{"id":"ac8f221e-08e6-415b-9c02-02a2142ab7c1","arxiv_id":"2508.21691","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The subleading terms in the densities of low-lying zeros of holomorphic cusp newforms depend on the factorization and growth rates of the level, differing from the prime-level case only when a prime factor stays fixed.","lead":"The paper computes the lower-order terms of the 1- and 2-level densities of zeros of holomorphic cusp newforms, pushing the error down to O(1/log^4 R). It shows these terms depend on how the prime factors of the level N approach infinity, so the universal Katz-Sarnak main term hides arithmetic information in its subleading corrections.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unevaluated root-number average in Eq. (1.19) leaves the claimed O(1/log^4 R) computation of the 2-level density incomplete.","rationale":"The paper's central novelty is the claim of family-dependent lower-order terms in both the 1- and 2-level densities of holomorphic cusp newforms. The 1-level part (Theorems 4.1, 7.1, 7.2) appears internally coherent: the distinction between 'fixed q1' and 'both factors large' is driven by the elementary fact λ_f(q1)^2 = 1/q1 for q1 || N, and the error analysis there is plausible assuming the cited trace-formula estimates and the support restriction σ < 0.22. I do not see a fatal flaw in that portion. The 2-level part, however, contains a term visible in the paper's own decomposition (1.19) that is never computed: the harmonic average of the root number ε_f. Inclusion-exclusion forces this term, and it is not absorbed into Ak,N, S1, or S2. Theorems 7.3–7.6 supply formulas for the S2 components but not for the ε-average. Therefore the statement that D2 is determined up to O(1/log^4 R) is not established. This is load-bearing because it affects the core 2-level claim: if the ε-average has family-dependent lower-order terms, the classification in the main theorems would need revision. The reader's weakest assumption was placed on the ILS/Barrett trace-formula error terms; I agree those error terms deserve scrutiny, but the root-number average is a more direct gap: it is an entire term in the formula, not merely an error bound. A concrete check is to evaluate this average using the same Petersson trace formula already employed in Section 6; this is a standard computation (cf. root-number averages in [ILS00a] or [Mil04]) and should settle whether the omission is benign. If E_F = O(1/log^4 R), the paper's 2-level conclusions are essentially correct and the conditional acceptance stands; if not, the theorems need to be supplemented. This is not a rejection of the arithmetic phenomenon, but a precise, fixable gap in the proof as written.","tokens_in":55181,"tokens_out":8789,"duration_ms":102130,"concrete_test":"Compute the harmonic average E_F = (1/WR(F)) Σ_{f∈H_k^*(N)} wR(f) ε_f for the four level regimes. For N prime, ε_f is determined by the Atkin–Lehner eigenvalue at N (ε_f = −i^k η_f(N) up to a known factor), so use the Petersson trace formula (Propositions 2.3/2.4) to bound Σ f wR(f) λ_f(N). For N = q1 q2, express ε_f in terms of η_f(q1)η_f(q2) and compute the average with the same trace formula. For N = p^2, use Barrett et al. Prop 2.5. If E_F = O(1/log^4 R) in all four cases, the omitted term is harmless; if not, add the explicit E_F term to Theorems 7.1–7.6 and re-check the claimed universality breakdown.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equation (1.19) expresses D2(F,φ1,φ2) as the sum of the computed S1 and S2 terms plus φ1(0)φ2(0) × (Σ_{f∈F} wR(f)(1−ε_f))/(2 WR(F)). This last factor is the harmonic average of the root number ε_f, and it is nowhere evaluated or bounded in Sections 4–7. Theorems 7.1–7.6 give explicit formulas for SA′, SA, SB″, SB′, SBf, and SB∞, but never for (Σ_{f∈F} wR(f) ε_f)/WR(F). Since the advertised result is a computation of the 2-level density to O(1/log^4 R) error, an unevaluated term that could be as large as a constant independent of R invalidates the claimed precision. This is independent of the trace-formula error estimates: even granting Propositions 2.3–2.5 exactly, D2 is not determined to the claimed accuracy. The root-number average may well be O(1/log^4 R) for all four level regimes, but that must be proved. If it instead has family-dependent lower-order terms (e.g., a contribution proportional to log(q1)/log R in the fixed-q1 case), the classification in Theorems 7.3–7.6 would be incomplete and the 'breaking universality' conclusion could shift.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the weighted 1- and 2-level densities for families of holomorphic cusp newforms of level N, with the harmonic weight w_R(f)=Z_N(1,f)/Z(1,f). Four level regimes are considered: N prime, N=q_1q_2 with q_1 fixed and q_2→∞, N=q_1q_2 with both primes growing at complementary power rates, and N=p^2. The main technical content is a computation of the terms S_1(F,φ) and S_2(F,φ_1,φ_2) up to O(log^{-4}R), followed by explicit lower-order terms in Theorems 7.1–7.6. The advertised conclusion is that the lower-order terms agree with the prime-level case whenever both prime factors go to infinity, but differ when one prime factor is fixed, with an explicit formula such as SA'(F) = -2 log(q_1)/log(R) · φ̂(0)/(q_1^2-1) - log(q_1)/log(R) · φ̂''(0)/(q_1^2-1) + O(log^{-5}R).","tokens_in":55519,"tokens_out":10364,"duration_ms":126834,"significance":"If the derivation were complete, the paper would give a genuine sharpening of Miller's 2009 lower-order-term analysis and an explicit family-dependent breaking of universality in the 1- and 2-level densities. The manuscript has real strengths: the skeleton via the Petersson trace formula is standard but carefully executed, the support restriction σ<0.22 is explicit and used in the tail estimates, and the appendices contain detailed proofs of several auxiliary lemmas. However, the central 2-level density claim is not established as written because a term in Eq. (1.19) involving the harmonic average of the root number ε_f is never evaluated or bounded. In addition, there is a concrete error in the principal fixed-q_1 formula, Eq. (7.1)/(7.3). These are load-bearing issues, not presentation problems.","major_comments":[{"comment":"The claimed O(log^{-4}R) computation of the 2-level density is incomplete. Equation (1.19) expresses D_2(F,φ_1,φ_2) as the computed S_1 and S_2 terms plus φ_1(0)φ_2(0) · (Σ_{f∈F} w_R(f)(1-ε_f))/(2W_R(F)). This last factor is the harmonic average of the root number ε_f. Sections 4–7 compute SA′, SA, SB″, SB′, SBf, and SB∞, but never evaluate or bound Σ_f w_R(f)ε_f / W_R(F). Since ε_f=±1, this term can be a nonzero constant independent of R unless a separate argument is given. Even granting Propositions 2.3–2.5 exactly, D_2 is not determined to the claimed precision. The authors must prove that the root-number average is 1+O(log^{-4}R), or else compute its family-dependent lower-order terms and include them in Theorems 7.3–7.6.","section":"Eq. (1.19); Theorems 7.3–7.6"},{"comment":"The second term in Eq. (7.1) does not follow from the expansion in Eq. (7.3). Set a=log(q_1)/log(R). The contribution of φ̂''(0) to SA′(F) is -2a Σ_{r≥1} q_1^{-2r} · (1/2)φ̂''(0)(2ra)^2 = -4a^3 φ̂''(0) Σ_{r≥1} r^2 q_1^{-2r}, which is O((log q_1)^3/log^3 R). The paper instead writes -a φ̂''(0)/(q_1^2-1), which is O(log q_1/log R). The two differ by a factor of order (log q_1/log R)^2, and the claimed O(log^{-5}R) error cannot absorb the discrepancy. This is a central formula for the fixed-q_1 case; it must be corrected or the notation clarified.","section":"Theorem 7.1, Eq. (7.1)/(7.3)"},{"comment":"The four main theorems on S_2 are only given proof sketches. The generic tail bounds in Lemmas 7.7–7.8 are useful, but the substitutions from Lemmas 6.4–6.7 into SB′, SBf, and SB∞ involve many mixed sums over p_1,p_2, including fixed-q_1 exceptional terms with implied constants O_{q_1,k} in Lemma 6.5 (e.g., B′_{r_1,r_2}(q_1,p_2) for odd r_1). The manuscript does not verify that every such error is O(log^{-4}R). Since these theorems are the central results, the proof needs to be expanded or a precise case-by-case table supplied. As written, the reader cannot check the claimed uniformity in the q_1-fixed regime.","section":"Theorems 7.3–7.6, proof sketches"}],"minor_comments":[{"comment":"Eq. (1.11) states α_f(n)β_f(n)=1 for all n, which is not the standard multiplicative relation for composite n. For prime p, α_f(p)β_f(p)=1 is correct; for general n the Satake parameters are indexed by prime powers and one uses multiplicativity. Please rephrase.","section":"Section 1, Eq. (1.11)"},{"comment":"Typographical error: 'Peterson trace formula' should be 'Petersson trace formula'.","section":"Section 6, opening paragraph"},{"comment":"The constant γPNT3 is listed as 1+∫_1^∞ E(t)/t^2 dt ≈ -1.33258. Since this same constant is used again in Theorems 7.4 and 7.5, it would help to state once in Section 7 that E(t)=θ(t)-t and that all integrals over E(t) are convergent by the prime number theorem.","section":"Theorem 7.2, Eq. (7.7)"},{"comment":"The functions φ_1,φ_2 are consistently assumed to be even Schwartz with Fourier transforms supported in [-σ,σ], but the notation sometimes writes φ̂''(0) for the second derivative of the Fourier transform and sometimes φ̂(0); this is standard but the dependence on the original φ should be stated more explicitly in the theorem statements.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The root-number average in Eq. (1.19) is the single most important gap: without it the 2-level density formulas are incomplete, and the advertised 'breaking universality' conclusion for the 2-level density is not proven. The error in Eq. (7.1) is also concerning because it is in the flagship formula for the fixed-q_1 regime. Both issues are local in the sense that they may be repairable within the same framework, but the authors need to supply a genuine proof or rigorous bound for the root-number term and correct the Taylor expansion in Theorem 7.1 before a resubmission is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper visibly extends Miller's 1-level density work: it treats non-prime levels, adds the 2-level density, and sharpens the error to O(log^{-4}R), and it gives a clean candidate mechanism—whether the smallest prime factor of the level stays fixed controls oldform/newform balance and hence the lower-order terms. The computations are long but the skeleton (Petersson trace formula, explicit formula, Abel summation, Catalan-number expansions) is standard, and the fixed-q1 correction formulas look plausible.\n\nThe problem is that the advertised 2-level computation is incomplete. Equation (1.19) writes D2 as the sum of the computed S1, S2, A_{k,N} terms plus φ1(0)φ2(0) * (Σ_{f∈F} w_R(f)(1-ε_f))/(2 W_R(F)). That root-number average is never evaluated or bounded anywhere in Sections 4-7. Theorems 7.3-7.6 give explicit formulas for SB″, SB′, SBf, SB∞, but none of those is the root-number term. So even if every trace-formula estimate in the paper is granted, D2 is determined only up to an unknown quantity that could be as large as a constant independent of R; if it has family-dependent lower-order terms, the classification of regimes would shift. This is a genuine gap, not a nitpick. The reader's report noted the root average was 'not explicitly handled'; I think that is the correct reading, and it is load-bearing.\n\nOther soft spots are more ordinary. Theorems 7.3-7.6 are proof sketches rather than complete proofs, and the error analysis relies on several Appendix D lemmas; a referee would need to check that all error terms from ILS/Barrett et al. actually survive the repeated prime sums. There are also typos and notational slips—Theorem 7.4's statement mislabels SB″/SB′, and some numerical constants (γ2, γ3 in Theorem 7.6) are duplicated. These are fixable.\n\nCredit where due: the citation pattern is fine; the self-citations to Miller 2009 and Barrett et al. 2016 are to genuinely prior, independent derivations. The paper is honest about its reliance on those estimates.\n\nBottom line: I would send this to a serious referee, but not accept it as is. The 1-level part may well survive; the 2-level claim needs the root-number average either evaluated or bounded at O(log^{-4}R) in each level regime. That is an extra section, not a cosmetic edit.","headline":"A substantial extension of Miller's lower-order-term program, but the 2-level density claim is incomplete because Eq. (1.19)'s root-number average is never evaluated or bounded.","tokens_in":56051,"tokens_out":3720,"would_cite":false,"duration_ms":43910,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11F72","11M26","11M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that the lower-order terms in the 1- and 2-level densities of holomorphic cusp newforms are not universal: they depend on how the prime factors of the level N grow, with explicit new corrections when one prime factor stays","keywords":["low-lying zeros","1-level density","2-level density","holomorphic cusp newforms","lower-order terms","level aspect","Petersson trace formula","universality breaking"],"falsifier":"For N=2q with q a large prime, fixed weight k, and an even Schwartz function phi whose Fourier transform is supported in [-0.1,0.1], compute the weighted 1-level density via the explicit formula and the trace formula to precision 1/log^5 R. After subtracting the universal term from Theorem 7.2, the remaining coefficient of hat phi(0)/log R should equal -2 log 2 / (2^2 - 1) = -(2/3) log 2; any other value would refute the fixed-prime correction.","tokens_in":55122,"feed_emoji":"🔢","tokens_out":8593,"duration_ms":90726,"temperature":0.7,"pith_summary":"The paper studies the distribution of low-lying zeros of L-functions of holomorphic cusp newforms as the level N grows. It shows that the main term is universal in the sense of the density conjecture, but that lower-order terms through precision 1/log^4 R depend on how the prime factors of N diverge. In the prime-level case, and when N is a product of two primes both growing as powers of N, the lower-order terms agree; when one prime factor is fixed, explicit corrections of size log(q1)/log R appear. The same phenomenon is established for the 2-level density, computed by inclusion-exclusion from the 1-level density.","feed_headline":"A fixed small prime changes zero-density lower-order terms","feed_subtitle":"New corrections at scale 1/log^4 R distinguish cusp-newform families with the same main term.","key_machinery":"The machinery is the averaged explicit formula, in which n-level densities are rewritten as sums over primes and Hecke eigenvalues, together with weighted moment sums A'_r, A_r, B''_{r1,r2}, B'_{r1,r2}, B_{r1,r2}, evaluated by the trace formula and its extension to arbitrary level. Rational-function identities convert shifted moment sums M_{c,k}(p) into closed forms in p and lambda_f(p); the surviving universal terms are controlled by Euler-type constants and by Catalan-number coefficients. Whether the level-divisor moments A'_r and B'' survive determines the regime dichotomy.","core_discovery":"The central discovery is that the factorization of the level controls lower-order terms in the weighted density of low-lying zeros. In four regimes—N prime, N=q1q2 with q1 fixed and q2 growing, N=q1q2 with both primes growing as powers of N, and N=p^2—the main terms coincide, but the lower-order terms agree with the prime-level case only when no prime factor is small. If q1 is fixed, the 1-level density gains explicit corrections such as SA'(F) = -2 (log q1)/(log R) * hat phi(0)/(q1^2-1) - (log q1)/(log R) * hat phi''(0)/(q1^2-1) + O(1/log^5 R), and analogous corrections appear in the 2-level density. The paper identifies the oldform space as the mechanism: a small prime factor leaves oldfor","pith_inferences":["The dichotomy likely persists for higher n-level densities: the inclusion-exclusion argument makes the n-level error inherit the 1-level error, so the same factor-growth regimes should produce family-dependent lower-order terms at every level.","The fixed-q1 corrections are proportional to log(q1) and to 1/(q1^2-1), so comparing levels with different small fixed primes (say q1=2 and q1=3) would provide a clean numerical test of the formula's shape.","A sharper threshold between 'small prime at most a given size' and 'largest factor at least a fixed power smaller than N' may depend on the error exponent; intermediate regimes where the small prime grows like log N, log^2 N, or log^3 N could map the boundary.","The same framework could be adapted to other GL(2) families, or to varying weight with fixed level, where analogous lower-order terms may show a complementary dichotomy."],"forward_implications":["If correct, the lower-order terms can serve as arithmetic fingerprints: two families with the same symmetry type can be distinguished by their 1- and 2-level densities at scale 1/log^4 R.","The explicit fixed-prime formula gives testable predictions, e.g. SA'(F) = -2 log(q1)/log(R) * hat phi(0)/(q1^2-1) - log(q1)/log(R) * hat phi''(0)/(q1^2-1) + O(1/log^5 R).","The N=p^2 case joins the prime-level and two-growing-factors cases, so square levels do not break universality at this precision.","The conjectured generalization implies that only prime factors growing slower than the reciprocal of the target error create genuinely new lower-order terms.","Because the 2-level density error is dictated by the 1-level density through inclusion-exclusion, sharper 1-level control automatically improves n-level computations."],"supporting_citations":[{"why":"Supplies the trace-formula estimates for squarefree level (Propositions 2.3 and 2.4) used to evaluate the weighted moments and the sum of weights.","marker":"[ILS00a]"},{"why":"Supplies the generalized trace formula (Proposition 2.5) that removes the squarefree restriction, covering the N=p^2 regime.","marker":"[Bar+16]"},{"why":"Establishes the lower-order-term framework and the 1-level formula for prime level that this paper extends to 1/log^4 R and to composite levels.","marker":"[Mil09]"},{"why":"Provides the Petersson trace formula underlying the trace estimates used throughout the moment computations.","marker":"[Pet32]"},{"why":"Provides the Abel-summation technique and constants used to evaluate the prime sums in the universal terms.","marker":"[You05]"},{"why":"States the density conjecture whose main terms are the baseline the lower-order corrections are compared against.","marker":"[KS99a]"},{"why":"Gives the newform theory separating newforms from oldforms, which the paper identifies as the mechanism behind the regime dichotomy.","marker":"[AL70]"},{"why":"Supplies the bound |lambda_f(p)| <= 2 used to control error tails and justify the large-m truncations.","marker":"[Del74]"}],"fun_headline_variants":["Small prime factor in level breaks zero-density universality","Level's prime factorization dictates lower-order zero-density terms","Oldform space breaks universality of low-lying zero densities","Fixed small prime factor shatters lower-order term universality","Zero-density lower terms lose universality when level has small prime"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The argument assumes that the error terms in the trace-formula estimates for the weighted harmonic sums are small enough to survive the repeated sums over primes at precision 1/log^4 R; if a hidden dependence on k or N invalidates this, the regime classification collapses.","fun_headline_variants_meta":{"raw":{"variants":["Small prime factor in level breaks zero-density universality","Level's prime factorization dictates lower-order zero-density terms","Oldform space breaks universality of low-lying zero densities","Fixed small prime factor shatters lower-order term universality","Zero-density lower terms lose universality when level has small prime"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000941,"raw_usage":{"total_tokens":3907,"prompt_tokens":842,"completion_tokens":3065,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":2984}},"tokens_in":586,"tokens_out":3065,"duration_ms":23968,"temperature":1.0,"reasoning_tokens":2984,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:02:16.578404+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For N=2q with q a large prime, fixed weight k, and an even Schwartz function phi whose Fourier transform is supported in [-0.1,0.1], compute the weighted 1-level density via the explicit formula and the trace formula to precision 1/log^5 R. After subtracting the universal term from Theorem 7.2, the remaining coefficient of hat phi(0)/log R should equal -2 log 2 / (2^2 - 1) = -(2/3) log 2; any other value would refute the fixed-prime correction.","supporting_citations":[],"review_version":1}