{"id":"b692cea1-b095-4449-bb78-0620e086af9a","arxiv_id":"2501.11675","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"All tournaments on at most five vertices that force quasirandomness in nearly regular tournaments are classified: eleven force it and nine do not.","lead":"The paper studies which small tournaments force a large tournament to look random when the large tournament is assumed to be nearly regular. It fully classifies all tournaments on at most five vertices under this stronger assumption, finding eleven forcing and nine non-forcing examples.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The positive half of Theorem 1.4 depends on unverified computer-generated flag-algebra certificates; a single error in the PSD matrices or Appendix A coefficients would invalidate the forcing results for H10, H11, H13, H14, H15.","rationale":"The reader's weakest_assumption already identifies the flag-algebra certificates, and I agree that this is the correct focus. The central claim is a full classification; the negative half is supported by explicit constructions (with the minor caveat that H18 relies on an approximate computer value, which should be made exact but is unlikely to fail given the large margin), while the positive half for the five genuinely new cases H10, H11, H13, H14, H15 rests entirely on Lemma 5.18 with the printed matrices and coefficient tables. The paper gives sample arithmetic for two tournaments per theorem, which builds confidence but does not verify the remaining cases or the PSD/kernel assertions. Because flag algebra is a standard and sound method, and because the displayed computations are consistent, I do not think the paper should be rejected; but the absence of certificates or code justifies a conditional verdict rather than full acceptance. An independent recomputation is a bounded, concrete task and would settle the question.","tokens_in":32567,"tokens_out":12029,"duration_ms":114177,"concrete_test":"Write or request a script that, from the printed tournament list and flag definitions, regenerates the Appendix A coefficient matrices via the handshaking identity (or parses the printed ones) and recomputes exactly, for every J ∈ {H8,...,H19}, the four sums in Theorems 5.4–5.7, checking equality with the stated constants. Separately compute exact eigenvalues (or certified floating-point intervals) of the printed A1, A2, A3 matrices to confirm PSD and the asserted kernels. Also recompute the exact rational value of t(H18, W_T) for the displayed 7-vertex regular tournament and compare it with 1/1024.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing assumption is correctness of the flag-algebra certificates behind Theorems 5.4–5.7. Each theorem applies Lemma 5.18 with explicit 8×8 matrices A2, A3 (and A1 in 5.7) and asserts, without proof or machine-checkable certificate, that these are positive semidefinite and that for every 5-vertex tournament J the printed expression—built from tinj values and the Appendix A coefficient matrices B^q_3(J)—equals the stated constant (−5/4, −5/4, 8/7, 6/5). The text demonstrates only two sample J per theorem (e.g., H8 and H12 for 5.4) and says the remaining cases are 'similar' or in the appendix. The equality-case arguments additionally rely on unproved kernel statements, e.g., 'the kernel of A2 is spanned by (1,1,1,1,1,1,1,1)^T' in 5.4 and 5.5, and the two-vector kernel in 5.6. These kernel claims are needed to derive t(TT4,W)=t(C4,W)=(1/8)t(TT3,W) and then apply Lemma 5.13. Unlike the negative results—where most constructions are explicit rational tournamentons (H9, H12, H19) and only H18 uses an approximate computer value—the positive classification of H10, H11, H13, H14, H15 has no independent verification. Since Theorem 1.4(ii) is the paper's main novelty, an error in any one matrix entry or coefficient table breaks the min-bound identity and with it the forcing conclusion. This is a verification gap, not evidence of a mistake; the flag-algebra framework itself is standard and the displayed sample arithmetic is consistent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces and studies a variant of quasirandom forcing for tournaments in which the host tournaments are assumed to be nearly regular. The main result, Theorem 1.4, classifies all tournaments on at most five vertices that force quasirandomness in regular tournaments: exactly eleven do (H4, H5, H6, H7, H8, H10, H11, H13, H14, H15, H17) and exactly nine do not (H0, H1, H2, H3, H9, H12, H16, H18, H19). The negative side is handled by explicit regular tournamenton constructions, including a neat hand computation for H9 and H16 and a computer-assisted example for H18. The positive side is proved by reducing the problem to inequalities on homomorphism densities of C3 or TT3 combined with H10, H11, H13, or H14, and then proving these inequalities with flag algebras. The paper also gives structural reductions (Propositions 3.1–3.10) that connect near regularity, density of TT3 and C3, and forcing in regular tournaments, and it closes with several natural open problems, including whether almost every tournament forces quasirandomness in regular tournaments.","tokens_in":33150,"tokens_out":5023,"duration_ms":50491,"significance":"If the central results are correct, this paper significantly enlarges the known family of quasirandom-forcing tournaments by adding a regularity assumption on the host sequence. This is a meaningful conceptual contrast to Theorem 1.2, where only one non-transitive tournament forces quasirandomness. The structural reductions are clean and appear sound, and the negative constructions are explicit and mostly hand-checkable. The positive half, however, relies on flag-algebra certificates that are only partially verified in the text: large matrices are asserted to be positive semidefinite, kernel claims are asserted without proof, and the coefficient tables in Appendix A are stated as computer-generated with only sample demonstrations. These are load-bearing for the main classification, but the issue is a verification gap rather than a detected mathematical error. The paper does not ship machine-checkable code or certificates, which would substantially strengthen confidence in the positive results.","major_comments":[{"comment":"","section":"Theorems 5.4–5.7 and Lemma 5.18"},{"comment":"","section":"Proofs of Theorems 5.4–5.7, equality cases"},{"comment":"","section":"Lemma 4.4, H18 construction"}],"minor_comments":[{"comment":"","section":"Proof of Theorem 5.5"},{"comment":"","section":"Appendix A"},{"comment":"","section":"Theorem 5.7"},{"comment":"","section":"Proposition 4.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the conceptual framework is appealing, but the main positive classification rests on flag-algebra certificates that are not fully verified in the manuscript. I would recommend major revision with a request for complete machine-checkable verification of the PSD matrices, the Appendix A coefficient tables, and the kernel claims, as well as an exact computation for the H18 example. The negative constructions and structural reductions are strong and should be highlighted in the final version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper does something genuinely useful. It weakens quasirandomness forcing by assuming the host tournaments are nearly regular, and then fully resolves the case of at most five vertices. The winning half—eleven forcing tournaments—contrasts sharply with the unrestricted case, where only transitive tournaments of size at least 4 and H17 force. The definition is natural, and the main theorem is the obvious next step after the work of Bucić–Long–Shapira–Sudakov and Hancock et al.\n\nWhat is good: the structural reductions in Section 3 are clean. Proposition 3.8, which equates forcing in regular tournaments with forcing in the presence of TT3 or C3, is a neat observation that makes the whole enterprise tractable. The negative constructions are mostly explicit rational tournamentons; the H9/H16 example is a straightforward hand computation, and the H6 proof via [4, Corollary 6] is elegant. The flag algebra part is standard but carefully laid out, with the equality-case arguments driven by kernel conditions rather than by fitting the matrices to the target densities.\n\nThe soft spot is the positive half of Theorem 1.4. The PSD assertions for the 8x8 matrices A2 and A3 (and A1 in Theorem 5.7) are stated without proof or certificate, and the identity for each 5-vertex tournament J is only checked for one or two sample J, with the rest delegated to the appendix. The appendix appears complete for the twelve 5-vertex tournaments, but the coefficient tables are themselves computer-generated and not independently machine-checked. This is a verification gap, not evidence of error—the sample arithmetic is consistent and the method is routine—but it is exactly the thing a referee should ask the authors to back up with an exact certificate, such as rational eigenvalues or floating-point SDP output in a reproducible form. A minor follow-on: the H18 negative construction uses an approximate numerical value for t(H18, WT); the margin over 1/1024 is large, so I am not worried, but a rational lower bound would tie it off cleanly.\n\nWho this is for: people working on quasirandomness, flag algebras, or tournament limits. It does not resolve the general classification or connect outside the field, but it opens a concrete new question—are almost all tournaments forcing under near-regularity?—and it provides a small complete case with a mix of methods.\n\nRecommendation: send it to a serious referee. The main theorem is plausible and the proof architecture is sound; the missing certificates are fixable and should be the main request. I would not desk reject.","headline":"A sensible new variant of quasirandomness forcing with a complete 5-vertex classification; the positive half rests on flag-algebra certificates that need machine-checkable backing.","tokens_in":33540,"tokens_out":3244,"would_cite":true,"duration_ms":37028,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C20","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Among the twenty tournaments on at most five vertices, exactly eleven force quasirandomness once the hosts are required to be nearly regular; the other nine do not.","keywords":["quasirandomness","tournaments","near-regularity","forcing property","homomorphism density","flag algebra","tournament limits","classification"],"falsifier":"Recompute the eigenvalues and kernels of the printed matrices $A_1,A_2,A_3$ and re-evaluate, for every five-vertex tournament $J$, the claimed constant expressions in Theorems 5.4–5.7; a matrix that is not positive semidefinite, a kernel different from the stated span, or a tournament $J$ whose expression differs from $5/4$, $8/7$ or $6/5$ would refute the positive half of the classification. An independent exhaustive search over regular tournamentons on a fine grid could look for a regular limit $W$ with t(H_i,W) equal to the random value for one of the eleven listed tournaments while W is not $1/2$ on a positive-measure set.","tokens_in":32421,"feed_emoji":"🎲","tokens_out":12975,"duration_ms":130104,"temperature":0.7,"pith_summary":"Random tournaments look the same at every scale: every finite pattern appears with its random probability, and a sequence of tournaments with these frequencies is called quasirandom. The paper asks which small patterns H can certify quasirandomness by themselves: if the frequency of H in a large tournament matches its random expectation, must all other pattern frequencies match too? In the unrestricted setting, only the transitive tournaments on four or more vertices and one exceptional five-vertex tournament have this property. The authors add a mild near-regularity assumption, that almost every vertex has out-degree close to half the number of vertices, and give the full picture for tournaments on at most five vertices: eleven force quasirandomness under this assumption and nine do not. The point is that the forcing family becomes substantially larger, so there are many more single-pattern tests for quasirandomness once near-regularity is known.","feed_headline":"Regularity assumption reveals 11 quasirandom-forcing tournaments","feed_subtitle":"Exactly eleven of the twenty small tournaments force quasirandomness once near-regularity is imposed.","key_machinery":"The central objects are tournamentons, measurable functions $W:[0,1]^2 \\to [0,1]$ with $W(x,y)+W(y,x)=1$, which are the continuous limits of tournament sequences; quasirandomness corresponds to $W(x,y)=1/2$ almost everywhere. The key reduction, Proposition 3.8, is that near-regularity is exactly the condition that the transitive triangle density or the cyclic triangle density sits at its random value, so H forces quasirandomness in regular tournaments precisely when the set {H, TT3} or {H, C3} forces it without regularity. On the positive side, the proof uses the flag algebra method, a calculus of rooted subpattern densities that turns density inequalities into positive semidefinite matrix certificates; the printed matrices A1, A2 and A3 yield inequalities such as $8t(C_3,W) + (1/4)1024t(H_{10},W) \\le 5/4$, and the equality analysis forces the two four-vertex densities $t(TT_4,W)$ and $t(C_4,W)$ to equal $(1/8)t(TT_3,W)$, which Lemma 5.13 converts into $W=1/2$ almost everywhere. On the negative side, Lemma 3.10 builds regular tournamentons by splicing two regular tournamentons and uses continuity to hit the random density while remaining non-quasirandom.","core_discovery":"On the paper's own terms, the discovery is the exact split of the twenty tournaments: H4, H5, H6, H7, H8, H10, H11, H13, H14, H15 and H17 force quasirandomness in regular tournaments, while H0, H1, H2, H3, H9, H12, H16, H18 and H19 do not. The positive half is proved by reducing regular forcing to ordinary forcing of a pair: by Proposition 3.8, H forces quasirandomness in regular tournaments if and only if {H, TT3}, equivalently {H, C3}, forces quasirandomness without the regularity assumption. The flag algebra inequalities of Theorems 5.4–5.7 then certify the needed pairs for H10, H11, H13 and H14, with H15 obtained by reversing arcs, and equality is shown to force the limit tournamenton to be $W(x,y)=1/2$ for almost all pairs. The negative half is witnessed by explicit regular tournamentons, including one-parameter interpolations between two regular limits and a regular seven-vertex tournament, for which t(H,W) reaches the random value while W stays far from the constant $1/2$.","pith_inferences":["This inference goes beyond the paper: the negative constructions all live on low-complexity regular limits built from small regular tournaments or interpolations between them, so the non-forcing phenomenon may be confined to a finite-dimensional slice of the space of tournamentons.","This inference goes beyond the paper: the same flag algebra pipeline could be run on the 112 six-vertex tournaments to test Question 6.3 at the next scale; the paper's method stops at five vertices.","This inference goes beyond the paper: for tournament data with approximately balanced outdegrees, matching any one of the eleven pattern counts is a practical quasirandomness certificate, provided near-regularity is checked separately."],"forward_implications":["In any nearly regular tournament sequence, matching the expected density of any one of the eleven listed tournaments forces every finite pattern density to match its random value.","The forcing list strictly enlarges the unrestricted one on five vertices: H5, H6, H7, H10, H11, H13, H14 and H15 are non-transitive tournaments that force only under near-regularity, while H4, H8 and H17 remain forcing without it.","Because near-regularity is equivalent to t(TT3) or t(C3) being at the random value, the classification can be restated as finitely many inequalities involving three- and five-vertex densities, all with equality only at the constant $1/2$ limit.","Any future resolution of the open classification problem must reproduce this exact split on five vertices; the paper's Questions 6.2 and 6.3 ask whether the family continues to infinity and whether it eventually contains almost all tournaments."],"supporting_citations":[{"why":"Establishes that no non-transitive tournament on seven or more vertices forces quasirandomness, the boundary result that the regular variant sits inside.","marker":"[2]"},{"why":"Supplies the C3/C4 density inequality whose equality characterization proves H6, the four-vertex cycle tournament, forces under regularity.","marker":"[4]"},{"why":"Provides the first non-transitive forcing example, H17, reused here as a positive case and benchmark for the regular variant.","marker":"[16]"},{"why":"Gives the flag algebra proof that transitive tournaments force quasirandomness, the baseline used for H4 and H8.","marker":"[17]"},{"why":"Completes the unrestricted classification for tournaments on at most six vertices, the result the paper extends by adding near-regularity.","marker":"[27]"},{"why":"Supplies the classical paired-comparison extremal fact behind the identity t(TT3,W) ≥ 1/8 and its equality characterization.","marker":"[29]"},{"why":"Records the classical exercise that TT_k forces quasirandomness for k ≥ 4, used to handle the transitive cases.","marker":"[35]"},{"why":"Introduces the flag algebra formalism used to produce the positive semidefinite certificates in Theorems 5.4–5.7.","marker":"[41]"}],"fun_headline_variants":["Regularity assumption: 11 tournaments force quasirandomness","Only 11 regular tournaments force quasirandomness","Near-regular tournaments: 11 forcing, 9 not","Quasirandom forcing in regular tournaments: 11 of 20"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the printed positive semidefinite matrices $A_1,A_2,A_3$ and the Appendix A coefficient tables are all correct; the paper gives no code or certificates, and a single wrong entry would break Theorems 5.4–5.7 and with them the positive half of the classification.","fun_headline_variants_meta":{"raw":{"variants":["Regularity assumption: 11 tournaments force quasirandomness","Only 11 regular tournaments force quasirandomness","Near-regular tournaments: 11 forcing, 9 not","Quasirandom forcing in regular tournaments: 11 of 20"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00069,"raw_usage":{"total_tokens":3138,"prompt_tokens":973,"completion_tokens":2165,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":2094}},"tokens_in":589,"tokens_out":2165,"duration_ms":16684,"temperature":1.0,"reasoning_tokens":2094,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:58:36.350639+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the eigenvalues and kernels of the printed matrices $A_1,A_2,A_3$ and re-evaluate, for every five-vertex tournament $J$, the claimed constant expressions in Theorems 5.4–5.7; a matrix that is not positive semidefinite, a kernel different from the stated span, or a tournament $J$ whose expression differs from $5/4$, $8/7$ or $6/5$ would refute the positive half of the classification. An independent exhaustive search over regular tournamentons on a fine grid could look for a regular limit $W$ with t(H_i,W) equal to the random value for one of the eleven listed tournaments while W is not $1/2$ on a positive-measure set.","supporting_citations":[{"cited_title":"Buci´ c, E","cited_arxiv_id":null,"evidence_quote":"Establishes that no non-transitive tournament on seven or more vertices forces quasirandomness, the boundary result that the regular variant sits inside."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the C3/C4 density inequality whose equality characterization proves H6, the four-vertex cycle tournament, forces under regularity."},{"cited_title":"Coregliano, R F","cited_arxiv_id":null,"evidence_quote":"Provides the first non-transitive forcing example, H17, reused here as a positive case and benchmark for the regular variant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the flag algebra proof that transitive tournaments force quasirandomness, the baseline used for H4 and H8."},{"cited_title":"Hancock, A","cited_arxiv_id":null,"evidence_quote":"Completes the unrestricted classification for tournaments on at most six vertices, the result the paper extends by adding near-regularity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical paired-comparison extremal fact behind the identity t(TT3,W) ≥ 1/8 and its equality characterization."},{"cited_title":"Lov´ asz","cited_arxiv_id":null,"evidence_quote":"Records the classical exercise that TT_k forces quasirandomness for k ≥ 4, used to handle the transitive cases."}],"review_version":1}