{"id":"44142237-383b-4589-8480-78138ee2aac7","arxiv_id":"2608.06238","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For tournaments on n vertices, the maximum number of directed 4k-cycles is asymptotic to (1 + 2 times the sum from i=1 to infinity of (2/((2i-1)pi))^(4k)) times the random tournament's count, attained by the carousel tournament.","lead":"The paper proves the exact asymptotic maximum number of directed cycles of length divisible by 4 in a tournament. It confirms that the carousel tournament, not the random tournament, is the extremal structure, resolving a conjecture by Bartley and Day.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reader's counterexample miscomputes the RHS (47, not 29), but Lemma 3.3's proof still skips a nontrivial rearrangement step; the conditional verdict stands.","rationale":"The reader correctly identifies Lemma 3.3 as the pivotal point of the paper, but the stated counterexample miscomputes the right-hand side of the manuscript's actual final inequality. After correcting that arithmetic, the lemma appears likely to be true: spot checks with transitive and carousel tournaments for n=3 through 6 satisfy the claimed weak majorization. Nevertheless, the proof as printed contains an unjustified substitution: it replaces the r selected entries x'_i by the r smallest global entries x_{n-r+i} in the second sum while simultaneously replacing the complementary sum by the largest n-r entries. The text's only justification, '|c_{r,i}|≤r', does not by itself establish that the total right-hand side is an upper bound; a rearrangement or majorization argument is needed. This is a genuine gap in the proof of Lemma 3.3, and since Corollary 3.4 and Theorem 1.4 depend on it, the upper-bound proof in Theorem 1.2 is conditional on repairing that step. I did not find any other load-bearing defect: Theorem 1.3's trace inequality appears sound, the eigenvalue computation for T_n in Lemma 4.2 checks out, and the counting reduction in Proposition 4.1 is standard. The equality case in Theorem 1.4 has a 'without loss of generality' step that deserves scrutiny, but it is not needed for the asymptotic upper bound. Thus the appropriate verdict is unchanged: conditional acceptance pending a rigorous proof of the missing rearrangement step.","tokens_in":11758,"tokens_out":52249,"duration_ms":502343,"concrete_test":"Exhaustively verify Lemma 3.3 for n=4,5,6: enumerate all skew-symmetric ±1 matrices A and all nonincreasing integer vectors x with entries in {0,...,5}; for each S and ε check both X ≤ ∑_{i=1}^{|S|}(K_n x)_i and the intermediate inequality r∑_{j∉S}x_j + ∑ c_i x'_i ≤ r∑_{j=1}^{n-r}x_j + ∑ c_i x_{n-r+i}. If any violation is found, the lemma is false; if none, the step is true but still requires a written proof before acceptance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's specific counterexample does not land: for n=4, x=(10,9,8,1), S={1,3}, and ε=(1,-1), the left side is 36, but the final right side in the paper is 2(x1+x2)+1*x3+1*x4=47, not 29. The quoted hybrid inequality with r∑_{j∉S}x_j is not the line actually written in the manuscript. However, Lemma 3.3 as written has a real proof gap. The proof derives X ≤ r∑_{j∉S}x_j + ∑ c_{r,i}x'_i, where x'_i are the entries of x on S sorted decreasing, and then asserts X ≤ r∑_{j=1}^{n-r}x_j + ∑ c_{r,i}x_{n-r+i}, saying only '|c_{r,i}|≤r'. The first term increases when passing from the complement to the largest n-r entries, but the second term decreases because x_{n-r+i}, the r smallest global entries, are coordinate-wise no larger than x'_i. The assertion that the net change is nonnegative is exactly a rearrangement/majorization step that is neither stated nor proved. Since Lemma 3.3 feeds into Corollary 3.4 and hence Theorem 1.4, the upper bound in Theorem 1.2 is not rigorously established until this step is justified or replaced by a correct proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to settle the remaining case of the Bartley--Day conjecture by showing that, for every fixed positive integer k, the maximum number of directed 4k-cycles in an n-vertex tournament is asymptotically (1 + 2 sum_{i>=1} (2/((2i-1)pi))^{4k} + o(1)) * (4k-1)! / 2^{4k} * binom(n,4k), with the carousel tournament attaining the bound. The proof encodes a tournament as a skew-symmetric +/-1 matrix A, reduces cycle counting to traces of (A+J)^{4k}, and derives two matrix inequalities: Theorem 1.3 bounds Tr[(A+J)^{4k}] in terms of Tr[A^{4k}], and Theorem 1.4 bounds Tr[A^{4k}] by the corresponding trace for the transitive tournament matrix T_n. The eigenvalues of T_n are then computed to obtain the constant.","tokens_in":12024,"tokens_out":46895,"duration_ms":411967,"significance":"If the proof can be completed, this is a substantial result: it confirms the Bartley--Day conjecture for all cycle lengths divisible by 4 and gives the first exact asymptotic constant for those cases, extending the work of Grzesik--Kral'--Lovasz--Volec. The trace-based strategy is attractive, and Theorems 1.3 and 1.4 are potentially of independent interest. The paper is also transparent about the sources of its lower bound and about the use of AI assistance. However, the main upper bound currently rests on Lemma 3.3, whose proof is incomplete at a load-bearing point, and on an unverified hypothesis in the application of Karamata's inequality.","major_comments":[{"comment":"The proof of Lemma 3.3 omits a decisive rearrangement argument. After the reduction to X <= r sum_{j not in S} x_j + sum_{i<j, i,j in S} |epsilon_i x_j - epsilon_j x_i|, the manuscript asserts two inequalities: (i) sum_{i<j, i,j in S} |epsilon_i x_j - epsilon_j x_i| <= sum_i c_{r,i} x'_i, and (ii) r sum_{j not in S} x_j + sum_i c_{r,i} x'_i <= r sum_{j=1}^{n-r} x_j + sum_i c_{r,i} x_{n-r+i}. Both are true but genuinely nontrivial. In particular, (ii) is not a consequence of the stated observation '|c_{r,i}| <= r': passing from x'_i to the smaller entries x_{n-r+i} decreases the second sum, while the first sum increases. The proof needs the sharper bound c_{r,i} <= r-1 together with an exchange argument showing that the maximum over S is attained when S is the set of the r smallest coordinates. Since Lemma 3.3 is the only bridge from arbitrary tournaments to K_n, and it is used to prove Corollary 3.4 and Theorem 1.4, the upper bound in Theorem 1.2 is not rigorously established as written.","section":"Section 3, Lemma 3.3"},{"comment":"Proposition 3.1 is applied to u = (A^{2k} e_j)^* and v = K_n^{2k} e_1, but Proposition 3.1 is stated only for pairs of nonnegative non-increasing vectors. The manuscript does not prove that K_n^{2k} e_1 is nonnegative or non-increasing. This is not automatic because K_n has negative entries. The authors should either prove that K_n^{2k} e_1 has these properties (for instance, by showing that K_n^2 preserves the cone of non-increasing nonnegative vectors) or state and prove the real-vector version of Karamata's inequality that the argument actually requires. Without this, the chain leading to (A^{4k})_{jj} <= (K_n^{4k})_{11} is missing a hypothesis.","section":"Section 3, proof of Theorem 1.4"}],"minor_comments":[{"comment":"In the definition of z_j, the text writes z_j = e^{(2j-1)pi/n}; the imaginary unit i is missing and should be z_j = e^{i(2j-1)pi/n}, the roots of z^n = -1.","section":"Section 4, Lemma 4.2"},{"comment":"The lower bound for the number of 4k-cycles in the carousel tournament is cited only to Bartley's PhD thesis [1]. Since this bound is half of the equality statement in the main theorem, the paper would be more self-contained if the computation were included in an appendix or at least stated with a precise pointer to the relevant result in the thesis.","section":"Section 4, proof of Theorem 1.2"},{"comment":"The equality-case proof is very compressed. In particular, the assertion that the first two entries of (B * 1)^* being n-2 forces an all-1 row and an all-(-1) row, and the subsequent induction, should be written out in full.","section":"Section 3, proof of Theorem 1.4, equality case"},{"comment":"The sentence 'Notice that w_{n-r,r} - w_{n-r+1,r} = r - (r-1) = 1 for r = 1, ..., n' has an index error: for r = n the term w_{0,n} is undefined. The range should be r = 1, ..., n-1, together with the separate statement for w_{n,2} - w_{n+1,2}.","section":"Section 3, Lemma 3.2"}],"recommendation":"major_revision","confidential_remarks":"The main claim is significant and the overall strategy is credible, but the proof is not yet complete: Lemma 3.3 needs a correct and complete rearrangement argument, and the application of Karamata's inequality needs a justification of the nonnegativity and monotonicity of K_n^{2k} e_1. These are fixable within the paper's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper closes the last open case of the Bartley-Day conjecture by proving the exact asymptotic maximum for directed 4k-cycles in tournaments. That is a real result, and the high-level strategy is attractive: the cycle count is reduced to a trace inequality, and the extremal matrix is shown to be the transitive tournament matrix T_n up to signed permutation. The two matrix theorems are new and give a clean explanation of why the carousel tournament beats random for lengths divisible by 4.\n\nWhat is actually new: the exact constant c(4k) for all k >= 3, previously only known for k = 1, 2. The proof combines a known reduction with two spectral/trace inequalities. Lemma 2.1 is a neat Schur-decomposition argument and looks correct. Lemma 4.2's spectrum of T_n is standard and correctly applied. The carousel lower bound is cited from Bartley's thesis, so the upper bound is the load-bearing part.\n\nThe soft spot is Lemma 3.3. The claim that (Ax)^* is weakly majorized by K_n x^* is plausible and probably true, but the proof as written has a gap. The first bound gives r times the sum over the complement of S plus a term involving the entries of x on S. The next line replaces the complement with the n-r largest entries of x, which increases the first term, and replaces the x'_i with the r smallest entries x_{n-r+i}, which decreases the second term. Since the coefficients c_{r,i} are nonnegative, this is not automatically an upper bound. The paper says \"It is easy to see that |c_{r,i}| <= r\", but that only controls the size of the coefficients, not the net change. A correct proof needs a rearrangement/majorization step showing the increase in the first term dominates the decrease in the second. The reader's specific counterexample does not land, by the way: for n=4, x=(10,9,8,1), S={1,3}, the paper's final RHS is 47, not 29. But the proof gap is genuine. Since Lemma 3.3 feeds Corollary 3.4 and Theorem 1.4, the upper bound in the main theorem is not fully rigorous as written.\n\nThe rest of the paper is in good shape: the reduction in Proposition 4.1 is correct, the equality cases in Theorem 1.4 are plausible, and the citations to Grzesik-Kral-Lovasz-Volec and Bartley's thesis are appropriate. The authors acknowledge using ChatGPT for parts of the proof; that is not itself a problem because the work is independently checked, but the referee should ask for the gap to be fixed by a human argument.\n\nWho should read this: anyone working on extremal combinatorics of tournaments, and the matrix inequalities may interest people working on trace majorization. It deserves a serious referee. I would suggest sending it to review with a request to fix Lemma 3.3 or provide a reference; the likely outcome is a correct proof of a significant theorem.","headline":"Resolves Bartley-Day for all 4k with a clean reduction, but Lemma 3.3 has a real proof gap that must be fixed before the upper bound is rigorous.","tokens_in":12587,"tokens_out":5670,"would_cite":false,"duration_ms":49173,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C20","05C35","05C38","15A18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The carousel tournament maximizes directed cycles of length divisible by 4.","keywords":["tournaments","directed cycles","carousel tournament","extremal combinatorics","trace inequalities","skew-symmetric matrices","weak majorization","Bartley-Day conjecture"],"falsifier":"Check the disputed inequality in Lemma 3.3 at n=4 with x=(10,9,8,1), A=T_4, and S={1,3}: the claimed bound would require 36 ≤ 29, which is false. This counterexample does not by itself disprove Theorem 1.2, but it pinpoints the exact step whose repair or replacement is needed for the upper bound.","tokens_in":11533,"feed_emoji":"🔄","tokens_out":5986,"duration_ms":55501,"temperature":0.7,"pith_summary":"This paper tries to settle the extremal question of which n-vertex tournament contains the most directed cycles of a fixed length. It claims that for every cycle length divisible by 4, the asymptotic maximum is achieved by the carousel tournament, with an explicit constant that is strictly larger than the random tournament's count. Combined with earlier work on lengths not divisible by 4, this would complete the Bartley-Day conjecture for all cycle lengths. The proof converts cycle counts into traces of powers of a skew-symmetric ±1 adjacency matrix and reduces the extremal problem to a linear-algebra inequality about those traces.","feed_headline":"Carousel tournaments maximize 4k-cycles","feed_subtitle":"For every cycle length divisible by 4, the carousel beats the random tournament, completing the Bartley-Day conjecture.","key_machinery":"The central object is the carousel tournament Car_n, the directed Cayley graph on Z/nZ in which each vertex points to the next (n−1)/2 vertices; for even n it is obtained by deleting one vertex from Car_{n+1}. On the matrix side, the argument runs through T_n, the skew-symmetric matrix with 1 above the diagonal and −1 below, and a closely related symmetric matrix K_n that preserves weak majorization. The mechanism: closed directed walks of length ℓ correspond to diagonal entries of (A+J)^ℓ, so bounding Tr[(A+J)^{4k}] bounds cycle counts; Theorem 1.4 shows the transitive sign matrix maximizes Tr[$A^{{4k}}$] via a weak-majorization comparison (Ax)^* ≺ K_n x^*, and Theorem 1.3 separates the contribution of the all-ones matrix J.","core_discovery":"The paper's central claim is Theorem 1.2: for each fixed positive integer k, the maximum number of directed 4k-cycles in an n-vertex tournament satisfies C(n,4k)/R(n,4k) → 1 + 2·Σ_{i≥1}(2/((2i−1)π))^{4k}, where R(n,4k) is the expected count in the random tournament, and the carousel tournament asymptotically attains this value. This confirms the Bartley-Day conjecture for all cycle lengths divisible by 4. The constant is strictly greater than 1, so the random tournament is not asymptotically extremal for these lengths. The proof proceeds through two matrix theorems: Theorem 1.3, which bounds Tr[(A+J)^{4k}] by Tr[$A^{{4k}}$] + Tr[$J^{{4k}}$], and Theorem 1.4, which bounds Tr[$A^{{4k}}$] by Tr[$T_n^{{4k}}$] where T_n is the sign matrix of the transitive tournament, with equality exactly for signed-permutation conjugates of T_n.","pith_inferences":["Editorial inference: the same trace-and-majorization framework may extend to other Cayley or circulant tournaments, potentially identifying carousel-type objects as extremal for other fixed orientations.","Editorial inference: the equality case in Theorem 1.4 suggests a stability version, namely that tournaments nearly extremal for 4k-cycles should be structurally close to a signed-permutation conjugate of the transitive tournament, hence close to a carousel in a suitable cut metric.","Editorial inference: a direct computational check for small n and k (especially k=3, where the paper leaves exact maximization open) could test whether the carousel is the exact finite-n maximizer or only asymptotically optimal."],"forward_implications":["Every cycle length divisible by 4 now has an identified asymptotic extremal tournament: the carousel, which beats the random tournament by the factor 1 + 2·Σ(2/((2i−1)π))^{4k}.","The Bartley-Day dichotomy is confirmed for all cycle lengths: the random tournament is asymptotically optimal when the length is not divisible by 4, and the carousel is optimal when it is.","The carousel's advantage over the random tournament decays like 2(2/π)^{4k} as k grows, so the effect is strongest for short 4k-cycles.","As a by-product, the paper establishes a linear-algebra extremal result: among all ±1 skew-symmetric matrices, the transitive-tournament matrix T_n maximizes the trace of the 4k-th power, with equality only for signed-permutation conjugates of T_n."],"supporting_citations":[{"why":"Proved the non-divisible-by-4 case and the value c(8), establishing the baseline and pattern this paper extends to all 4k.","marker":"[11]"},{"why":"Formulated the explicit conjectured constant 1 + 2·Σ(2/((2i−1)π))^{4k} for c(4k).","marker":"[1]"},{"why":"Conjectured, together with Bartley, that the carousel tournament maximizes cycles when the length is divisible by 4.","marker":"[8]"},{"why":"Established the seed case c(4)=4/3 with the carousel as the extremal tournament.","marker":"[2]"},{"why":"Supplied the early result c(3)=1 that anchors the random-tournament side of the dichotomy.","marker":"[13]"},{"why":"Proved c(5)=1, supporting the divisibility-by-4 pattern in the conjecture.","marker":"[14]"},{"why":"Supplies the Karamata-style inequality that converts weak majorization into the trace comparison used in Theorem 1.4.","marker":"[15]"}],"fun_headline_variants":["Carousel beats random for 4k-cycles","For 4k-cycles, carousel wins","4k-cycles: carousel outshines random","Carousel maximizes 4k-cycles, beats random"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper bound depends on the technical claim that every tournament sign matrix stretches any vector less aggressively than one fixed matrix K_n, when both are measured by their largest partial sums; if that claim is false, the paper's proof of the maximum does not go through.","fun_headline_variants_meta":{"raw":{"variants":["Carousel beats random for 4k-cycles","For 4k-cycles, carousel wins","4k-cycles: carousel outshines random","Carousel maximizes 4k-cycles, beats random"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1511,"prompt_tokens":884,"completion_tokens":627,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":562}},"tokens_in":500,"tokens_out":627,"duration_ms":5134,"temperature":1.0,"reasoning_tokens":562,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:47:06.203840+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the disputed inequality in Lemma 3.3 at n=4 with x=(10,9,8,1), A=T_4, and S={1,3}: the claimed bound would require 36 ≤ 29, which is false. This counterexample does not by itself disprove Theorem 1.2, but it pinpoints the exact step whose repair or replacement is needed for the upper bound.","supporting_citations":[{"cited_title":"Lov´ asz, and Jan Volec, Cycles of a given length in tournaments, J","cited_arxiv_id":null,"evidence_quote":"Proved the non-divisible-by-4 case and the value c(8), establishing the baseline and pattern this paper extends to all 4k."},{"cited_title":"Bartley,Topics in extremal and probabilistic combinatorics, Ph.D","cited_arxiv_id":null,"evidence_quote":"Formulated the explicit conjectured constant 1 + 2·Σ(2/((2i−1)π))^{4k} for c(4k)."},{"cited_title":"Day,A collection of problems in extremal combinatorics, Ph.D","cited_arxiv_id":null,"evidence_quote":"Conjectured, together with Bartley, that the carousel tournament maximizes cycles when the length is divisible by 4."},{"cited_title":"Beineke and Frank Harary, The maximum number of strongly connected subtournaments, Canadian Mathematical Bulletin8(1965), no","cited_arxiv_id":null,"evidence_quote":"Established the seed case c(4)=4/3 with the carousel as the extremal tournament."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplied the early result c(3)=1 that anchors the random-tournament side of the dichotomy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proved c(5)=1, supporting the divisibility-by-4 pattern in the conjecture."},{"cited_title":"Marshall and Ingram Olkin,Inequalities: Theory of majorization and its applications, Academic Press, New York, 1979","cited_arxiv_id":null,"evidence_quote":"Supplies the Karamata-style inequality that converts weak majorization into the trace comparison used in Theorem 1.4."}],"review_version":1}