{"id":"d0edc3aa-4b5a-4a23-a673-a4dcaea5a8d4","arxiv_id":"2610.26575","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditionally on RH, per-zero exponential sums attached to automorphic L-function zeros have Gaussian pair-correlation limits and Laplace versus chi-squared triple-correlation limits depending on two scale parameters.","lead":"This paper studies the statistical distribution of exponential sums built from spacings between zeros of automorphic L-functions. Under the Riemann Hypothesis, these per-zero sums are shown to converge to Gaussian, Laplace, or chi-squared limits, a new layer of structure beyond classical pair-correlation results.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Advertised phase transition and surface are only proved in the α_j→0 limit; for fixed α, Theorem 1.6 covers at most finitely many moments, and for ζ at α1=α2=1/2 it covers none.","rationale":"Reader's verdict CONDITIONAL is appropriate. I did not find a defect in the moment computations as applied to the stated α_j→0 regime: the explicit-formula passage (Lemma 3.3), Landau–Gonek Corollaries 3.5/3.6, and the prime-power bounds in Lemma 4.4 are internally consistent for the stated ranges, and the construction of V_j in the proof of Theorem 1.1 is a legitimate way to make error terms negligible relative to the α_j-dependent normalization. The most load-bearing weakness is the mismatch between the unqualified 'surface/phase transition' language and the α_j→0 restriction. For fixed α, the moments that are actually proved are finite in number, and in the ζ case α1=α2=1/2 none are proved, so the distributional phase transition at fixed α is not a theorem. This is distinct from the reader's identified weakest assumption (RH), though both support CONDITIONAL: RH is explicit and cannot be removed, while the α→0 restriction limits the geometric interpretation advertised in the abstract. I also note two smaller presentation issues consistent with the same verdict: Lemma 8.6's displayed sign for ReSav appears to conflict with Corollary 8.4 and the centering in Theorem 1.5, and reference [4] lacks a usable URL while the 'Claude [1]' mention in Section 2 does not match reference [1]. None of these changes the verdict.","tokens_in":54303,"tokens_out":64780,"duration_ms":654586,"concrete_test":"Verify the scope claim by checking Theorem 1.6's hypothesis at the parameter point depicted in Figure 3: take m=1, π=ζ, θ_1=0, α1=α2=1/2. The condition 0<α1+α2<1/(m r(1+θ_m)) becomes 1<1/r, which has no solution in positive integers r, so Theorem 1.6 supplies no moment of Re bS at this fixed α. If the phase-transition statement is intended only in the α_j→0 regime of Theorem 1.7, the abstract should say so; if it is intended for fixed α, an additional argument covering this case is needed. This single check determines whether the advertised scope is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 1.1 and 1.7 are the distributional backbone of the paper. Both require α_j (and α_{1j}, α_{2j}) to tend to 0, with T_j chosen after α_j. The abstract and Section 1.5, however, present the Gaussian surface and the Laplace/chi-squared phase transition as features of fixed-α correlations. The gap is not cosmetic: for fixed α, Theorem 1.6 only controls moments of order r satisfying α1+α2 < 1/(m r (1+θ_m)). Thus for fixed positive α1, α2 the number of controlled moments is finite and can be zero. Taking ζ (m=1, θ_1=0), α1=α2=1/2, the hypothesis demands 1 < 1/r, so no moment of any order is proved; this is precisely the region near α2=1/2 shown in Figure 3. Consequently the claimed Laplace/chi-squared phase transition is not established at any fixed positive α: the limiting results describe only the α→0 endpoint, where the normalization depends on α_j and V_j is chosen after α_j. This is a scope gap between the advertised central claim and the proved statement, not an internal contradiction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper studies per-zero exponential sums S_pi(X,T,rho') attached to the zeros of automorphic L-functions and their normalized versions Q_pi, eQ_pi, and bQ_pi. Assuming the Riemann Hypothesis for L(s,pi), the authors compute even and odd moments of these sums over primes and prime powers (Theorems 1.2 and 1.5), and use the method of moments to prove Gaussian limits for the normalized per-zero sums along sequences alpha_j tending to 0, with T_j chosen after alpha_j (Theorem 1.1). Under a weaker zero-density Hypothesis Z_pi with B_pi<3, analogous results are proved for shifted sums at the short scale X=(log T)^{alpha m} (Theorems 1.3 and 1.4). For triple correlations, Theorem 1.6 gives moment asymptotics with Laplace moments when alpha_1 != alpha_2 and chi-squared_2 moments when alpha_1 = alpha_2, and Theorem 1.7 derives the corresponding distributional limits, again only along sequences alpha_{1j}, alpha_{2j} tending to 0. The paper also proposes Conjectures 3--5 asserting fixed-alpha Gaussian, Laplace, and chi-squared behavior.","tokens_in":56322,"tokens_out":1504,"duration_ms":67987,"significance":"If the endpoint statements are read as the main theorem, the paper contains a substantial and carefully executed calculation: the normalization constants are derived from prime-sum second moments rather than fitted, the limiting distributions are universal in pi, and the Laplace/chi-squared transition is a genuinely interesting new phenomenon. The authors are explicit about the Riemann Hypothesis and Hypothesis Z_pi assumptions, and the unconditional cases m=1,2 for Theorems 1.3--1.4 are correctly flagged. The main weakness is scope: the distributional theorems are proved only in the alpha_j -> 0 endpoint regime, whereas the abstract and parts of Section 1.5 present the fixed-alpha Gaussian surface and the Laplace/chi-squared phase transition as established facts. The fixed-alpha statements, including the numerically illustrated case alpha_1=alpha_2=1/2 for zeta, remain conjectural in the sense that only finitely many moments are controlled for any fixed positive alpha.","major_comments":[{"comment":"Theorem 1.6 as stated is internally inconsistent: it fixes alpha_1, alpha_2 and imposes 0<alpha_1+alpha_2<1/(mr(1+theta_m)) with r in the hypothesis, but Conclusion (2) then claims the formula for any r in N. For zeta (m=1, theta_1=0) and alpha_1=alpha_2=1/2 the hypothesis becomes 1<1/r, which no r satisfies, so no moment of any order is covered in exactly the region shown in Figure 3. The theorem should be reformulated with r fixed and the admissible range stated for that r, or the claim must be explicitly restricted to the moments for which the range condition holds.","section":"Section 1.5, Theorem 1.6","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The advertised Laplace/chi-squared phase transition is not established at any fixed positive alpha. Theorem 1.7 requires alpha_{1j}, alpha_{2j} decreasing to 0 and T_j >= V_j chosen after alpha_j; even the underlying Theorem 1.6, after repairing its statement, gives only moments up to an order depending on alpha_1+alpha_2. Consequently the sentence in Remark 1.5.1 that the distribution of Re bS_pi,psi(T^{alpha_1 m}, T^{alpha_2 m}, T, rho') 'exhibits the characteristics' of Laplace or chi-squared distributions, together with the abstract's claim of an 'unexpected phase transition', overstates what is proved. The fixed-alpha transition should be presented as a conjecture or as proved only in the alpha_j -> 0 endpoint regime, and the abstract should be reworded accordingly.","section":"Section 1.5 and Theorem 1.7","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The same scope gap occurs in the pair-correlation results. Theorem 1.1 is properly stated with alpha_j -> 0, and Remark 1.3.1 explicitly acknowledges that no fixed alpha is covered; however the abstract's statement that surfaces 'display Gaussian behavior' and Conjectures 3--4 concern fixed alpha in R\\{0}. Since Theorem 1.2 controls only moments with alpha < 1/(mr(1+(4/3)theta_m)), for any fixed positive alpha only finitely many moments are proved. The authors should make the distinction between the proved alpha_j -> 0 endpoint and the fixed-alpha conjectures prominent in the introduction and abstract, rather than leaving it to remarks placed after the main theorems.","section":"Abstract and Sections 1.3, 1.5","authors_acknowledge":"Remark 1.3.1.In some sense, Theorem 1.1 is as close as possible to Conjecture 3, while failing to prove it for any single fixed α.","authors_acknowledge_p":0.75}],"minor_comments":[{"comment":"The sentence 'where L_{2d} is given by (1.20)' should refer to (1.35), the definition of L_r.","section":"Proof of Theorem 1.6, page 43","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The notation defines Q_1 twice; the second factor should presumably be Q_2 = product_j q_{j2}.","section":"Equation (9.9)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The displayed CDF, written as (1/2) integral from 0 to lambda of e^{-u/2} du, evaluates to a negative number for lambda<0; the chi-squared_2 CDF should be 0 for lambda<0 and 1-e^{-lambda/2} for lambda>=0. The formula should be corrected.","section":"Theorem 1.7, part (2)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The 'stronger' near-alpha_1=alpha_2 statements at the thresholds |alpha_1-alpha_2| >= loglog T / log T and <= 1/((loglog T)(log T)) should be labelled as moment-range results for a fixed r, since they do not by themselves yield distributional limits at fixed alpha.","section":"Section 1.5, Remark 1.5.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The data companion is cited without a URL; if a stable link exists it should be included so the numerical evidence in Figures 1--4 is reproducible.","section":"Reference [4]","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The technical core appears sound and the endpoint theorems are meaningful. My main concern is that the abstract and Section 1.5 advertise fixed-alpha distributional behavior (Gaussian surface, Laplace/chi-squared phase transition) that the theorems do not prove, and Theorem 1.6 as stated contains an internal range inconsistency. These issues are fixable by rewriting the theorem statements and sharply separating proved endpoint results from conjectures; if the authors are unwilling to narrow the advertised claims, the overstatement would be grounds for rejection, but a careful revision should be sufficient."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The new object here is the per-zero exponential sum S_pi(X,T,rho'), and the paper proves, assuming RH, that its normalized fluctuations are Gaussian when the scaling exponent alpha_j tends to 0. The triple-correlation analogue is the real news: Laplace when the two exponents differ, chi-squared with two degrees of freedom when they match. That phase transition is not in Montgomery or Rudnick-Sarnak, and the moment computations behind it are substantial.\n\nCredit where due: the proof machinery is serious. The authors push Landau-Gonek formulas, handle prime powers carefully, and improve the needed second-moment asymptotics for Lambda_pi(n) beyond what Rudnick-Sarnak used. The universality in pi, traced to the shared second moment of the lambda_pi(p)'s, is well motivated and clearly explained. Theorems 1.2, 1.5, and 1.6 are explicit conditional statements, and the zero-density variants in Theorems 1.3 and 1.4 are a real addition, especially since they are unconditional for m=1,2.\n\nThe soft spot is the gap between what is advertised and what is proved. The abstract and Section 1.5 present the Gaussian surface and the Laplace/chi-squared transition as properties of fixed-alpha correlations. The theorems require alpha_j -> 0, with T_j chosen after alpha_j. The stress-test note is correct: for fixed positive alpha, Theorem 1.6 controls at most finitely many moments, and for zeta at alpha1=alpha2=1/2 it covers none. So the phase transition is established only at the alpha-to-0 endpoint, not at any fixed positive alpha. Remark 1.3.1 acknowledges this for pair correlation; Section 1.5 should say the same for triple correlation. I do not see an internal contradiction, just a real scope gap between the headline claims and the theorem statements.\n\nOne smaller issue: the numerical data companion [4] is cited as support, but it has no URL and Section 10.2 only points back to [4]. As submitted, the data is not independently accessible. That is fixable and should be fixed.\n\nWho should read this: analytic number theorists working on fine-scale zero statistics and comparisons with random matrix models. The paper deserves a serious referee. My recommendation is to send it onward, with the clear instruction that the authors align the abstract and Section 1.5 with the alpha-to-0 range and provide a working data link.","headline":"Per-zero zero-spacing sums give a genuinely new Gaussian surface under RH and a Laplace/chi-squared triple-correlation phase transition, but only in the alpha-to-0 regime; the fixed-alpha claims remain conjectural.","tokens_in":55173,"tokens_out":1967,"would_cite":true,"duration_ms":19395,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11M26","11F66","11M41","11L20"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Assuming the Riemann Hypothesis, per-zero sums over zeros of automorphic $L$-functions have a universal Gaussian limit, with a Laplace-to-chi-squared phase transition in triple correlations.","keywords":["Riemann zeta function","automorphic L-functions","non-trivial zeros","correlation surfaces","Gaussian distribution","Laplace distribution","chi-squared distribution","pair correlation"],"falsifier":"Using zeros from the paper's own computational companion, one can settle the main claims numerically. For $\\zeta(s)$ at a fixed scale such as $X=T$, compute $Q_\\pi(T,T,\\rho')$ over zeros up to $T=10^6$: Theorem 1.2 predicts the fourth centered moment of $\\mathrm{Re}\\,S_\\pi-\\alpha$ to be $3(\\alpha m\\log T/4)^2$ up to $O((\\log T)^{3/2})$, and the empirical CDF of $Q_\\pi$ should approach the standard normal. For the phase transition, compute the fourth empirical moment of the normalized triple sums $\\widehat{Q}_{\\pi,\\Psi}$ with $\\alpha_1,\\alpha_2$ clearly unequal and with $\\alpha_1=\\alpha_2$: the paper predicts 24 in the first case and 384 in the second, so values far from 24 at unequal scales, or far from 384 at equal scales, would falsify the corresponding law, and sweeping $\\alpha_2$ across $\\alpha_1$ tests the claimed transition window of width about $1/\\log T$.","tokens_in":55798,"feed_emoji":"📈","tokens_out":25446,"duration_ms":170289,"temperature":0.7,"pith_summary":"The paper studies, for an automorphic $L$-function $L(s,\\pi)$, the per-zero exponential sum $S_\\pi(X,T,\\rho'_\\pi)=\\sum_{|\\mathrm{Im}\\,\\rho_\\pi|\\le T}X^{\\rho_\\pi-\\rho'_\\pi}w(\\rho_\\pi-\\rho'_\\pi)$ with $w(u)=4/(4+u^2)$, whose phases are the spacings between a fixed zero $\\rho'_\\pi$ and its neighbours. The central claim is that, assuming the Riemann Hypothesis for $L(s,\\pi)$, the normalized values of these sums converge in distribution to the standard Gaussian as $\\rho'_\\pi$ varies over zeros up to height $T$: along any sequence $\\alpha_j\\searrow 0$ there are heights $V_j$ such that for $T_j\\ge V_j$ the quantities $Q_\\pi(T_j^{\\alpha_j m},T_j,\\rho')$ tend to $N(0,1)$, and the limit is universal in $\\pi$ (Theorem 1.1). For triple correlations the same construction splits: with two scales $X_1=T^{\\alpha_1 m}$ and $X_2=T^{\\alpha_2 m}$, the limit is Laplace(0,1) when $\\alpha_1\\ne\\alpha_2$ and chi-squared with two degrees of freedom ($\\chi^2_2$) when $\\alpha_1=\\alpha_2$ (Theorem 1.7), an unexpected phase transition driven by the diagonal combinatorics of the moment computation. If the theorems are right, the fine-scale statistics of zeros of every automorphic $L$-function collapse to one common object, the Pair Correlation Surface with height $g_\\pi(\\alpha,\\lambda)=(2\\pi)^{-1/2}e^{-\\lambda^2/2}$.","feed_headline":"Zeros of L-functions obey Gaussian, Laplace and chi-squared laws","feed_subtitle":"Pair sums trace a universal Gaussian surface; triple sums flip from Laplace to chi-squared at a sharp threshold.","key_machinery":"The load-bearing object is the explicit formula of Lemma 3.3, which converts the zero sum $S_\\pi(X,T,\\rho')$ into a sum over primes and prime powers, $S_\\pi(X,T,\\rho')=-\\sum_n a_n\\Lambda_\\pi(n)n^{-1/2-i\\gamma'}+\\frac{m}{2}\\log(|\\gamma'|+3)X+\\text{error}$, valid for all but $O_\\pi(\\sqrt{T}\\log T)$ zeros once RH holds. With zeros replaced by primes, the distribution of $S_\\pi$ is read off from mixed moments over primes, and Landau-Gonek type estimates (Corollaries 3.5 and 3.6) show that every off-diagonal term with unequal prime products is negligible; only the diagonal $P=Q$ survives. The diagonal contributes $d!(\\sum_{p\\le V}a_p^2|\\Lambda_\\pi(p)|^2/p)^d$, and Lemma 3.9 evaluates that prime sum as $\\alpha m\\log T/4+O(1)$, the single point where universality in $\\pi$ enters, since the second moment of the coefficient distribution is shared by all $\\pi$. Prime-power contributions are controlled by the admissible exponent $\\theta_m$ toward the Ramanujan conjecture together with Hypothesis $H_\\pi$, and the moment method with the Berry-Esseen theorem converts the moment estimates into the distributional limits. In the triple-correlation case the same diagonal analysis bifurcates: for $\\alpha_1\\ne\\alpha_2$ the relation $P_2Q_1=P_1Q_2$ forces $P_1=Q_1$ and $P_2=Q_2$ separately, producing the Laplace moments $r!$ for even $r$, while for $\\alpha_1=\\alpha_2$ all permutations among the four products contribute, producing the $\\chi^2_2$ moments $2^r r!$. The paper names the resulting object the Pair Correlation Surface, defined pointwise as $g_\\pi(\\alpha,\\lambda)=(2\\pi)^{-1/2}e^{-\\lambda^2/2}$.","core_discovery":"On its own terms, the paper discovers that the local configuration of zeros around a fixed zero, encoded in the exponential sum $S_\\pi(X,T,\\rho'_\\pi)$ with the weight $w(u)=4/(4+u^2)$, is statistically universal once the zero $\\rho'_\\pi$ is allowed to vary. Theorems 1.1 and 1.2 establish, under RH for $L(s,\\pi)$, that the centered real parts of $S_\\pi(T^{\\alpha m},T,\\rho')$ have even moments $\\mu_{2d}(\\alpha m\\log T/4)^d+O((\\log T)^{d-1/2})$ with $\\mu_{2d}=1\\cdot3\\cdots(2d-1)$, exactly the moments of a Gaussian of variance $\\alpha m\\log T/4$, whence the standard-normal limit; universality in $\\pi$ enters through the shared Rankin-Selberg second moment $\\sum_{n\\le X}|\\Lambda_\\pi(n)|^2/n=\\log^2 X/2+O_\\pi(1)$, which is identical for every $\\pi$. Theorems 1.3 and 1.4 replace RH by the zero-density hypothesis $Z_\\pi$ for the short-range regime $X=(\\log T)^{\\alpha m}$, where a secondary term $X^{-2}L'/L(\\rho'-2,\\pi)$ is shifted out and the variance scale becomes $\\log\\log T$; for $m=1,2$ these results are unconditional. Theorems 1.6 and 1.7 treat double sums over pairs of neighbouring zeros: the limiting law is Laplace(0,1) when $\\alpha_1\\ne\\alpha_2$ and $\\chi^2_2$ when $\\alpha_1=\\alpha_2$, the two regimes corresponding to whether the diagonal of the moment computation forces the prime products to match separately or allows full permutation symmetry.","pith_inferences":["One testable extension: at the critical window $|\\alpha_1-\\alpha_2|\\asymp 1/\\log T$ the moments should interpolate between $r!$ and $2^r r!$; the natural guess is a mixture of the Laplace and $\\chi^2_2$ laws whose mixing weight depends on $|\\alpha_1-\\alpha_2|\\log T$, a question the paper's Remark 1.5.2 leaves open.","The diagonal-counting mechanism suggests a general rule for higher-order correlation surfaces: the limiting law for sums over $k$-tuples of zeros should be indexed by the equivalence classes of the scales $\\alpha_i$ under additive relations, with the Gaussian, Laplace and $\\chi^2_2$ laws as the first three members of a family.","Conjecture 3 postulates the Gaussian limit at a single fixed $\\alpha$, whereas the theorems only reach it along sequences $\\alpha_j\\to 0$; the gap is the lack of uniformity in $\\alpha$, so an $\\alpha$-uniform error term in the moment estimates would make the Pair Correlation Surface literally the graph of the Gaussian density over $\\mathbb{R}^2$."],"forward_implications":["Universality of fine-scale statistics: if Theorem 1.1 is right, the per-zero sums of every automorphic $L$-function converge to one and the same Gaussian law regardless of the arithmetic of its coefficients, so the Pair Correlation Surface has the same shape for all $\\pi$.","A phase transition in the triple-correlation regime: the limit law jumps between Laplace(0,1) and $\\chi^2_2$ exactly according to whether $\\alpha_1=\\alpha_2$, so the relation between the two averaging scales, not the underlying $L$-function, decides the entire distribution.","The transition is quantitative: the paper proves Laplace behavior when $|\\alpha_1-\\alpha_2|\\ge\\log\\log T/\\log T$ and chi-squared behavior when $|\\alpha_1-\\alpha_2|\\le 1/((\\log\\log T)(\\log T))$, locating the switch in a window of width about $1/\\log T$ around equality.","For the Riemann zeta function and for $\\mathrm{GL}_2$ $L$-functions the short-range Gaussian of Theorems 1.3 and 1.4 holds unconditionally, so part of the Gaussian surface is verifiable without any unproved hypothesis.","Smooth compactly supported weight functions (Theorem 1.5) widen the admissible range of $\\alpha$ in the moment estimates, because their rapid decay removes the $\\xi>4/3$ truncation constraint forced by the weight $w(u)=4/(4+u^2)$."],"supporting_citations":[{"why":"Supplies the weight $w(u)=4/(4+u^2)$, the function $F(\\alpha)$, and the pair-correlation framework that the per-zero sums extend.","marker":"[41]"},{"why":"Gives the $n$-level correlation theory for automorphic $L$-functions, including Hypothesis $H_\\pi$ and the Rankin-Selberg second-moment mechanism behind universality.","marker":"[50]"},{"why":"Provides the explicit formulas for pair correlation that underlie the Landau-Gonek estimates in Corollaries 3.5 and 3.6.","marker":"[45]"},{"why":"Supplies the refined sums over zeros (Lemma 3.4) with the error terms needed in the moment computations.","marker":"[13]"},{"why":"Its zero-free region for $L(s,\\pi)$ lets Lemma 3.8 prove the $O_\\pi(1)$ second-moment asymptotic without the self-contragredient condition.","marker":"[24]"},{"why":"Gives the asymptotic $\\sum_{n\\le X}|\\Lambda_\\pi(n)|^2/n=\\log^2 X/2+O(1)$ used in Lemma 3.8.","marker":"[16]"},{"why":"Establishes the admissible exponent $\\theta_m$ toward the Ramanujan conjecture that controls the prime-power contributions.","marker":"[38, 39]"},{"why":"The moment-uniqueness theorem invoked in the proof of Theorem 1.7 to conclude the Laplace and $\\chi^2_2$ laws from their moments.","marker":"[6]"},{"why":"Proves the zero-density hypothesis $Z_\\pi$ for $\\mathrm{GL}_2$ forms, making Theorems 1.3 and 1.4 unconditional for $m=2$.","marker":"[5]"}],"fun_headline_variants":["L-function zero pairs trace Gaussian surfaces; triples flip from Laplace to chi-squared","Zeros of automorphic L-functions: Gaussian surfaces, then Laplace vs chi-squared","Gaussian surfaces and a Laplace-to-chi-squared transition in L-function zeros","Pair sums form Gaussian surfaces; triple sums switch laws at a sharp threshold"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The Riemann Hypothesis for $L(s,\\pi)$ is the load-bearing premise: Lemma 3.3 uses it to place every zero on the critical line when replacing $S_\\pi$ by its prime sum, and the Gaussian, Laplace and $\\chi^2_2$ limits of Theorems 1.1, 1.2, 1.5, 1.6 and 1.7 are all derived from that replacement, so a single zero off the line would undo the mean and variance normalization and every stated limit. Theorems 1.3 and 1.4 instead rest on the zero-density hypothesis $Z_\\pi$ with $B_\\pi<3$, which, as the paper notes, is known only for the Riemann zeta function, Dirichlet $L$-functions and $\\mathrm{GL}_2$ forms.","fun_headline_variants_meta":{"raw":{"variants":["L-function zero pairs trace Gaussian surfaces; triples flip from Laplace to chi-squared","Zeros of automorphic L-functions: Gaussian surfaces, then Laplace vs chi-squared","Gaussian surfaces and a Laplace-to-chi-squared transition in L-function zeros","Pair sums form Gaussian surfaces; triple sums switch laws at a sharp threshold"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000132,"raw_usage":{"cost":0.000132314,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000132314,"upstream_inference_prompt_cost":3.5854e-05,"upstream_inference_completions_cost":9.646e-05},"total_tokens":1624,"prompt_tokens":1094,"completion_tokens":530,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":875,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":456}},"tokens_in":1094,"tokens_out":530,"cost_source":"openrouter_usage","duration_ms":6666,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":456,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-20T08:31:41.513968+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Using zeros from the paper's own computational companion, one can settle the main claims numerically. For $\\zeta(s)$ at a fixed scale such as $X=T$, compute $Q_\\pi(T,T,\\rho')$ over zeros up to $T=10^6$: Theorem 1.2 predicts the fourth centered moment of $\\mathrm{Re}\\,S_\\pi-\\alpha$ to be $3(\\alpha m\\log T/4)^2$ up to $O((\\log T)^{3/2})$, and the empirical CDF of $Q_\\pi$ should approach the standard normal. For the phase transition, compute the fourth empirical moment of the normalized triple sums $\\widehat{Q}_{\\pi,\\Psi}$ with $\\alpha_1,\\alpha_2$ clearly unequal and with $\\alpha_1=\\alpha_2$: the paper predicts 24 in the first case and 384 in the second, so values far from 24 at unequal scales, or far from 384 at equal scales, would falsify the corresponding law, and sweeping $\\alpha_2$ across $\\alpha_1$ tests the claimed transition window of width about $1/\\log T$.","supporting_citations":[{"cited_title":"The Pair Correlation of Zeros of the Zeta Function","cited_arxiv_id":null,"evidence_quote":"Supplies the weight $w(u)=4/(4+u^2)$, the function $F(\\alpha)$, and the pair-correlation framework that the per-zero sums extend."},{"cited_title":"Zeros of Principal L-functions and Random Matrix Theory","cited_arxiv_id":null,"evidence_quote":"Gives the $n$-level correlation theory for automorphic $L$-functions, including Hypothesis $H_\\pi$ and the Rankin-Selberg second-moment mechanism behind universality."},{"cited_title":"Explicit Formulas for the Pair Correlation of Zeros of Functions in the Selberg Class","cited_arxiv_id":null,"evidence_quote":"Provides the explicit formulas for pair correlation that underlie the Landau-Gonek estimates in Corollaries 3.5 and 3.6."},{"cited_title":"On the Distribution of Imaginary Parts of Zeros of the Riemann Zeta Function. II","cited_arxiv_id":null,"evidence_quote":"Supplies the refined sums over zeros (Lemma 3.4) with the error terms needed in the moment computations."},{"cited_title":"Towards aGLn variant of the Hoheisel phenomenon","cited_arxiv_id":null,"evidence_quote":"Its zero-free region for $L(s,\\pi)$ lets Lemma 3.8 prove the $O_\\pi(1)$ second-moment asymptotic without the self-contragredient condition."},{"cited_title":"The prime number theorem and Hypothesis H with lower-order terms","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic $\\sum_{n\\le X}|\\Lambda_\\pi(n)|^2/n=\\log^2 X/2+O(1)$ used in Lemma 3.8."},{"cited_title":"Billingsley.Probability and measure","cited_arxiv_id":null,"evidence_quote":"The moment-uniqueness theorem invoked in the proof of Theorem 1.7 to conclude the Laplace and $\\chi^2_2$ laws from their moments."},{"cited_title":"A zero density estimate and fractional imaginary parts of zeros for GL2 L-functions","cited_arxiv_id":null,"evidence_quote":"Proves the zero-density hypothesis $Z_\\pi$ for $\\mathrm{GL}_2$ forms, making Theorems 1.3 and 1.4 unconditional for $m=2$."}],"review_version":1}