in Section 5) and labels the computational finding as an aside. I see no circularity and no hidden free parameters.","major_comments":[{"comment":"D_{W,k} is defined as a quotient with denominator N^+_{W,k} N^-_{W,k}, but this denominator vanishes on a non-trivial subset of [0,1]^k for many tournamentons. Later identities and the proof of Lemma 6 treat the product N^+ D N^- as 0 on that set, but the convention is never stated. Please define D_{W,k} (or the product) explicitly on the null set so that (4), (12), and (15) are pointwise meaningful.","section":"Section 4, definition before Lemma 6; Eqs. (4), (12), (15)"},{"comment":"The generalized Hölder step sets p_2=ab/(ab-b) and p_3=ab/(b-a), which are undefined or infinite when a=1 or a=b. These cases are included in the theorem (e.g., T[1,2,c] and T[a,a,c]), so the proof as written does not cover the full statement. A short limiting argument for the infinite-exponent cases, or a separate treatment, is needed. The statement of generalized Hölder in Section 2 should also be adjusted to allow this convention.","section":"Section 4, Theorem 7 proof, inequality (15)"}],"minor_comments":[{"comment":"The sentence \"every triple of vertices of V_i induces a transitive tournament (otherwise, the triple and the vertex u_{i+1} would form W4)\" appears to name the wrong tournament: with the established orientation of edges from V_i to V_{i+1}, u_{i+1} would be a sink, hence L4, not W4. Since both W4 and L4 are forbidden, the conclusion is unaffected. Also, \"for every i∈V_i\" should read \"for every i∈[3].\"","section":"Section 3, Lemma 3, last paragraph"},{"comment":"The statement says p_1,...,p_k are non-negative reals, but reciprocals 1/p_i appear. Please state p_i>0 and allow p_i=∞ with the usual convention, or restrict to the finite-exponent cases actually needed after the limiting argument in Theorem 7.","section":"Section 2, generalized Hölder inequality"},{"comment":"Regularity of the constructed tournamenton is asserted implicitly by the displayed definition but not verified. It is a short integral check and should be included. The inequality 2(2n)^{-n} > 2^{-binom(n,2)} for n≥10 should also be justified in one line.","section":"Section 5, Proposition 12"},{"comment":"The sentence \"every quasirandom forcing tournament has either the Sidorenko property or the anti-Sidorenko property\" is used to reduce Problem 1 to finitely many cases, but no proof or reference is given. Please add a reference or a short argument; as written, this is an unsupported step in the secondary claim about Problem 6.1.","section":"Section 5, paragraph before Proposition 12"},{"comment":"The claim that three 6-vertex tournaments were \"computationally verified\" to have the constant tournamenton as the unique maximizer among regular tournamentons is not reproducible without code or a description of the computation. Since this is an aside, it could be documented or moved to an appendix; otherwise it should be labelled as an unverified computational remark.","section":"Section 5, computational aside"}],"recommendation":"minor_revision","confidential_remarks":"I agree with the conditional assessment: the core chain is sound and the classification is almost certainly correct. The undefined quotient in Section 4 and the finite-exponent gap in Theorem 7 must be fixed before publication, but both are routine. The W4/L4 typo in Lemma 3 and the unsupported Sidorenko/anti-Sidorenko assertion in Section 5 should also be corrected. Once these are addressed, the paper is suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers what the abstract promises: a full characterization of tournaments whose homomorphism density is minimized by the random tournament among all regular tournamentons. The two families are transitive tournaments and T[a,b,c] blow-ups of the cyclic triangle with transitive parts. That closes two Noel–Ranganathan–Simbaqueba problems and settles Problem 1 up to finitely many cases. I think the result is new, and the proof is in good shape.\n\nThe main line is Theorem 4 -> Lemma 5 -> Lemma 6 -> Theorem 7 -> Lemma 8 -> Theorem 9 -> Corollaries 10/11. I checked the displayed inequalities (10), (15), (16), (17) and they are right. The structural dichotomy in Lemma 3 is the load-bearing step; the partition by in-neighbor count relative to a cyclic triangle and the four-case edge check genuinely force the T[a,b,c] form. I do not see a missing case there. The use of subgraph monotonicity in Theorem 4 (zero density for W4/L4/C5 kills any supergraph) is correct, though the prose could say it explicitly. The self-citations are background; no circularity.\n\nSoft spots are minor and mostly cosmetic. D_k is undefined when N+ or N- vanish; the fix is to declare the product zero off the support, which the proof effectively does. In Theorem 7, the generalized Hölder step uses p3=∞ when a=b (and p2=p3=∞ for a=b=1), but those factors have exponent zero, so a standard limiting argument covers it. The three 6-vertex anti-Sidorenko tournaments are reported without code or data; this is an aside and does not affect any theorem. Section 5 honestly flags that the entropy method only gives c=1, which is a limitation of an alternative proof, not of the main argument. I would not hold these against acceptance beyond asking for a few sentences of cleanup.\n\nThe paper is for people working in quasirandomness, Sidorenko-type inequalities, and tournament limits. It deserves a serious referee. I would send it out; with the small technical fixes it should be accepted.","headline":"A complete and essentially correct Sidorenko classification for regular tournamentons; the proof is sound and only needs small technical cleanups.","tokens_in":18799,"tokens_out":7367,"would_cite":true,"duration_ms":66711,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C20","05C35","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, among regular tournaments, the random tournament minimizes the density of a tournament H exactly when H is a transitive tournament or a blow-up of a cyclically oriented triangle with transitive parts.","keywords":["tournaments","Sidorenko property","quasirandom forcing","tournament limits","tournamentons","regular tournaments","homomorphism density","blow-ups of cyclic triangle"],"falsifier":"Run an exhaustive search over small tournaments (up to 8 or 9 vertices) that are non-transitive but contain neither W4, L4 nor C5; if any of them is not isomorphic to some T[a,b,c], Lemma 3 (and the 'only if' direction of Corollary 10) fails. Alternatively, construct a regular tournamenton W and a tournament T outside the listed families with t(T,W) < 2^{-binom(|V(T)|,2)}.","tokens_in":17851,"feed_emoji":"🎯","tokens_out":10882,"duration_ms":108761,"temperature":0.7,"pith_summary":"Tournaments are complete directed graphs. A tournament has the Sidorenko property if, among all large tournaments whose vertices are all balanced (out-degree about half the number of vertices), the random tournament minimizes the number of copies of it. The paper characterizes this property completely: it holds exactly for transitive tournaments and for blow-ups of a cyclic triangle in which each part is transitive. As a consequence, it also characterizes which tournaments force quasirandomness in this regular setting, resolving two open problems about quasirandom forcing in nearly regular tournaments.","feed_headline":"Transitive and cyclic-triangle blow-ups are the only minimizers","feed_subtitle":"These two families, and only these, minimize tournament copy density, answering two open problems.","key_machinery":"Tournament limits are represented by measurable functions W:[0,1]^2→[0,1] with W(x,y)+W(y,x)=1, called tournamentons; regular tournamentons model nearly regular sequences. The paper's positive direction uses three specially designed digraphs: the blow-up C[a,b,c] of a cyclic triangle with independent twin parts, its sibling T[a,b,c] with transitive parts, and the auxiliary digraph B[c] (a single source directing c paths to a single sink). For regular W, the identity t(C[1,1,c],W)=t(B[c],W)/2 reduces the problem to a Jensen/Hölder inequality for B[c], which is then lifted to all C[a,b,c] via Lemma 6 and Hölder's inequality, and finally to T[a,b,c] via Lemma 8 (adding a transitive tournament t","core_discovery":"The paper's central result, Corollary 10, is: for a tournament T, the inequality t(T,W) ≥ 2^{-binom(|V(T)|,2)} holds for every regular tournamenton W if and only if T is transitive or T is isomorphic to T[a,b,c] for natural numbers a,b,c — the tournament obtained from the cyclically oriented triangle by replacing each vertex with a transitive tournament. Corollary 11 sharpens this: the constant tournamenton W≡1/2 is the unique minimizer of t(T,W) among regular tournamentons precisely when T is transitive with at least four vertices or T[a,b,c] with a+b+c≥4. Together these give a complete description of the Sidorenko property and of quasirandom-forcing tournaments in the class of nearly regul","pith_inferences":["The structural dichotomy in Lemma 3, if correct, suggests a general phenomenon: requiring a Sidorenko-type inequality in a degree-regular setting forces the extremal objects into iterated blow-up forms, which may guide conjectures for other regular limits such as regular permutation or hypergraph limits.","The entropy-method proof sketched for the case c=1 could, if completed for all parameters, provide a conceptual information-theoretic explanation of Theorem 7 and possibly extend the inequality to other tripartite blow-ups beyond the cyclic triangle.","A finite exhaustive search over tournaments with up to nine vertices could settle the remaining open case: which of them are maximizers among regular tournamentons, thereby completing the full classification of quasirandom forcing tournaments in the regular setting.","The strong difference between the global forcing list (only two tournaments up to isomorphism) and the regular forcing list (infinitely many) suggests that imposing degree regularity dramatically enriches the forcing phenomenon; quantifying this gap for other structures could be a fruitful direction."],"forward_implications":["There are infinitely many non-transitive tournaments that are quasirandom forcing for nearly regular tournaments: all T[a,b,c] with a+b+c≥4.","Almost every tournament is not quasirandom forcing for nearly regular tournaments, since forcing tournaments must lie in a sparse list and any anti-Sidorenko candidate has at most nine vertices.","The classification of the Sidorenko property among regular tournaments is exact; no other tournament has the random tournament as a density minimizer over the regular class.","The random tournament is the unique minimizer among regular tournamentons precisely for transitive tournaments with at least four vertices and for T[a,b,c] with a+b+c≥4, completing the forcing list on the minimizer side."],"fun_headline_variants":["Only two families minimize hom densities in regular tournaments","Transitive and cyclic triangle blow-ups: complete characterization","Answering two open problems on tournament quasirandomness","Which tournaments are Sidorenko in regular setting? Solved","Full description of Sidorenko tournaments for nearly regular ones"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire negative direction of the characterization rests on Lemma 3, which asserts that every non-transitive tournament that contains none of the three small tournaments W4, L4 and C5 is necessarily a blow-up of the cyclic triangle; if that dichotomy is false, the list in the main theorem is incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Only two families minimize hom densities in regular tournaments","Transitive and cyclic triangle blow-ups: complete characterization","Answering two open problems on tournament quasirandomness","Which tournaments are Sidorenko in regular setting? Solved","Full description of Sidorenko tournaments for nearly regular ones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000962,"raw_usage":{"total_tokens":3873,"prompt_tokens":625,"completion_tokens":3248,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":369,"completion_tokens_details":{"reasoning_tokens":3184}},"tokens_in":369,"tokens_out":3248,"duration_ms":23546,"temperature":1.0,"reasoning_tokens":3184,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T23:47:16.155121+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an exhaustive search over small tournaments (up to 8 or 9 vertices) that are non-transitive but contain neither W4, L4 nor C5; if any of them is not isomorphic to some T[a,b,c], Lemma 3 (and the 'only if' direction of Corollary 10) fails. Alternatively, construct a regular tournamenton W and a tournament T outside the listed families with t(T,W) < 2^{-binom(|V(T)|,2)}.","supporting_citations":[],"review_version":1}