{"id":"36d4dfb5-498d-403f-9278-dae7282a1ccf","arxiv_id":"2607.21568","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Tight Hamilton cycles in (p,μ)-dense 3-graphs are forced iff the minimum codegree exceeds ((1−√((4p−1)/3))/2)^2 n for p > 1/3, and the p,α > 1/4 conjecture fails at p0 ≈ 0.3177.","lead":"This paper finds the exact minimum-codegree condition under which a quasirandom 3-uniform hypergraph must contain a tight Hamilton cycle, and disproves a conjecture that density above 1/4 would suffice. The sharp threshold formula δ0(p) covers densities p > 1/3, and matching examples show no weaker codegree bound works.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.3's local cleaning and the repeated-cluster convention are only sketched; the parameter hierarchy linking M0 to the fixed walk length L may be circular.","rationale":"I read the paper in good faith and verified the main structural components: the extremal constructions in Section 2, the reduction of Theorem 1.3 to the connecting lemma via the absorption framework, the reduced scalar lemma (Lemma 5.4) and its application to strong connectivity and aperiodicity, and the codegree/density inheritance lemmas. These parts are mathematically coherent and I found no internal inconsistency. The scalar lemma's algebra checks out, and the strong-connectivity/aperiodicity argument correctly reduces to it. The residual risk is concentrated in the regularity infrastructure: Lemma 4.3 is stated as a standard consequence of [3] but the local cleaning property is stronger than the usual global bound, and the proof sketch does not resolve the dependency between the regularity parameters, the returned bound M0, and the pattern length L needed for the extension lemma. The repeated-cluster convention in Section 4.2 is similarly load-bearing and unproved. These are genuine technical gaps, but they are of the kind that can often be filled without changing the main theorem, so a CONDITIONAL verdict is appropriate. I therefore do not change the reader's verdict.","tokens_in":20552,"tokens_out":43901,"duration_ms":390899,"concrete_test":"Provide a fully formal version of Lemma 4.3 from the error-function regular slice lemma, specifying ε(t) and proving that the returned M0 satisfies L(M0^3)+2 ≤ B for the pattern-size threshold in Lemma 4.2 (or restructure the proof to fix the number of clusters m before choosing δ2, δ3, r). Additionally, give an explicit proof of the repeated-cluster convention for a concrete tight-path pattern of length 5 that revisits the same cluster, showing that refined pair-cells retain positive relative density and regularity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The connecting lemma (Lemma 1.4) depends on the locally cleaned regular slice (Lemma 4.3) and on the repeated-cluster convention of Section 4.2. Lemma 4.3's local cleaning property (3) is genuinely stronger than the global irregularity bound of the error-function regular slice lemma; the sketch ('ε(t) ≪ η t^{-C}') does not track how the returned M0 interacts with the fixed walk length L from Lemma 5.2. Since the number of states is at most M0^3, L is a function of M0, but the regularity parameters δ2, δ3 in Lemma 4.3 are fixed before M0 is produced. The extension lemma (Lemma 4.2) requires δ2, δ3 to be small enough for pattern size r = L+2, so if the constants are chosen in the order stated, there is no guarantee that the slice is regular enough to lift the reduced walk. The repeated-cluster convention is also asserted without proof: when a tight-path pattern revisits a cluster, splitting clusters into subclusters is claimed to preserve all regularities and positive densities, but this is essential for Lemma 5.3 to apply the extension lemma. If either step fails, the reduced pair-state theorem and hence Lemma 1.4 collapse.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies tight Hamilton cycles in 3-graphs that are linearly quasirandom in the sense of (p,μ)-density, under a minimum codegree condition. It first constructs, for p0≈0.317672, (p0−ε,μ)-dense 3-graphs with minimum codegree at least (p0−ε)n and no tight Hamilton cycle, disproving a conjecture of Araújo, Piga and Schacht (Problem 1.1). It then proves for every p>1/3 that the sharp asymptotic minimum-codegree threshold is δ0(p)=((1−√((4p−1)/3))/2)^2: every sufficiently large (p,μ)-dense 3-graph H with δ2(H)≥αn, α>δ0(p), contains a tight Hamilton cycle, and a biased construction shows the bound is best possible. The positive theorem is deduced from the absorption framework of Han–Shu–Wang via a fixed-length connecting lemma. The connecting lemma is proved through a regular slice, a directed pair-state graph whose states are ordered pair-cells, a finite scalar lemma ruling out non-trivial closed sets, and a lifting argument from reduced walks to tight paths. The paper also gives a construction showing that below p=1/3, minimum codegree slightly above n/3 does not force arbitrary ordered pairs to be connectable.","tokens_in":20793,"tokens_out":22542,"duration_ms":192921,"significance":"If the proof of the connecting lemma is made fully rigorous, the results constitute a substantial advance: they solve the sharp minimum-codegree threshold for tight Hamilton cycles in linearly quasirandom 3-graphs for all p>1/3, give an explicit counterexample to a natural conjecture, and identify p=1/3 as a genuine boundary for the arbitrary-pair connecting method. The constructions in Section 2 are explicit, self-contained, and correct; the scalar lemma (Lemma 5.4) is elegant and its proof is convincing. The paper is honest about the limitations of the method below p=1/3. However, the central proof currently rests on a regularity claim (Lemma 4.3) and a repeated-cluster convention that are only sketched, and the parameter hierarchy linking the size of the regular slice to the length of the reduced walks appears circular as written. These issues are load-bearing and require substantial clarification before the main theorem can be considered established.","major_comments":[{"comment":"The quantifier order in the proof of Lemma 1.4 is circular. Lemma 4.3 takes δ2, δ3, and a regularity parameter r as inputs and returns M0; then L=L(M0) is defined via Lemma 5.2, and t=L+2 becomes the pattern size in Lemma 5.3. Lemma 5.3 applies Lemma 4.2 with pattern size L+2, and Lemma 4.2 requires δ2, δ3 to be very small and the triad-regularity parameter to be at least some function of L. Since L is not known until M0 is produced, the stated order — first choose \"all regularity parameters sufficiently small\", then define M and L — does not guarantee that the slice is regular enough for the extension lemma. A rigorous hierarchy must be supplied, e.g. by proving a version of Lemma 4.3 that explicitly bounds M0 in terms of the input parameters and then choosing parameters by a fixed-point argument, or by an iterative refinement argument. As written, the reduced walk may be too long for t","section":"§5, proof of Lemma 1.4; §4.2, Lemma 4.3"},{"comment":"Property (3) of Lemma 4.3 is genuinely stronger than the global irregularity bound in the standard regular-slice lemma of [3], because it asserts a local bound over every pair-cell of weight at least 1/M0. The proof sketch ('ε(t) ≪ η t^{-C}') does not quantify C, nor how the returned M0 interacts with the pair-cell weight lower bound, nor how the bounded refinements for reversal coherence affect the local estimates. Since Lemma 4.5 (reduced codegree inheritance) relies precisely on property (3) to feed the reduced extension inequalities and hence the scalar lemma, this lemma is load-bearing and must be proved in full or replaced by a precisely stated reference.","section":"§4.2, Lemma 4.3"},{"comment":"The repeated-cluster convention asserts that splitting clusters into subclusters preserves all regularities and positive densities, with losses 'absorbed into the parameter hierarchy'. This is used essentially in Lemma 4.4 and in Lemma 5.3 when a reduced walk revisits clusters many times. In Lemma 5.3, the walk length r=L can be a growing function of M0, so a single cluster may be used O(L) times. The number of required subclusters and the resulting degradation of δ2, δ3 (and of the positive densities of pair-cells and dense triads) are not quantified. Without this quantification, the application of Lemma 4.2 in Lemma 5.3 is not fully justified.","section":"§4.2, repeated-cluster convention; used in §5.3, Lemma 5.3"}],"minor_comments":[{"comment":"The displayed identity for the total reduced weight of a subfamily of triads is stated as an equality. When two or three indices coincide, the product of pair-cell weights counts ordered triples with repeated vertices, whereas the support K3(T) excludes them. The identity is therefore only true up to o(1), and the statement should say so to avoid a formal gap.","section":"§4.3, Lemma 4.4(2)"},{"comment":"The conclusion that the constructed H is (p0−ε, μ)-dense uses the definition's additive error μ n^3, not a relative error. This is correct, but the argument would be clearer if the definition of (p,μ)-density were explicitly recalled at that point, since for very small subsets X,Y,Z the error term dominates.","section":"§1.2 / Theorem 1.2"},{"comment":"The absorption framework of [20] is used as a black box, and [20] is by the same authors. The theorem is stated in full, which is helpful; a sentence on where the theorem is proved or published would aid readers in verifying that the stated hypotheses (in particular the reservoir condition) match the published result.","section":"§3, Theorem 3.1"}],"recommendation":"major_revision","confidential_remarks":"The main technical risk is the regularity hierarchy: the circularity between M0, L, and the regularity parameters in the proof of Lemma 1.4 must be resolved, and Lemma 4.3 needs a real proof. The constructions and the scalar lemma appear correct, so the result is plausible and the paper is a significant contribution if the gaps are filled. The self-citation to [19,20] is heavy but appropriate given the framework; the editor may wish to confirm that Theorem 3.1 is indeed published in the cited form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Just read Shu's arXiv:2607.21568. The headline: this is a real advance. It kills the Araújo–Piga–Schacht conjecture (Problem 1.1) with the p0≈0.3177 example, and gives a sharp codegree threshold for p>1/3. The biased construction and the scalar lemma are the goods—I checked them, they work. The directed pair-state graph idea is a natural but clever way to convert density/codegree constraints into a finite system of scalar inequalities.\n\nWhere I'd push back: the paper as written is not fully checkable at the connecting-lemma core. Lemma 4.3 (locally cleaned regular slice) is asserted as a standard consequence of [3], but it is stronger than the global irregularity bound and the proof sketch doesn't track the constants. More serious is the hierarchy: you choose regularity parameters δ2, δ3 before M0 exists, then M0 bounds the number of states, then L=L(M0) is the walk length, then the extension lemma (Lemma 4.2) needs δ2, δ3 to be small relative to L+2. That's a circular dependency unless there is a fixed-point argument, and I don't see one in the text. For the exponential/tower bounds typical of the regular slice lemma, the inequality δ ≪ 1/L(δ)^C is not obviously satisfiable. This needs to be fixed, either by a two-stage slice with back-refinement or by a different bound on L. The repeated-cluster convention in Section 4.2 is also asserted rather than proven; maybe standard, but it's load-bearing for lifting.\n\nThe absorption framework [20] is taken as a black box from the author's own prior work. That's acceptable if the theorem is right, but it's another unproven external dependency.\n\nIf the hierarchy is fixed, I'd expect the main theorem to hold. The constructions are rigorous, the scalar lemma is self-contained, and the p≤1/3 discussion is honest. The τ vs τ′ parameter inconsistency should be cleaned up.\n\nBottom line: this deserves a serious referee, but the referees should be asked to nail down Section 4 and the constant order in Section 5. I'd send it to review.","headline":"A real advance that likely settles the p>1/3 threshold and disproves the 1/4 conjecture, but the parameter hierarchy in the connecting lemma is not closed and one core regularity lemma is only sketched.","tokens_in":21374,"tokens_out":4997,"would_cite":true,"duration_ms":50045,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C45","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a 3-graph of density p > 1/3, minimum codegree above δ0(p)n forces a tight Hamilton cycle, and this bound is sharp; the earlier 1/4 conjecture fails at p0 ≈ 0.318.","keywords":["tight Hamilton cycles","3-uniform hypergraphs","quasirandom hypergraphs","minimum codegree","regular slice","absorption method","pair-state graph","scalar lemma"],"falsifier":"A numerical search could look for a non-trivial matrix (s_ij) ∈ [0,1]^{I×I} and weights (λ_i) summing to one satisfying the three scalar inequalities of Lemma 5.4 for some τ > 1/3 with Δ > a(τ)^2 (where a(τ) is the smaller root of a^3 + (1−a)^3 = τ); finding one would falsify the scalar lemma and hence the connecting argument. Alternatively, exhibiting, for some p > 1/3 and ε > 0, a (p, μ)-dense 3-graph on arbitrarily large vertex sets with δ2 ≥ (δ0(p)+ε)n and no tight Hamilton cycle would falsify Theorem 1.3 directly.","tokens_in":20324,"feed_emoji":"🔄","tokens_out":11979,"duration_ms":89911,"temperature":0.7,"pith_summary":"The paper settles, for densities above 1/3, how much common-neighbour structure a quasirandom 3-uniform hypergraph must have to be guaranteed to contain a tight Hamilton cycle — a cyclic ordering in which every three consecutive vertices form an edge. It first disproves the natural conjecture that density and minimum codegree both above 1/4 suffice: a balanced split of the vertex set into two parts of sizes x0n and p0n (where x0^3 + x0 = 1, p0 = x0^3 ≈ 0.318) yields a (p0−ε, μ)-dense 3-graph with minimum codegree at (p0−ε)n and no tight Hamilton cycle. For every p > 1/3 the paper then determines the exact asymptotic threshold: if minimum codegree exceeds δ0(p)n = ((1−√((4p−1)/3))/2)^2 n, a tight Hamilton cycle is forced, and a biased version of the earlier construction shows no smaller codegree bound can work. The proof reduces the problem to a finite system of scalar inequalities and shows that above the threshold only the trivial solutions exist, so a directed 'pair-state' graph is strongly connected and aperiodic, enabling fixed-length connecting paths.","feed_headline":"Above density 1/3, codegree δ0(p)n forces a tight Hamilton cycle","feed_subtitle":"A construction at p0 ≈ 0.318 voids the old 1/4 conjecture; the sharp curve δ0(p) fills all p > 1/3.","key_machinery":"The proof rests on a regular slice that partitions every ordered pair of vertices into a bounded number of 'pair-cells', each with bounded relative weight. Dense regular triads of pair-cells define transitions in a directed pair-state graph whose vertices are the pair-cells; a walk in this graph tracks the terminal ordered pair of a growing tight path. The central reduced statement is that this digraph is strongly connected and aperiodic when p > 1/3 and α > δ0(p). To prove this, a non-trivial closed set of states is shown to produce a weight matrix (s_ij) satisfying the scalar inequalities Φ(s_ij, s_jk, s_ki) ≥ τ and Σ_k λ_k s_jk s_ki ≥ Δ (plus a mirrored inequality for 1−s), and a convexit","core_discovery":"The central claim is that the map p ↦ δ0(p) = ((1−√((4p−1)/3))/2)^2 is the asymptotically sharp minimum-codegree threshold for tight Hamilton cycles in (p, μ)-dense 3-graphs whenever p > 1/3: every sufficiently large such n-vertex 3-graph with δ2(H) ≥ αn for α > δ0(p) contains a tight Hamilton cycle, while for every ε > 0 there are (p, μ)-dense 3-graphs with δ2 ≥ (δ0(p)−ε)n and no tight Hamilton cycle. A second claim is that the previously proposed p, α > 1/4 condition is false; the number p0 = max_{0≤x≤1} min{x^3, 1−x} ≈ 0.317672 supplies a counterexample that is both (p0−ε, μ)-dense and has minimum codegree at least (p0−ε)n yet lacks a tight Hamilton cycle because of a global imbalance bet","pith_inferences":["The balanced constant p0 arises from equating two competing obstructions (x^3 and 1−x); analogous 'balancing constants' may appear in the threshold functions for tight Hamilton cycles in k-uniform quasirandom hypergraphs under other minimum-degree conditions, a pattern worth testing numerically.","The pair-state digraph plus scalar-lemma template is a general recipe: it reduces a global spanning problem to a finite convexity statement, and the same template could plausibly be adapted to tight paths of fixed length or to loose Hamilton cycles in quasirandom hypergraphs.","The threshold function h(p) may have a genuine phase transition at or near p0, since the two constructions presented give different extremal mechanisms on different sides of p0; deciding whether h(p) drops sharply to the right of p0 would clarify the structure of extremal quasirandom 3-graphs.","One concrete testable extension is to check numerically whether the scalar obstruction at Δ = a(τ)^2 from Remark 5.5 is the only obstruction for τ slightly above 1/3, which would indicate whether the connecting lemma is tight exactly at the stated boundary."],"forward_implications":["The conjectured condition p, α > 1/4 does not force tight Hamilton cycles; the honest threshold only exists for p > 1/3.","For every p > 1/3, δ0(p)n is the asymptotically sharp minimum-codegree threshold, so the answer to the problem of the full threshold function h(p) is now known on that whole range.","The arbitrary-pair connecting method cannot be pushed below p = 1/3: even minimum codegree slightly above n/3 does not guarantee that every two ordered pairs can be joined by a tight path once density drops below 1/3.","Any sharp Hamiltonicity result for p ≤ 1/3 will need to control the endpoints of absorbers and path covers within a selected family of connectable pairs, since a global connecting lemma fails there.","The full threshold function h(p) for 0 < p ≤ 1/3 is left open; for the diagonal value p_diag (the first p with h(p) ≤ p), the two theorems bracket p0 ≤ p_diag ≤ 1/3, and the paper asks whether in fact p_diag = p0."],"fun_headline_variants":["Sharp codegree threshold for p>1/3 in quasirandom 3-graphs","Counterexample voids 1/4 conjecture; exact threshold for p>1/3","δ0(p) is the minimum codegree for tight Hamilton cycles when p>1/3","No tight Hamilton cycle at density 0.318 despite high codegree","Quasirandom 3-graphs: sharp codegree bound for Hamilton cycles"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes the hypergraph can be sliced into pair-cells so that every needed triad is regular and every positive density survives the splitting of clusters into subclusters; this 'regular slice with local cleaning' step is cited as a standard consequence of the hypergraph regularity lemma rather than proved in detail, and if it fails the reduced pair-state digraph and the entire connecting argument collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sharp codegree threshold for p>1/3 in quasirandom 3-graphs","Counterexample voids 1/4 conjecture; exact threshold for p>1/3","δ0(p) is the minimum codegree for tight Hamilton cycles when p>1/3","No tight Hamilton cycle at density 0.318 despite high codegree","Quasirandom 3-graphs: sharp codegree bound for Hamilton cycles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000711,"raw_usage":{"total_tokens":3149,"prompt_tokens":970,"completion_tokens":2179,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":714,"completion_tokens_details":{"reasoning_tokens":2067}},"tokens_in":714,"tokens_out":2179,"duration_ms":15600,"temperature":1.0,"reasoning_tokens":2067,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:07:25.507705+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerical search could look for a non-trivial matrix (s_ij) ∈ [0,1]^{I×I} and weights (λ_i) summing to one satisfying the three scalar inequalities of Lemma 5.4 for some τ > 1/3 with Δ > a(τ)^2 (where a(τ) is the smaller root of a^3 + (1−a)^3 = τ); finding one would falsify the scalar lemma and hence the connecting argument. Alternatively, exhibiting, for some p > 1/3 and ε > 0, a (p, μ)-dense 3-graph on arbitrarily large vertex sets with δ2 ≥ (δ0(p)+ε)n and no tight Hamilton cycle would falsify Theorem 1.3 directly.","supporting_citations":[],"review_version":1}