{"id":"110735a5-5e05-4dc5-a37c-32582ab55c51","arxiv_id":"2502.05106","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any sequence satisfying two axioms, long-term averages of its Montgomery-type form factor lie between explicit constants computed from reproducing kernels; the method refutes Gonek-Ki's conjecture for fixed c under RH.","lead":"The authors build a general framework that turns long-run averages of Montgomery-type pair-correlation form factors into two Fourier extremal problems, and they compute the relevant constants with reproducing-kernel Hilbert spaces. Applied to zeros of L-functions and of Re and Im zeta, the method gives explicit average bounds and shows that a 2018 conjecture of Gonek and Ki fails when the parameter c is fixed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified. The disproof of Gonek-Ki Conjecture 1 follows from A1/A2 plus the quoted Gonek-Ki asymptotic; the remaining unverified numerical supremum is not load-bearing for the central claim.","rationale":"The reader's conditional verdict is appropriate. The paper's main claim is the disproof of Gonek-Ki Conjecture 1 via long-interval averages of a form factor. This claim rests on A1 (counting function), A2 (weak-* convergence of FΓ dα), and the averaging mechanism of Theorem 1. A2 is supplied by the quoted theorem of Gonek-Ki under RH, so it is a known conditional input rather than a gap. The RKHS computations, while intricate, are not needed for the qualitative contradiction: Theorem 2's lower bound 1/2 already contradicts the conjecture's prediction of averages tending to 0, once A2 ensures bounded symmetric averages. The numerical constants in Corollary 10 (0.7467, 2.1659) are consistent with the stated theorems; a spot-check of the c3=0 case reproduces the expected Montgomery-style constants. The only unverified item, the supremum 0.586..., is used to establish a sufficient parameter range and is not load-bearing for the applications, which fall well inside the range. Overall, the argument is coherent and no fatal or even significant flaw was identified; the conditional verdict should remain unchanged.","tokens_in":28280,"tokens_out":42110,"duration_ms":392883,"concrete_test":"Re-derive the weak-* limiting measure for the Gonek-Ki form factor directly from Gonek-Ki Theorem 3 by integrating FΓ(α,T) against a smooth compactly supported test function concentrated near α=0; verify that the contribution of the term 2T^{-4|α|}log T is exactly a Dirac mass of coefficient 1 as T→∞. If the coefficient is not 1, A2 as stated in Corollary 10 would fail and the bounds would need revision; if it is 1, the contradiction with Conjecture 1 stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim — that Gonek-Ki Conjecture 1 fails for fixed c>0 — is supported by the paper's own Theorem 2 (which requires only A1 and a bounded symmetric average supplied by A2) and by Corollary 10's stronger quantitative bound. The proof of Theorem 1 is internally consistent: the convolution/averaging mechanism transfers control of ∫ ĝ dFΓ from the Fourier-support window to arbitrary long intervals, and the use of s0 is sound. A2 is indeed the hinge, but in the applications it is supplied by external theorems (RH plus Gonek-Ki Theorem 3, GRH plus Hypothesis MP, etc.), so it is not an unsupported assumption. The only item not fully derived is the numerical supremum sup_{t>0} (2−2cos t−2t sin t)/t² = 0.586..., used to set the range c2/c1 Δ² ≤ 5/3; however, all applications use parameter values well inside this range (e.g., Δ=1/2, c2/c1=1/4), so an error in the third decimal of this constant would not change any stated corollary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an axiomatic framework, Assumptions A1 and A2, for the form factors F_Γ(α,T) of a general family of sequences Γ(T), and proves that long-interval averages of F_Γ are controlled by the Fourier-optimization constant C_ν attached to the limiting measure ν. Theorem 1 gives the bounds 1+s0(C_ν−1)−ε+o(1) < (1/ℓ)∫_b^{b+ℓ}F_Γ dα < C_ν+ε+o(1), and Theorem 2 gives a universal lower bound of 1/2 via the complete solution of a second extremal problem (EP2). For measures of the form dν=c1δ+c2|α|e^{-c3|α|}dα, the paper constructs the reproducing kernel Hilbert space H_ν, computes K_ν(0,0) (Theorems 5 and 6), and derives explicit average bounds for zeros of Selberg-class L-functions, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, Corollary 10 yields a positive lower bound for averages of the form factor, which is incompatible with Gonek and Ki's Conjecture 1 when c>0 is fixed.","tokens_in":28526,"tokens_out":27812,"duration_ms":260024,"significance":"If correct, the paper provides a clean and broadly applicable transfer principle from Fourier optimization to pair-correlation averages, and the disproof of Gonek–Ki Conjecture 1 is a notable result. The solution of EP2 is complete and elementary, and the RKHS approach gives explicit, reproducible constants. The framework is transparently axiomatic: the burden is placed on external theorems that supply A2, and the paper is careful to state this dependence. The numerical lower bound 0.7467 for the averages attached to Re ζ and Im ζ is a concrete, falsifiable consequence, and the internal consistency of the numerical constants in Corollaries 8–10 is a strength.","major_comments":[{"comment":"The displayed formula for K_ν(0,0) is inconsistent with the rest of the paper as written. The formula appears to state K_ν(0,0)=√(2/(c1c2)) sin θ cos θ + θ sin θ, where θ=√(c2/(2c1))Δ. For Δ=1, c1=c2=1 this would give 1/K_ν(0,0)≈0.864, contradicting Theorem 2's lower bound of 1/2 and the known Montgomery–Carneiro–Milinovich–Ramos bounds for the zeta function. The numerical values in Corollaries 8–10 correspond instead to K_ν(0,0)=√(2/(c1c2)) sin θ / (cos θ + θ sin θ). Since Theorem 5 is the basis for the explicit constants in all applications, the displayed formula must be corrected and checked.","section":"§1.5, Theorem 5"}],"minor_comments":[{"comment":"The class ABL_Δ is used in the proof of Theorem 1 before being defined; please state explicitly that ABL_Δ consists of those f∈A_Δ with supp(f̂) compact.","section":"§2.1"},{"comment":"The numerical bound sup_{t>0}(2−2cos t−2t sin t)/t^2 = 0.586... is load-bearing for the admissibility range c2/c1 Δ^2 ≤ 5/3, but no proof or reference is supplied; include a short derivation or a citation.","section":"§3.1"},{"comment":"There is a typographical artifact in the definition of r(z): the formula ends with a stray semicolon, and the surrounding display would be clearer with consistent notation for the limit cases.","section":"§4.1, Theorem 13"},{"comment":"In the sentence introducing Gonek and Ki's Theorem 3, there is a stray '1.' after 'established the following'; this should be removed.","section":"§1.6.3"},{"comment":"The statement 'where the numbers η1 and η2 are roots of the equation ... such that η1+η2≠0' leaves the choice of roots ambiguous; the proof uses one root from each of the two pairs {±η1}, {±η2}, and this convention should be stated in the theorem.","section":"§1.5, Theorem 6"},{"comment":"The displayed lower bounds are difficult to parse because of missing parentheses and line breaks (e.g., '− 2m − 1 2m' should presumably be '− (2m−1)/(2m)'). Please typeset these expressions with clear fraction bars and parentheses.","section":"Corollaries 8 and 9"}],"recommendation":"major_revision","confidential_remarks":"The central argument appears sound, and the disproof of Gonek–Ki Conjecture 1 is convincing assuming the external theorems that provide A2. The main concern is the Theorem 5 formula, which as displayed is inconsistent with the paper's own corollaries; I have treated the text literally. If this is a conversion artifact rather than the actual arXiv formula, the correction is routine, but it must be fixed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the paper carefully. The main new thing is Theorem 1: from two axioms, one about the counting function and one about weak-* convergence of the form factor to a measure ν, you get explicit long-term average bounds for FΓ in terms of a Fourier optimization constant Cν. That's a genuine unification: Montgomery, Selberg-class, Dedekind zeta, and Gonek-Ki's Re/Im ζ cases all fall out of the same machine. The proof is a clean adaptation of the Carneiro-Milinovich-Ramos averaging mechanism, and I see no gap there.\n\nThe exact solution Dβ = β for the second extremal problem is a nice little result, and Theorem 2's unconditional 1/2 lower bound is a useful byproduct. The RKHS section is heavier. I was initially startled by the closed form in Theorem 5 because the text extraction shows a plus sign where a fraction bar belongs. Working from the boundary-value problem gives Kν(0,0) = √(2/(c1c2)) sin x / (cos x + √(c2/(2c1)) Δ sin x), with x = √(c2/(2c1)) Δ, and that exactly reproduces the constants in Corollaries 8 and 10 (2.1659 and 0.7467 for Δ=1/2, c1=c2=1). So the paper is internally consistent; someone with the PDF should just confirm the displayed formula isn't garbled in the original. The c3>0 formula in Theorem 6 is a serious ODE computation; I didn't verify every line, but the appendix's non-vanishing lemma covers the parameter range actually used, and the limiting cases are handled.\n\nThe refutation of Gonek-Ki Conjecture 1 is the headline. The limiting measure from their own Theorem 3 is δ + |α|e^{-4c|α|}, and Corollary 10 gives a lower bound that stays above 0.7467 for fixed c, while the conjectured expression averaged over long intervals [b,b+ℓ] tends to 0 as b or ℓ grows. The contradiction is real. It is conditional on RH because the input asymptotics are, but the conjecture lives in that same conditional world, so that's not a flaw.\n\nSoft spots: the supremum sup_{t>0}(2-2cos t-2t sin t)/t² = 0.586... is quoted without derivation. It's elementary and not load-bearing, since all applications have c2/c1 Δ² well below 5/3. Theorem 6's proof delegates substantial algebra to ODE solving; an independent check would be welcome, but I found no internal contradiction and the numerics match.\n\nI'd send this to a serious referee. It's a real contribution, not a routine extension. My own verdict would be conditional accept, pending a clean statement of Theorem 5 and a verification of the numerical constants in the c3>0 case.","headline":"Genuinely new axiomatic framework for form-factor averages, with a clean conditional refutation of Gonek-Ki Conjecture 1; the core transfer is solid and the paper deserves serious peer review.","tokens_in":29073,"tokens_out":26510,"would_cite":true,"duration_ms":224876,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A30","46E22","11M06","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Fourier-optimization framework bounds pair-correlation averages and refutes a 2018 conjecture in zeta-zero cases.","keywords":["pair correlation","form factor","Fourier optimization","reproducing kernel Hilbert space","Riemann zeta function","Selberg class","Dedekind zeta function","extremal problems"],"falsifier":"Numerically compute, for fixed $c>0$ and large $b,\\ell,T$, the averages $(1/\\ell)\\int_b^{b+\\ell}F_\\Gamma(\\alpha,T)\\,d\\alpha$ for zeros of $\\operatorname{Re}\\zeta$ or $\\operatorname{Im}\\zeta$ on the line $a=1/2-c/\\log T$, using rigorous high-precision zero data; if they converge to $0$ as the conjecture predicts, the lower constant $0.7467$ is false. A more direct check is to test (A2) itself by confirming whether $F_\\Gamma(\\alpha,T)\\,d\\alpha$ on $[-1/2,1/2]$ converges weak-* to $\\delta(\\alpha)+|\\alpha|e^{-4c|\\alpha|}d\\alpha$.","tokens_in":28073,"feed_emoji":"📈","tokens_out":10071,"duration_ms":95913,"temperature":0.7,"pith_summary":"The paper tries to convert pair-correlation questions for arbitrary sequences of real numbers into two extremal problems in Fourier analysis. Under a density axiom (A1) and a weak-* convergence axiom (A2) for the associated form-factor measure, it shows that long averages of the form factor are sandwiched between $1+s_0(C_\\nu-1)$ and $C_\\nu$, where $s_0=-0.217\\ldots$ is the minimum of $\\sin x/x$ and $C_\\nu$ is an optimization constant depending only on the limiting measure $\\nu$. For the measures that occur for zeta-family zeros, $d\\nu(\\alpha)=c_1\\delta(\\alpha)+c_2|\\alpha|e^{-c_3|\\alpha|}d\\alpha$, the paper computes an upper bound for $C_\\nu$ through reproducing-kernel Hilbert spaces, yielding explicit numerical intervals for Selberg-class zeros, Dedekind zeta zeros, and zeros of the real and imaginary parts of the Riemann zeta function. In the last case the long averages stay above $0.7467$, which contradicts the 2018 conjecture that they should tend to zero for fixed $c>0$. A second, exactly solved extremal problem gives a universal lower bound of $1/2$ for symmetric averages, independent of the limiting measure.","feed_headline":"Lower bound 0.7467 breaks a zeta-zero correlation conjecture","feed_subtitle":"A general Fourier-optimization framework traps long averages of zero-pair form factors with explicit numerical constants.","key_machinery":"The load-bearing object is the extremal constant $C_\\nu=\\inf_{g\\in A_\\Delta,\\,g(0)>0}\\Phi_\\nu(g)/g(0)$, where $A_\\Delta$ consists of continuous even functions $g$ with $g,\\hat g\\in L^1$, $g\\ge 0$, and $\\hat g(\\alpha)\\le 0$ outside $[-\\Delta,\\Delta]$, and $\\Phi_\\nu(g)=\\int_{-\\Delta}^{\\Delta}\\hat g(\\alpha)\\,d\\nu(\\alpha)$. For measures $\\nu=c_1\\delta+c_2|\\alpha|e^{-c_3|\\alpha|}d\\alpha$ supported in $[-\\Delta,\\Delta]$, the paper works in the reproducing kernel Hilbert space $H_\\nu$ of entire functions of exponential type at most $\\pi\\Delta$ with inner product weighted by $\\hat\\nu$; a reproducing kernel Hilbert space is one in which each point evaluation is a continuous functional, represented by a kernel function $K_\\nu$. Lemma 4 gives $C_\\nu\\le 1/K_\\nu(0,0)$, and Theorems 5 and 6 compute $K_\\nu(0,0)$ by solving the integral equation $(\\hat f*\\nu)=e^{-2\\pi i w\\xi}$ with ordinary differential equations. The auxiliary problem EP2, maximizing $g(0)$ subject to $\\hat g\\le \\chi_{[-\\beta,\\beta]}$ and $g\\ge 0$, is solved exactly by $D_\\beta=\\beta$ using Poisson summation, producing the universal $1/2$ lower bound.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1: any sequence satisfying (A1) and (A2) has form-factor averages obeying $1+s_0(C_\\nu-1)-\\varepsilon+o(1)<(1/\\ell)\\int_b^{b+\\ell}F_\\Gamma(\\alpha,T)\\,d\\alpha<C_\\nu+\\varepsilon+o(1)$ as $T\\to\\infty$, with $\\ell$ large depending on $\\varepsilon$ and, for the lower bound, on $b$. The constant $C_\\nu$ is defined by the extremal problem over even nonnegative test functions whose Fourier transform is nonpositive outside $[-\\Delta,\\Delta]$. For the applied measures $\\nu=c_1\\delta+c_2|\\alpha|e^{-c_3|\\alpha|}d\\alpha$ with $c_2\\Delta^2/c_1\\le 5/3$, the paper constructs the reproducing kernel Hilbert space $H_\\nu=PW(\\pi\\Delta)$ and proves $C_\\nu\\le 1/K_\\nu(0,0)$, then solves the kernel's differential equation to get explicit values. Feeding in previously established form-factor asymptotics for primitive Selberg-class functions, Dedekind zeta functions, and the real and imaginary parts of zeta, it obtains Corollaries 8 through 10. Corollary 10's lower bound $0.7467\\ldots$ is uniformly positive on long intervals, so the conjecture's predicted vanishing of the averages for fixed $c>0$ cannot hold.","pith_inferences":["Editorial inference: the same reproducing-kernel computation should apply, with the same explicit constants, to any sequence whose limiting measure has the form $c_1\\delta+c_2|\\alpha|e^{-c_3|\\alpha|}d\\alpha$, once an (A2)-type input is available.","Editorial inference: the universal $1/2$ lower bound suggests a rigidity of pair correlations that might hold under weaker hypotheses than (A2), perhaps whenever a positive proportion of the form factor's mass stays near $\\alpha=0$.","Editorial inference: a numerical experiment comparing the Re/Im zeta averages with the conjecture's predicted value would locate the breakdown, either in the conjectured form factor or in the RH-supplied input, and so would tell future work where to focus.","Editorial inference: optimizing the kernel parameter $c_3$ and the interval $[-\\Delta,\\Delta]$ is a natural next step, since the bounds are valid for all admissible parameters and their tightness varies with them."],"forward_implications":["Any sequence satisfying (A1)–(A2) has long form-factor averages trapped between $1+s_0(C_\\nu-1)$ and $C_\\nu$, forcing $C_\\nu\\ge 1$ as a necessary condition on any limiting measure.","For symmetric intervals, the density axiom alone forces long averages to be at least $1/2$, regardless of the size of the interval or the shape of the limiting measure.","For primitive Selberg-class functions of degree $m$, the averages are explicitly bounded in terms of $m$; for Dedekind zeta functions of abelian degree-$n$ fields, in terms of $n$.","For zeros of the real and imaginary parts of zeta with $a=1/2-c/\\log T$, the averages lie between $0.7467$ and $2.1659$, so the addressed conjecture cannot hold for fixed $c>0$; whether it can hold when $c=o(1)$ is left open."],"supporting_citations":[{"why":"introduces the form factor whose long averages are the paper's object of study.","marker":"[18]"},{"why":"supplies the averaging inequality (2.1) and the mechanism that Theorem 1 adapts to the axiomatic setting.","marker":"[5]"},{"why":"provides the Hilbert-space method, including Krein decomposition and the bound $C_\\nu\\le 1/K_\\nu(0,0)$, which Lemma 4 adapts.","marker":"[4]"},{"why":"defines the class $A_\\Delta$ of test functions used in the extremal problem EP1.","marker":"[7]"},{"why":"establishes that finite-length integrals of the zeta form factor are bounded, used to control remainder intervals.","marker":"[12]"},{"why":"provides the form-factor asymptotic for zeros of the real and imaginary parts of zeta that supplies (A2), and states the conjecture addressed.","marker":"[15]"},{"why":"supplies the Selberg-class form-factor asymptotic under GRH and the relevant coefficient hypothesis that gives (A2) for primitive elements.","marker":"[20]"},{"why":"provides the Dedekind zeta form-factor asymptotic under GRH that gives (A2) for abelian extensions.","marker":"[8]"}],"fun_headline_variants":["Zeta zero pair correlations: new lower bound kills conjecture","Fourier method yields 0.7467 bound that sinks Gonek-Ki","Explicit constant refutes zeta-zero correlation conjecture","Pair correlation form factors: new bounds rule out vanishing","0.7467 breaks Gonek-Ki: Fourier framework traps form factors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The hinge is Assumption (A2): the form-factor measures $F_\\Gamma(\\alpha,T)\\,d\\alpha$ must converge weak-* to a finite limiting measure $\\nu$; in the number-theoretic applications this convergence is imported from unproved input such as the Riemann Hypothesis, so if that input is wrong the bounds and the disproof of the conjecture do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Zeta zero pair correlations: new lower bound kills conjecture","Fourier method yields 0.7467 bound that sinks Gonek-Ki","Explicit constant refutes zeta-zero correlation conjecture","Pair correlation form factors: new bounds rule out vanishing","0.7467 breaks Gonek-Ki: Fourier framework traps form factors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2884,"prompt_tokens":976,"completion_tokens":1908,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":1819}},"tokens_in":592,"tokens_out":1908,"duration_ms":12113,"temperature":1.0,"reasoning_tokens":1819,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:16:56.860353+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically compute, for fixed $c>0$ and large $b,\\ell,T$, the averages $(1/\\ell)\\int_b^{b+\\ell}F_\\Gamma(\\alpha,T)\\,d\\alpha$ for zeros of $\\operatorname{Re}\\zeta$ or $\\operatorname{Im}\\zeta$ on the line $a=1/2-c/\\log T$, using rigorous high-precision zero data; if they converge to $0$ as the conjecture predicts, the lower constant $0.7467$ is false. A more direct check is to test (A2) itself by confirming whether $F_\\Gamma(\\alpha,T)\\,d\\alpha$ on $[-1/2,1/2]$ converges weak-* to $\\delta(\\alpha)+|\\alpha|e^{-4c|\\alpha|}d\\alpha$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the form factor whose long averages are the paper's object of study."},{"cited_title":"Carneiro, M","cited_arxiv_id":null,"evidence_quote":"supplies the averaging inequality (2.1) and the mechanism that Theorem 1 adapts to the axiomatic setting."},{"cited_title":"Carneiro, V","cited_arxiv_id":null,"evidence_quote":"provides the Hilbert-space method, including Krein decomposition and the bound $C_\\nu\\le 1/K_\\nu(0,0)$, which Lemma 4 adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the class $A_\\Delta$ of test functions used in the extremal problem EP1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes that finite-length integrals of the zeta form factor are bounded, used to control remainder intervals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the form-factor asymptotic for zeros of the real and imaginary parts of zeta that supplies (A2), and states the conjecture addressed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the Selberg-class form-factor asymptotic under GRH and the relevant coefficient hypothesis that gives (A2) for primitive elements."},{"cited_title":"Pair correlation for Dedekind zeta functions of abelian extensions","cited_arxiv_id":"1908.04876","evidence_quote":"provides the Dedekind zeta form-factor asymptotic under GRH that gives (A2) for abelian extensions."}],"review_version":1}