{"id":"0e676ed2-092b-4917-9b81-da1a293790f0","arxiv_id":"2610.36069","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The number of Markov triples with largest entry below x is C(log(3x))^2 plus an error term that is o(log x), improving previous bounds.","lead":"This paper improves the error term in the asymptotic count of Markov triples, showing it is o(log x) rather than O(log x log log x). The proof connects Markov triples to simple geodesics on a punctured torus and uses geometric estimates to refine lattice point counting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Central argument relies on Theorem 2 of an unpublished preprint (Doan–Li–Nguyen, arXiv:2605.14574v3); any error in that result would propagate directly into the main error estimate.","rationale":"The reader identified the reliance on Theorem 2 of Doan–Li–Nguyen as the weakest assumption. I concur: the entire geometric ingredient for the improved lattice-point count rests on that theorem. Without it, the analysis of normal turn and the estimates in Lemmas 6–8 and Corollary 11 lose their quantitative foundation. The paper cites the preprint fairly, but since it is unpublished, the stress-test must flag this dependency. However, the paper makes its own contributions (algebraic separation in Lemma 5, convex cap arguments) and handles all subsequent steps rigorously. The parameter choices in Theorem 12 are consistent with the existence of positive constants from Theorem 2, and the unconditional bounds using Vinogradov–Korobov show that some sub-log bound is achievable even without the theorem. Therefore I do not recommend changing the verdict, but the concern is real and warrants a specific verification test. The proposed concrete test targets the exponential decay claim directly and is feasible (e.g., via computational geometry or author inquiry). No other internal logical gap or algebraic error was found; the reasoning apart from the external dependency is sound and well-structured.","tokens_in":21129,"tokens_out":5685,"duration_ms":50450,"concrete_test":"Independent verification of Theorem 2 for the modular torus: compute the total normal turn (μ_turn) on boundary arcs between consecutive primitive lattice points of height up to H=10 (or larger, via symmetry) using numerical integration of supporting normals derived from the known trace–length relation, and measure the total turn on the complement. If the total turn on arcs whose endpoints have heights > H does not decay at least as fast as e^{-c H} for some c>0, the theorem would be contradicted. Alternatively, obtain a direct confirmation from the authors of the preprint that their constants hold for the modular torus with explicit bounds, and check that those bounds satisfy the inequality me^{-KH} ≥ (2+ε)η used in Lemma 6 with the parameters chosen in Theorem 12.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1 depends on Theorem 2 of Doan, Li, and Nguyen (arXiv:2605.14574v3), which states exponentially small turn after removing corners of the McShane–Rivin ball for any once-punctured torus. This theorem is not yet published in a peer-reviewed venue; its proof has not been independently verified. The constants κ, K, m, M are stated to depend on the surface X, and the subsequent lemmas (especially Lemma 6, Lemma 7, and Corollary 11) hinge on these constants being positive and on inequalities such as (15). If any of these constants are incorrect or if the exponential decay rate is misestimated for the modular torus, the parameter choices in Theorem 12 (e.g., H = L/(4K), η = R^{-1/2}) could fail to satisfy the required conditions, thereby invalidating the claimed error term. The paper itself does not provide independent confirmation of Theorem 2 for the modular case, and the numerical verification in §1.1, while consistent, does not test the geometric estimates directly. This is the single most load-bearing concern because it is an external dependency that, if unsound, collapses the entire counting argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper gives an asymptotic formula for the number of Markov triples with largest entry below x, improving the error term to o(log x) (specifically, O(log x (log log log x)^{1/10} / (log log x)^{3/10})). The proof exploits the correspondence between Markov triples and simple closed geodesics on the modular punctured torus, the geometry of the McShane–Rivin norm, lattice-point counting with smoothing and Poisson summation, and Möbius inversion linked to zero-free regions of the Riemann zeta function. Under the quasi-Riemann hypothesis a sharper error is given. The unconditional error relies on the classical Vinogradov–Korobov zero-free region.","tokens_in":21534,"tokens_out":4237,"duration_ms":42748,"significance":"If the result holds, it represents the first sub-log x error term for the Markov triple counting function, improving on the O(log x log log x) bound due to McShane and Rivin. The method combines geometric estimates for the McShane–Rivin ball with analytic number theory in a novel way and may be applicable to other arithmetic surfaces. The explicit constant C is expressed in geometric and arithmetic terms.","major_comments":[{"comment":"The proof of Theorem 1 depends crucially on Theorem 2 of Doan, Li, and Nguyen (arXiv:2605.14574v3), an unpublished preprint that provides the exponential turn estimates for the McShane–Rivin ball of any once-punctured torus. The constants κ, K, m, M and the decay estimates from that result enter every subsequent geometric lemma (Lemmas 3, 6, 7, Corollary 11, Theorem 12). The paper does not supply an independent verification for the modular torus, nor does it discuss the current status of that preprint. If Theorem 2 contains an error, the entire counting argument, including the main error bound, would be invalidated. The authors should either (a) provide a self-contained proof of the needed estimates for the modular case, or (b) clearly state that the main result is conditional on Theorem 2 and indicate the verification status of that preprint.","section":"§3.1, Theorem 2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The paper claims to have computed M(10^k) for every integer 1 ≤ k ≤ 30 000 and to have verified the uniqueness conjecture up to 10^{30000}, all in about 70–90 minutes on a standard laptop. No algorithm, implementation, or computational details are given. Given that numbers up to 30 000 digits are involved, this claim is not credible as stated and undermines the reader's confidence in the manuscript's reporting standards. The authors should provide a detailed explanation of the algorithm, the data structures used, and the actual computational environment, or else retract the numerical claims altogether. The numerical verification does not affect the theoretical result, but the lack of credibility in this section is a serious concern for the paper's overall rigor.","section":"§1.1, Numerical verification","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The lower bound in Lemma 5 is |det(λ_{u|v}, d)| ≥ exp(-M_1 |d| A^2). The proof uses the fact that Δ A_{\\pm} are algebraic integers in the real quadratic field K = Q(√Δ). This step requires verification that Δ = t_u^2 - 4 is not a perfect square (true because t_u ≥ 3 and t_u is not a Lucas number that would make Δ a square? Actually, for the modular torus t_u are Markov numbers, and it is known that t_u^2-4 is never a perfect square, but the proof in the text seems to rely on the field being real quadratic; it might be worth a brief justification to ensure completeness.","section":"§4, Lemma 5","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The constant C in the abstract and Theorem 1 is given as C ≈ 0.180717..., but the derivation at the end of §7 expresses C = 2 vol(B_X)/π^2. The value of vol(B_X) is not provided; the reader must compute it from the references. A short remark or a numerical value would improve readability.","section":"§1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The factor 6(M(T)-2) + 2·3 yields 6M(T) - 6, not 6M(T) - 12 + 6 = 6M(T) - 6; then dividing by 12 gives M(T)/2 - 1/2, but eq. (5) says M(T) = (1/12)#(L_T B_X ∩ Z^2_{prim}) + 1. Please check the arithmetic. The final formula seems correct because the exceptional orbits are handled, but the step from (4) to (5) could be written more clearly.","section":"§1, eq. (4)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The second sentence of the proof claims that 'the four lines x=0, y=0, y=x, and y=-x divide R^2 into eight coordinate sectors.' This is correct; but the following paragraph states that every boundary ray of these sectors is represented by a primitive vector of height 1. For the sector {(p,q): 0 ≤ q ≤ p}, the boundary rays have directions (1,0) and (1,1), both height 1. This is fine.","section":"§3.1, proof of Lemma 3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the display under (11), the expression log(r^{d(v)} A_+^{-d(u)}) appears. There is a missing absolute value inside the logarithm? The text says 'if |det(λ_{u|v}, d)| = 0, then r^{d(v)} = A_+^{d(u)}' which is correct if the absolute value of the argument is 1. The subsequent argument about norms uses equality to 1. However, the equation (11) gives 2|log(r^{d(v)} A_+^{-d(u)})|, so |det|=0 implies r^{d(v)} A_+^{-d(u)} = ±1. The text uses =1; one should argue that a negative sign would be impossible because both sides are positive? Actually, r>0, A_+>0, so the expression is positive, so =1. That's okay.","section":"§3.1, Lemma 5","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The function ρ is defined as radially symmetric with support in the Euclidean unit ball, but then ρ_Q is supported in the ball of radius 1/Q. However, (19) requires R > C/Q. The constant C from (6) is the upper bound of the norm relative to the Euclidean norm, so |x| ≤ C^{-1}||x||_X? Actually, from (6): c|x| ≤ ||x||_X ≤ C|x|, so the unit ball B_X is contained in the Euclidean ball of radius 1/c. The convolution approximation uses the Euclidean norm; the condition for 1_{S B_X} * ρ_Q(x) to be 1 when ||x||_X ≤ S - C/Q appears to assume that the support radius 1/Q of ρ_Q is measured in the Euclidean norm. Since the norm equivalence gives B_X ⊂ {|x| ≤ 1/c}, the estimate should use the Euclidean radius of the dilate. The text writes 'if ||x||_X ≤ S - C/Q' but earlier C is defined as the constant in (6) that gives the upper bound, so ||x||_X ≤ C|x|. To guarantee that the support of ρ_q(x-y) is within the dilate, one would need a condition like ||x||_X ≤ S - C'/Q for some constant C'. The argument as written appears to swap the direction of the inequality. This should be checked and corrected.","section":"§5, smoothing and Poisson summation","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The declaration acknowledges the use of AI tools for exposition and exploration. While this is acceptable, the paper does not specify which parts were generated or checked by AI. Given the unconventional numerical claims, the editor may wish to request a more detailed disclosure.","section":"Declaration of AI use","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"References [26] and [27] are given as OpenAI web pages with dates in 2026. These are not standard scholarly publications, and their content may not be verifiable. The paper should clarify that the quasi-Riemann hypothesis is not a proven theorem but a conditional assumption that the author uses in the statement of the improved error bound.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The numerical verification claims in §1.1 appear highly implausible. Computing M(10^k) for k up to 30,000 would require handling numbers with 30,000 digits and enumerating an astronomically large set of Markov triples. The stated runtime of 70 minutes on a standard laptop is not credible. I recommend that the editor request the authors to provide a detailed explanation of the algorithm and to confirm the reproduction of the numerics, or to remove the numerical section entirely. If the numerics cannot be substantiated, this may affect the editor's assessment of the manuscript's reliability, even though the main theorem does not depend on these computations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The paper does what it says: it improves the error term in the asymptotic count of Markov triples from O(log x log log x) to o(log x), and gives a sharper conditional bound under a quasi-Riemann hypothesis. That is a real advance on a problem that has been open since McShane and Rivin's work in the mid-1990s. The method is novel and the execution is careful — the combination of Farey fractions, convex geometry, Möbius inversion, and zero-free region estimates is well-integrated. The numerical verification is a nice bonus, though it doesn't test the geometric estimates directly.\n\nThe main soft spot is the reliance on Theorem 2 of Doan–Li–Nguyen (arXiv:2605.14574v3). That preprint is not yet peer-reviewed, and the constants involved in the exponential decay bound are stated to depend on the surface in a way that is not made uniform for the modular case. The paper's entire lattice-point argument hinges on those bounds. The author is transparent about using this theorem, but there is no independent check or adaptation here — it is a pure black box. If that theorem had a hidden flaw, the main result would not hold.\n\nThat said, the concern is proportionate: the cited result seems plausible and the authors of that preprint are established mathematicians. The risk is real but not obviously high. The paper itself is otherwise rigorous and well-structured. The conditional bound under quasi-RH is clearly labelled and doesn't depend on that theorem in the same way.\n\nWho this is for: anyone working on Markov numbers, lattice-point counting in hyperbolic geometry, or the geometry of once-punctured tori. It will be of interest to a moderately specialized audience.\n\nRecommendation: this paper deserves serious peer review. The central vulnerability should be checked by a reviewer familiar with the Doan–Li–Nguyen result, but the rest of the argument is solid and the advance is clear. A desk reject would be inappropriate.","headline":"First o(log x) error term for Markov triple counting, built on a clever geometric lattice-point argument — but the core estimate leans on an unpublished theorem that hasn't been independently verified.","tokens_in":21980,"tokens_out":951,"would_cite":false,"duration_ms":11439,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11D45","11P21","11D25","11H06","53C22"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"First sub-logarithmic error for Markov triple counting","keywords":["Markov triples","simple geodesics","modular torus","McShane-Rivin norm","lattice point counting","Riemann zeta function","primitive lattice points","quasi-Riemann hypothesis"],"falsifier":"Compute $M(x)$ for $x$ up to $10^{5000}$ (beyond current numerical range) with high precision and check whether the error term $|M(x) - C(\\log(3x))^2| / \\log x$ decays to zero. If it stabilizes at a positive constant, the $o(\\log x)$ claim would be falsified. Alternatively, if the Doan–Li–Nguyen turn estimate is shown to be incorrect for the modular torus (e.g., corners of height $H$ produce turn decay only at a polynomial rate), the entire geometric argument collapses.","tokens_in":22672,"feed_emoji":"","tokens_out":5363,"duration_ms":53820,"temperature":0.7,"pith_summary":"The paper proves that the number of Markov triples with largest entry at most $x$ is $C(\\log(3x))^2 + o(\\log x)$, the first time the error has been shown to be sub-logarithmic. The previous best error was $O(\\log x \\log\\log x)$. The proof converts the counting problem into counting primitive lattice points in dilations of the McShane–Rivin unit ball on the modular punctured torus. A detailed geometric analysis of this ball, combined with number-theoretic estimates from the Riemann zeta function's zero-free region, yields the improved error. Under the quasi-Riemann hypothesis the error can be further reduced to $O(\\log x / (\\log\\log x)^{1/2})$.","feed_headline":"First sub-logarithmic error for Markov triple counting","feed_subtitle":"Uses the modular torus's arithmetic geometry to refine the count of Markov triples and simple geodesics.","key_machinery":"The key objects are (1) the McShane–Rivin norm $\\|\\cdot\\|_X$ on $\\mathbb{R}^2$, whose unit ball $B_X$ has strictly convex boundary with corners at rational directions, and (2) the Doan–Li–Nguyen theorem (Theorem 2) stating that after removing corners coming from primitive lattice points of height at most $H$, the remaining normal turn on $\\partial B_X$ is exponentially small in $H$. The arithmetic structure of the modular torus (Lemma 5) gives a quantitative separation between supporting normals and rational directions, enabling a refined lattice-point estimate via a convex-cap diameter bound (Lemma 8). The Möbius inversion step uses zero-free regions of $\\zeta(s)$ via the Mertens function to pass from full lattice points to primitive lattice points.","core_discovery":"The central result is Theorem 1: $M(x) = C(\\log(3x))^2 + O\\big(\\log x\\, (\\log\\log\\log x)^{1/10} / (\\log\\log x)^{3/10}\\big)$ unconditionally, and with error $O(\\log x / (\\log\\log x)^{1/2})$ under the quasi-Riemann hypothesis. This improves on the previous best error $O(\\log x \\log\\log x)$ due to McShane and Rivin. The constant $C$ is explicit and equals $\\operatorname{vol}(B_X)/(12\\,\\zeta(2)) = 2\\operatorname{vol}(B_X)/\\pi^2$, where $B_X$ is the McShane–Rivin unit ball for the modular torus. The proof also gives an improved error term for the number of simple closed geodesics on the modular punctured torus (Theorem 13).","pith_inferences":["The paper's approach opens the possibility of extending sub-logarithmic error bounds to other surfaces whose McShane–Rivin ball has arithmetic properties similar to the modular torus, such as congruence covers.","The reliance on Doan–Li–Nguyen's theorem means that any uniform improvement of that theorem (e.g., replacing exponential decay by super-exponential decay) would directly sharpen the error in Theorem 1.","The numerical data hint that the true error may be as small as $O((\\log x)^{1/2})$, which would match the McShane–Rivin conjecture; proving this would likely require fundamentally new ideas beyond the current geometric and analytic toolbox."],"forward_implications":["The error term for Markov triple counting is now known to be $o(\\log x)$, ending a decade of stagnation at $O(\\log x \\log\\log x)$.","If the quasi-Riemann hypothesis is true, the error improves to $O(\\log x / (\\log\\log x)^{1/2})$, which is closer to the conjectured $O_\\varepsilon((\\log x)^{1/2+\\varepsilon})$.","The same method yields an improved error for the number of simple closed geodesics on the modular punctured torus: $P_X(R) = (\\operatorname{vol}(B_X)/\\zeta(2)) R^2 + O(R / (\\log R)^{3/10})$ unconditionally.","The numerical verification of the Frobenius uniqueness conjecture up to $10^{30000}$ suggests that the main term captures the count accurately, with a difference of only 35 triples at that height."],"supporting_citations":[{"why":"Supplies the original asymptotic $C(\\log(3x))^2$ and error $O(\\log x (\\log\\log x)^2)$ that the paper improves upon.","marker":"[36]"},{"why":"Established the McShane–Rivin norm ball and proved the correspondence between Markov triples and simple geodesics, giving the previous best error $O(\\log x \\log\\log x)$.","marker":"[21, 22]"},{"why":"Provides Theorem 2 on exponentially small turn on $\\partial B_X$ after removing corners, the geometric engine for the refined lattice-point estimate.","marker":"[7]"},{"why":"Gives the Vinogradov–Korobov zero-free region for $\\zeta(s)$ used in the unconditional error estimate.","marker":"[35]"},{"why":"Classical de la Vallée Poussin zero-free region used for the weaker prime-number-theorem-based bound in Remark 14.","marker":"[6]"},{"why":"Reference for the bound on the Mertens function under a zero-free half-plane, used in Theorem 13.","marker":"[24]"}],"fun_headline_variants":["Modular torus sharpens Markov triple counting","Sub-logarithmic error for Markov triple count","Improved bound for simple geodesics on punctured torus","Markov triple count error reduced to o(log x)"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof depends on the uniform geometric constants in Theorem 2 of Doan, Li, and Nguyen applying to the modular torus with the same exponential decay rate; any hidden dependence on the surface's geometry that worsens those constants would break the final error estimate.","fun_headline_variants_meta":{"raw":{"variants":["Modular torus sharpens Markov triple counting","Sub-logarithmic error for Markov triple count","Improved bound for simple geodesics on punctured torus","Markov triple count error reduced to o(log x)"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000116,"raw_usage":{"cost":0.0001157184,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001157184,"upstream_inference_prompt_cost":3.273984e-05,"upstream_inference_completions_cost":8.297856e-05},"total_tokens":1841,"prompt_tokens":812,"completion_tokens":1029,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":966}},"tokens_in":812,"tokens_out":1029,"cost_source":"openrouter_usage","duration_ms":10325,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":966,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-28T16:44:55.993417+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute $M(x)$ for $x$ up to $10^{5000}$ (beyond current numerical range) with high precision and check whether the error term $|M(x) - C(\\log(3x))^2| / \\log x$ decays to zero. If it stabilizes at a positive constant, the $o(\\log x)$ claim would be falsified. Alternatively, if the Doan–Li–Nguyen turn estimate is shown to be incorrect for the modular torus (e.g., corners of height $H$ produce turn decay only at a polynomial rate), the entire geometric argument collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original asymptotic $C(\\log(3x))^2$ and error $O(\\log x (\\log\\log x)^2)$ that the paper improves upon."},{"cited_title":"Stable-norm boundary geometry on once-punctured tori","cited_arxiv_id":"2605.14574","evidence_quote":"Provides Theorem 2 on exponentially small turn on $\\partial B_X$ after removing corners, the geometric engine for the refined lattice-point estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Vinogradov–Korobov zero-free region for $\\zeta(s)$ used in the unconditional error estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classical de la Vallée Poussin zero-free region used for the weaker prime-number-theorem-based bound in Remark 14."},{"cited_title":"Montgomery and Robert C","cited_arxiv_id":null,"evidence_quote":"Reference for the bound on the Mertens function under a zero-free half-plane, used in Theorem 13."}],"review_version":1}