{"id":"80040c9d-bedd-43b7-931d-624e292c7b57","arxiv_id":"2411.13452","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-linear Hamilton ℓ-cycles in random hypergraphs appear exactly when the expected count diverges, with a lognormal fluctuation law in the special case ℓ=2.","lead":"This paper proves that non-linear Hamilton cycles in random hypergraphs appear as soon as the expected number of such cycles tends to infinity, confirming a conjecture of Narayanan and Schacht. It also shows that for 2-cycles the number of cycles follows a lognormal distribution, a new and surprising law in this setting.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ℓ=2 lognormal law depends on Lemma 3.2's planted and double-planted joint CLT, which is asserted after only first- and second-moment checks; any non-Gaussianity or extra covariance would break Lemma 3.3's variance cancellation and Theorem 1.2's lognormal limit.","rationale":"I read the paper as aiming to confirm the Narayanan-Schacht conjecture by showing concentration for ℓ≥3 and a lognormal limit for ℓ=2. The proof outline is coherent: the second-moment analysis cleanly separates ℓ≥3 from ℓ=2, and the small-subgraph conditioning construction with Y=Σ t_k Y(P_k) is a natural way to absorb the constant factor in the second moment. The most delicate and load-bearing point is indeed Lemma 3.2's joint CLT under the planted and double-planted measures. The reader's weakest_assumption identifies the same step, and I agree: the manuscript computes means and covariances but does not supply a full CLT argument or a citation for the tilted measures. The paper's own text flags this ('the convergence follows once we check...'), so this is an omitted proof rather than a demonstrated contradiction. The concern is decisive for the ℓ=2 main theorem because Lemma 3.3 and Lemma 4.1 both use the Gaussian form of the MGFs to cancel the extra second-moment factor. I do not see a reason to change the reader's CONDITIONAL verdict: the gap is real but plausibly repairable, and I would not reject the paper or mark it unverdictable on this basis. No machine-checked or reproducible evidence is present, so the conditional assessment remains appropriate.","tokens_in":16922,"tokens_out":15772,"duration_ms":178585,"concrete_test":"Complete the proof of Lemma 3.2 for P* and P*2_t: write each Y(P_j) as its deterministic planted contribution plus a polynomial in the independent Bernoulli variables on the complement of the planted cycle(s), and verify the hypotheses of a multivariate CLT for generalized subgraph counts, e.g. Janson's orthogonal decomposition or a dependency-graph/Stein bound. Concretely, compute the limiting joint cumulants of order at least three, or otherwise show they vanish; if the limiting covariance under P* or P*2_t is not I_k, recompute the MGF ratio in Lemma 3.3 and the second-moment ratio in Lemma 4.1 to test whether the variance cancellation still forces X to concentrate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Lemma 3.2: under the planted measure P* and the double-planted measure P*2_t, the vector (Y(P_1),...,Y(P_k)) must converge jointly to N(µ,I_k) and N(2µ,I_k), respectively. The null case is cited to Janson, but the P* and P*2_t cases only compute E*[Y(P_j)], Var*(Y(P_j)), and Cov*(Y(P_{j1}),Y(P_{j2})), and then state that 'the convergence follows once we check that the dominant contribution comes from connected subgraphs.' Computing means and pairwise covariances is not a CLT: joint Gaussianity requires control of all higher-order cumulants or a genuine multivariate limit theorem for the relevant polynomial functionals of the independent edge indicators on the complement of the planted cycle(s). Lemma 3.3 then evaluates E*[e^{-Y_N}] and E*2_t[e^{-2Y_N}] by Gaussian moment generating functions, so any failure of joint Gaussianity, or any nonzero limiting covariance, changes the exponential factor that cancels the extra constant in Proposition 2.5. The theorem's lognormal conclusion in the ℓ=2 case therefore rests directly on this omitted proof. The gap is plausibly fillable via Janson's orthogonal decomposition, but it is exactly the assertion that the manuscript does not justify.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the number Z of Hamilton ℓ-cycles in the r-uniform Erdős–Rényi hypergraph G_r(n,p), for integers r > ℓ > 1 with (r − ℓ) | n, at densities p = (c + o(1))p*(r,ℓ). The main results are: for ℓ ≥ 3, Z/E[Z] → 1 in probability whenever E[Z] → ∞; for ℓ = 2, Z/E[Z] converges to an explicit lognormal distribution; and in the constant-expectation regime, Z converges to a Poisson law for ℓ ≥ 3 and to a lognormal mixture of Poissons for ℓ = 2. This confirms the conjecture of Narayanan and Schacht. The proof combines a refined second-moment computation (Section 2) with small-subgraph conditioning (Sections 3–4) and a subsampling argument for the constant-expectation case (Section 5).","tokens_in":17172,"tokens_out":17411,"duration_ms":192982,"significance":"If the proof is completed, the paper resolves a conjecture and gives the first sharp distributional characterization for non-linear Hamilton cycles at the threshold. The second-moment case analysis is detailed, the dichotomy ℓ ≥ 3 versus ℓ = 2 is clean, and the constants A_k entering the lognormal parameters are derived from automorphism counts rather than fitted to data; the choice of coefficients t_k is a standard variance-cancellation step, not a circular definition of the answer. The central unresolved point is the planted and double-planted central limit theorem asserted in Lemma 3.2, which is load-bearing for the ℓ = 2 results.","major_comments":[{"comment":"The planted and double-planted joint Gaussianity assertions are not proved. The proof verifies only the first two moments under P* and P*2_t and then states that 'the convergence follows once we check that the dominant contribution comes from connected subgraphs.' First- and second-moment computations do not imply joint asymptotic normality for a polynomial functional of independent edge indicators; higher-order cumulants or a genuine multivariate normal approximation theorem are required. Lemma 3.3 subsequently evaluates the moment generating functions by Gaussian integration, including the identity covariance and means μ and 2μ, and the cancellation of the extra constant in Proposition 2.5 depends on these exact exponential factors. Without a complete proof of Lemma 3.2, the lognormal limit in Theorem 1.2(2) and the mixed-Poisson limit in Theorem 1.4(2) are not established. The gap appears localized and plausibly fillable, for instance by showing that under P* the vector equals its null counterpart on the complement plus a deterministic shift plus o_p(1), or by a full adaptation of Janson's orthogonal decomposition to the planted measures, but it must be supplied.","section":"§3, Lemma 3.2"},{"comment":"The application of Lemma 3.3 to the truncated variables ~Y_N is justified only by the sentence that truncation 'changes the first and second moments by o(1) factors.' Since the entire small-subgraph conditioning cancellation requires the exponential factors to be correct to within 1 + o(1), this step needs a quantitative tail estimate. The needed estimate is standard for a Gaussian variable with bounded variance at truncation level M = min{log log E[Z], log log n}, but it should be written out so that the reader can verify that the error is o(1) uniformly in the subsequent sums.","section":"§4, Lemma 4.1"}],"minor_comments":[{"comment":"The provided text contains numerous typographical and OCR artifacts, such as 'Ham ilton', 'Hamil ton', and a missing word in the abstract; these should be corrected in the final version.","section":"Throughout"},{"comment":"The sentence that A_k is 'constant' for sufficiently large k is not immediate from the definition A_k = Aut(P_k)/(t!(s−t)!)^k; only boundedness is needed for the argument. Please clarify the intended statement.","section":"§1.3, Eq. (1.1)"},{"comment":"The summation in Case 2 is written with the condition v(F) ≤ log n, although the case under discussion is log n < v(F) < n; this should be corrected to avoid confusion.","section":"§4, Lemma 4.1, Case 2"},{"comment":"The proof says 'Choose N = N0(ε, δ)' but then uses N in the subsequent estimates; the notation should be aligned.","section":"§4, Lemma 4.1"},{"comment":"The notation |C1 ∩ C2| should be defined explicitly as the number of common edges, since for hypergraphs vertex overlap and edge overlap are both meaningful.","section":"§5, Lemma 5.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a strong candidate for publication if the planted and double-planted CLT in Lemma 3.2 is supplied. I agree with the stress-test note that this is the main load-bearing gap; it is real and localized, and I do not see circularity or fitted parameters. The citation to Janson for the null case is appropriate, but the planted and double-planted cases need either a citation or a proof. The relationship to Narayanan and Schacht's conjecture is stated clearly, and the novelty is appropriate for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a serious paper and I think the main results are probably true, but the ℓ=2 lognormal theorem is not fully proved as written. The stress-test is right: Lemma 3.2 asserts joint Gaussianity of the Y(P_k) under the planted and double-planted measures after computing only means and covariances. That is not a CLT. The null case has Janson's theorem behind it; the planted cases do not. Since Lemma 3.3 uses the Gaussian moment generating function to cancel the extra second-moment factor, Theorem 1.2(2) and the Poisson-lognormal mixture in Theorem 1.4(2) both depend on this unproved joint limit.\n\nWhat is genuinely new and good: for ℓ≥3, the sharp second moment actually works and gives concentration, which already confirms the Narayanan–Schacht conjecture in that case. The case analysis in Section 2—small connected overlaps, large subcritical overlaps, spanning connected, and spanning disconnected—is well organized and the estimates look right. The small subgraph conditioning setup with explicit constants A_k is natural, and the variance cancellation is exactly the right mechanism. No fitted parameters, no self-citation; the constants come from the combinatorics. The subsampling argument for the constant-expectation regime is sketched but plausible, and I did not find a hidden circularity.\n\nOn the gap: I do not think it is fatal to the truth of the results. The planted and double-planted models are local perturbations of the null model, and a multivariate CLT should follow from a Janson-style orthogonal decomposition or a cumulant argument. But the manuscript does not supply that argument; the line that convergence follows once the dominant contribution comes from connected subgraphs is not a proof. For a theorem at this level, the planted CLT is exactly the step a referee should ask to see.\n\nWho this is for: random graph theorists, probabilists working on small subgraph conditioning, and combinatorialists interested in sharp thresholds. It deserves a serious referee, not a desk reject. I would condition acceptance on a complete proof of Lemma 3.2, or a precise citation to a theorem that covers the planted and double-planted joint Gaussian limits.","headline":"Good paper, probably right, but the lognormal case needs a real proof of the planted CLT.","tokens_in":17731,"tokens_out":3103,"would_cite":true,"duration_ms":31015,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","05C65","05C45","60C05","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random hypergraphs gain non-linear Hamilton cycles exactly once the expected count diverges, and the limiting count is lognormal at overlap 2.","keywords":["random hypergraphs","Hamilton cycles","threshold","second moment","small subgraph conditioning","lognormal distribution","Poisson mixture","subgraph counts"],"falsifier":"Compute the third and fourth moments of the normalized short-path count $Y(P_1)$ under the planted measure for a concrete parameter pair such as $r=3,\\ell=2$ and moderately large $n$; a deviation from the Gaussian values (third moment 0, fourth moment 3) beyond $o(1)$ would refute Lemma 3.2 and the lognormal limit it implies.","tokens_in":16675,"feed_emoji":"🔄","tokens_out":16608,"duration_ms":147553,"temperature":0.7,"pith_summary":"This paper settles when a random hypergraph contains a spanning non-linear Hamilton cycle: an $r$-uniform $\\ell$-cycle that visits every vertex, with consecutive edges overlapping in exactly $\\ell$ vertices, for $r>\\ell\\ge 2$. It proves that such a cycle appears with probability tending to one as soon as the expected number of copies tends to infinity, confirming a conjecture from earlier work. The proof also identifies the limiting distribution of the number of cycles: for overlap size $\\ell\\ge3$ the count concentrates around its mean, while for $\\ell=2$ it converges to an explicit lognormal law (the exponential of a normal random variable) once the expectation diverges. At the constant-expectation boundary, the count is asymptotically Poisson for $\\ell\\ge3$ and a Poisson mixture with a lognormal rate for $\\ell=2$. These results pin down the sharp threshold for a large family of spanning structures in random hypergraphs.","feed_headline":"Random hypergraphs gain Hamilton cycles at the exact threshold","feed_subtitle":"The cycle appears as soon as the expected number of copies tends to infinity; for overlap 2 the count is lognormal.","key_machinery":"The argument runs through a refined second-moment calculation combined with small subgraph conditioning, a technique that removes the random fluctuation contributed by small subgraphs. The paper introduces the variable $X = \\frac{Z(C^{(r)}_{n,\\ell})}{E[Z(C^{(r)}_{n,\\ell})]}e^{-Y}$, where $Y=\\sum_{k\\ge1} t_k Y(P_k)$ is a weighted sum of normalized counts of short $\\ell$-paths $P_k$ in the random hypergraph, with $t_k=\\sqrt{A_k c^{-k}e^{-k(r-\\ell)}}/(r-\\ell)$. These short paths are precisely the small subgraphs whose random fluctuations inflate the ordinary second moment by a constant factor; multiplying by $e^{-Y}$ cancels that inflation and makes the second moment sharp. Under the original measure, the measure with one planted cycle, and the measure with two planted cycles of a given overlap, the vector of path counts is shown to be jointly Gaussian, which lets the paper evaluate the cancellation factor explicitly. For $\\ell\\ge3$ the corrected second moment is $1+o(1)$; for $\\ell=2$ it fails by a factor that the Gaussian structure turns into the lognormal law.","core_discovery":"On the paper's own terms, the central discovery is that the first-moment threshold is exact for Hamilton $\\ell$-cycles in the random $r$-uniform hypergraph $G_r(n,p)$ for all $r>\\ell>1$: whenever $E[Z(C^{(r)}_{n,\\ell})]\\to\\infty$, the probability of finding a copy of $C^{(r)}_{n,\\ell}$ tends to 1. More precisely, the normalized count $Z/E[Z]$ is $1+o(1)$ with high probability when $\\ell\\ge3$, and for $\\ell=2$ it converges in distribution to a lognormal random variable with explicit parameters built from the constants $A_k$ (the numbers of automorphisms of length-$k$ $\\ell$-paths). In the complementary regime where $E[Z]=m$ stays bounded, the count converges to a Poisson$(m)$ law for $\\ell\\ge3$ and to a Poisson mixture whose random rate is lognormal for $\\ell=2$. Together these statements characterize the limiting behaviour of the cycle count in every parameter regime.","pith_inferences":["This suggests that other spanning structures whose ordinary second moment is inflated by a family of small connected subgraphs may also have lognormal counts, and that the same $e^{-Y}$ correction could be used to expose them.","One testable extension is to check whether the lognormal behaviour at $\\ell=2$ persists for $\\ell=1$ (loose cycles), where the paper's assumptions exclude the case; the mechanism here is tied to the 'thick' two-edge overlap, so a different limit, perhaps a Poissonian one, is plausible.","The paper's method also yields a Poisson mixture at constant expectation, which suggests that in the $\\ell=2$ case the conditional law of the cycle count given the short-path fluctuations $Y$ is approximately Poisson, with the lognormal randomness entering only through $Y$."],"forward_implications":["The conjectured sharp threshold is confirmed: if $E[Z(C^{(r)}_{n,\\ell})]\\to\\infty$ then a Hamilton $\\ell$-cycle exists with probability tending to 1, with no extra logarithmic or constant factor.","For $\\ell\\ge3$ the cycle count is asymptotically deterministic relative to its mean whenever the mean diverges; at bounded expectation it is Poisson$(m)$, so the full distribution is now known.","For $\\ell=2$ the count is lognormal with explicit series parameters when the expectation diverges, and a Poisson mixture with lognormal rate when the expectation is constant; the lognormal occurrence is a direct consequence of the small subgraph conditioning correction.","The constants $A_k$ are explicit and uniformly bounded, so the lognormal variance and mixture rate can be evaluated numerically for any fixed $r$ and $\\ell$."],"supporting_citations":[{"why":"It supplies the sharp-threshold baseline and the conjecture that this paper confirms, and its second-moment estimates are refined to an exact constant.","marker":"[18]"},{"why":"It provides the orthogonal-decomposition central limit theorem for subgraph counts that underpins the null-model Gaussianity of the path counts.","marker":"[12]"},{"why":"It formulates the small-subgraph conditioning method that yields the lognormal correction for $\\ell=2$.","marker":"[13]"},{"why":"It introduces the conditioning idea that the paper adapts to the dense random-hypergraph setting.","marker":"[22]"}],"fun_headline_variants":["Exact threshold for non-linear Hamilton cycles in random hypergraphs","Overlap-2 Hamilton cycles: lognormal count revealed","First-moment threshold proven exact for hypergraph cycles","Hamilton cycles in random hypergraphs: Poisson and lognormal limits","Conjecture confirmed: exact threshold for non-linear Hamilton cycles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The lognormal conclusion for overlap size $\\ell=2$ depends on the claim that, even after planting one cycle or two overlapping cycles in the random hypergraph, the normalized counts of short paths still converge jointly to a normal distribution; the proof verifies only the first two moments of this limiting distribution.","fun_headline_variants_meta":{"raw":{"variants":["Exact threshold for non-linear Hamilton cycles in random hypergraphs","Overlap-2 Hamilton cycles: lognormal count revealed","First-moment threshold proven exact for hypergraph cycles","Hamilton cycles in random hypergraphs: Poisson and lognormal limits","Conjecture confirmed: exact threshold for non-linear Hamilton cycles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1251,"prompt_tokens":943,"completion_tokens":308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":225}},"tokens_in":559,"tokens_out":308,"duration_ms":3878,"temperature":1.0,"reasoning_tokens":225,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:26:09.859847+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the third and fourth moments of the normalized short-path count $Y(P_1)$ under the planted measure for a concrete parameter pair such as $r=3,\\ell=2$ and moderately large $n$; a deviation from the Gaussian values (third moment 0, fourth moment 3) beyond $o(1)$ would refute Lemma 3.2 and the lognormal limit it implies.","supporting_citations":[{"cited_title":"1, 2, 3, 5","cited_arxiv_id":null,"evidence_quote":"It supplies the sharp-threshold baseline and the conjecture that this paper confirms, and its second-moment estimates are refined to an exact constant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the orthogonal-decomposition central limit theorem for subgraph counts that underpins the null-model Gaussianity of the path counts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It formulates the small-subgraph conditioning method that yields the lognormal correction for $\\ell=2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It introduces the conditioning idea that the paper adapts to the dense random-hypergraph setting."}],"review_version":1}