{"id":"96f9e056-96e4-41f9-bcd4-4f8170ea7500","arxiv_id":"2607.24060","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For large random hyperbolic surfaces, the pointwise variance of a smoothed local Weyl law equals the Berry random-wave value, up to explicit O(1/(L²g)+1/g²) errors.","lead":"This paper computes the variance of smoothed eigenfunction densities on random large-genus hyperbolic surfaces, showing it matches Berry's random wave model. The result gives a precise asymptotic in terms of the energy τ and the window width L.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.18's replacement of the compactly supported ˆf factors by ˆf(ℓ)² requires an unproved uniform bound against the K-kernel; if the (y+v)/L error is not integrable, the main term gains an uncontrolled L-dependent correction.","rationale":"The reader's weakest assumption correctly identifies Lemma 4.18's stationary-phase/uniform-asymptotics step as the most load-bearing analytic point. My own tracing of the proof confirms that the main constant of Theorem 1.1 passes through this step: the replacement of the two ˆf factors by ˆf(ℓ)², the evaluation of the double integral via Lemma 4.4, and the Plancherel factor all produce the stated τtanhπτ/(4πL)||f||². The constants appear internally consistent; I also verified Lemma 4.4's s=0 value π³/4 is the correct limit of the s≠0 formula, so the reader's 'misprint' claim there is not correct. The genuinely missing piece is the explicit uniform error estimate for the y,v integration. The paper's phrase 'exponential decay in both y,v variables' is too terse: the K-kernel has a logarithmic singularity near v=0, and the transition from the compactly supported fhat domain to the infinite y,v integral needs a quantitative tail bound. My estimates suggest the error is controllable, but the printed proof does not demonstrate it; this is exactly the sort of gap that justifies a conditional verdict rather than acceptance. The final-line typo O(1/(Lg)) instead of O(1/(L²g)) in the proof of Theorem 1.1 should also be fixed. There is independent support from Mirzakhani's integration formula, standard volume estimates, and Monk--Thomas's tangle-free bound, all of which are used in a recognizable way; the novelty and structure of the argument are credible.","tokens_in":34984,"tokens_out":35519,"duration_ms":290100,"concrete_test":"For a concrete bump, e.g. ˆf(s)=(1−s²)²1_{[-1,1]}(s), evaluate numerically at τ=1 for L=10, 100, 1000 the error integral R(L)=∫∫∫ sinh²(Lℓ/2) A_{1,1}(Lℓ,y,v)[ˆf(ℓ+y/L)ˆf(ℓ+(y+v)/L)−ˆf(ℓ)²] cos(τv)dℓ dy dv, using the exact A expression (Lemma 4.14/4.17) with a cutoff near ℓ=0. If LR(L) does not tend to 0, the O(L^{-2}) claim fails. Analytically, prove J=∫∫ (y+v)K(√((e^y−1)/(e^{y+v}−1)))/√(e^{y+v}−1) dy dv <∞ using K(x)≤π/(2√(1−x²)) and splitting the quadrant into {y+v≤1}, {1≤y+v≤L}, {y+v≥L}; the last region must yield O(e^{-cL}), not merely O(1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 4.10 (Section 4.5, Lemma 4.18) extracts the main constant via the approximation ˆf((ℓ+y)/L)ˆf((ℓ+(y+v))/L) = ˆf(ℓ)² + O_f((y+v)/L), then integrates against K(√((e^y−1)/(e^{y+v}−1)))/√(e^{y+v}−1) over all y,v≥0. The paper asserts this error is O_{f,τ}(L^{-2}) after the 1/L prefactor, citing 'exponential decay' in y,v, but no uniform-in-L bound is written. Two concrete dangers: (i) near v=0 the K-kernel has a logarithmic singularity in y when y≫v, and the (y+v) weight must be checked in the corner y,v→0; (ii) the original fhat factors restrict the domain to y+v ≲ L, so passing to the infinite integral requires the tail y+v>L to contribute O(L^{-2}) after the 1/L prefactor. If the doubled error integral J(L)=∫∫ ((y+v)/L)K(...)/√(...) dy dv over the relevant domain were not O(1/L), the leading constant τtanhπτ/(4πL)||f||² would be contaminated. The printed proof also contains an error-term typo: the proof of Theorem 1.1 at the end of Section 3 states O(1/(Lg)) while the theorem statement and Proposition 3.3 give O(1/(L²g)). Note that Lemma 4.4's s=0 value π³/4 is consistent with the limit of π²/(4s)tanhπs, so I do not count that as a misprint.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the variance of a smoothed pointwise local Weyl law N(X,z)=Σ h(r_j)|φ_j(z)|² on Weil–Petersson random hyperbolic surfaces of large genus, with the basepoint z uniformly distributed. Theorem 1.1 claims that this variance is (4π(g−1))⁻¹ · τ tanh(πτ)/(πL) · ‖f‖²_{L²} plus O_{f,τ}(1/(L²g)) and O_{f,L,τ}(1/g²), in agreement with Berry's random wave model. The proof uses Selberg's pre-trace formula to write N as a constant plus a sum over based geodesic loops, splits loop pairs into diagonal and off-diagonal terms, evaluates the diagonal term via Mirzakhani's integration formula and a stationary-phase argument (Section 4), and bounds the off-diagonal term using new geometric results on length-minimising loops (Theorem 5.3, Theorem 5.5, Theorem 1.2) plus work of Monk–Thomas (Section 6). Appendix A checks compatibility with an i.i.d. Gaussian model.","tokens_in":35444,"tokens_out":14663,"duration_ms":127544,"significance":"If Theorem 1.1 is correct, it is a substantial parameter-free confirmation of Berry's random wave model for pointwise eigenfunction densities on large random hyperbolic surfaces. The leading constant is derived, not fitted, from Selberg's pre-trace formula and Mirzakhani's integration formula, and the Gaussian comparison in Appendix A is explicitly post-hoc, so it does not enter the main proof. The new geometric notion of length-minimising loops and the simplicity theorems are of independent interest. However, the proof as printed has a missing factor π in the central constant, a gap in the uniform-asymptotics step in Lemma 4.18, and an incorrect argument in the proof of Theorem 5.3; these issues are technical rather than conceptual and appear repairable.","major_comments":[{"comment":"The step replacing f̂((ℓ+y)/L) f̂((ℓ+(y+v))/L) by f̂(ℓ)² + O_f((y+v)/L) and then integrating against K(√((e^y−1)/(e^{y+v}−1)))/√(e^{y+v}−1) needs a uniform bound. The text cites 'exponential decay in both the y,v variables' for the integral of (y+v) times this kernel. Near v=0 the kernel is not exponentially decaying in v; it behaves like ~ const·log(1/v)/√(e^y−1), and the required O(1) bound for the error integral is not demonstrated. Please provide a complete estimate of J(L)=∫∫ ((y+v)/L) K(...)/√(...) dy dv over the relevant support, with explicit logarithmic integrability near v=0. This is load-bearing for Proposition 4.10 and hence for Theorem 1.1.","section":"§4.5, Lemma 4.18, around (4.20)"},{"comment":"There is a missing factor π in the main constant. Lemma 4.18 states ∫ F_{1,1}^{(0)} ... = τ tanh(πτ)/(4L) ‖f‖² + O(1/L²), but Proposition 4.10 and Theorem 1.1 require τ tanh(πτ)/(4πL) ‖f‖². The proof of Proposition 4.10 follows Lemma 4.18 and uses 1/(4L), so the printed chain of arguments would be off by π. The calculation after Plancherel and Lemma 4.4 gives the factor 1/(4πL), so this is a correction of a typo, but it must be fixed both in Lemma 4.18 and in the proof of Proposition 4.10.","section":"§4.5, Lemma 4.18 and proof of Proposition 4.10"},{"comment":"In the case p=z, the proof states δ = b·η₂ ∈ G, but the ALA decomposition gives δ = η₁·b·η₂, and η₂ is not equal to η. The conclusion can be repaired: η₁, b, and η₂ are all loops based at z of length strictly less than ℓ(δ), so each lies in G, and therefore δ ∈ G. As printed, however, the argument is incorrect. Since Theorem 5.3 underlies Theorem 1.2 and the off-diagonal estimate in Proposition 3.4, this proof must be corrected.","section":"§5.2, proof of Theorem 5.3"}],"minor_comments":[{"comment":"The error term is printed as O_{f,τ}(1/(Lg)) in the last display, while Theorem 1.1 and Proposition 3.3 give O_{f,τ}(1/(L²g)). This appears to be a typo.","section":"§3, proof of Theorem 1.1"},{"comment":"The phrase 'exponential decay in both the y,v variables' is misleading near v=0; the kernel is logarithmically integrable there rather than exponentially decaying. The estimate is plausible, but a precise split of the integration domain and a bound using Lemma 4.2 are needed.","section":"§4.5, proof of Lemma 4.18"},{"comment":"The notation N(X,z) is reused for the Gaussian surrogate model, creating possible confusion with the original local Weyl law. Consider using a different symbol, e.g. N_G(X,z).","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the central result is likely correct after the listed corrections. The missing π factor and the faulty p=z case in Theorem 5.3 are concrete and fixable. The uniform-bound gap in Lemma 4.18 is the main technical risk; if the author supplies the missing estimate, the proof should be complete. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives the first pointwise local-Weyl-law variance asymptotics for Weil–Petersson random surfaces. If Theorem 1.1 is right, it confirms Berry’s random wave prediction at the second-moment level. I think it is likely right, but the printed proof has a gap in the analytic core that needs to be filled before I would bet on it.\n\nWhat is genuinely new: the variance formula for N(X,z) with the explicit leading constant τ tanh(πτ)/(4π²(g−1)L) ‖f‖², the length-minimising loop theorem (they are simple and mutually trivially intersecting), and the exploring loops. The strategy is sound: Selberg pre-trace, split into diagonal and off-diagonal, Mirzakhani integration, stationary phase for the diagonal main term, and Monk–Thomas for the off-diagonal. The geometric part, especially the loop collar argument giving a lower bound on injectivity radius when two short primitive loops exist, is clean and clever. The paper is honest about using Appendix A only as a post-hoc consistency check, not as an input.\n\nThe soft spots are concentrated in Section 4.5. The step in Lemma 4.18 where Ŷ(ℓ + y/L)Ŷ(ℓ + (y+v)/L) is replaced by Ŷ(ℓ)² with error O((y+v)/L), then integrated against the elliptic kernel over y,v, is not fully justified. The paper asserts exponential decay in y,v, and the kernel does decay like e^{-(y+v)/2}, and the support of Ŷ forces y+v ≲ L, so the tail is exponentially small. Still, a uniform-in-L bound on the doubled error integral is not written down. I believe it is true and fillable, but as printed it is a real gap in the proof of the main term. The other flags are minor: the proof of Theorem 1.1 at the end of Section 3 writes O(1/(Lg)) where the statement and Proposition 3.3 give O(1/(L²g)) — clearly a typo. Lemma 4.4 is not misprinted: the s=0 value π³/4 is exactly the limit of the s≠0 formula. There are also small notation slips (e.g., H vs T in Lemma 4.5).\n\nOverall: this is a substantial, well-researched paper with a plausible main theorem and a novel geometric component. It deserves a serious referee. I would send it to review, and I would want the referee to push for a complete proof of the stationary-phase estimate in Lemma 4.18 before accepting.","headline":"A credible, significant computation of the pointwise local Weyl law variance on random hyperbolic surfaces, with a real but likely fixable gap in one stationary-phase step and several typos.","tokens_in":35892,"tokens_out":3194,"would_cite":true,"duration_ms":27937,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J50","30F60","53C22","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"For large random hyperbolic surfaces, the pointwise local Weyl law has variance (τ tanh(πτ))/(4π²(g−1)L)‖f‖², matching the random wave prediction.","keywords":["local Weyl law","random hyperbolic surfaces","Weil-Petersson","random wave model","eigenfunction variance","geodesic loops","length-minimising loops","pre-trace formula"],"falsifier":"Evaluate the remainder integral in Lemma 4.18 with the explicit error O_f((y+v)/L) and test numerically whether the y,v integral converges to O(1/L²); if it instead saturates at O(1/L), the leading constant is not established.","tokens_in":34890,"feed_emoji":"📐","tokens_out":10139,"duration_ms":78834,"temperature":0.7,"pith_summary":"This paper establishes a precise asymptotic for the variance of a smoothed local Weyl law on random hyperbolic surfaces of large genus. The variance is proportional to (τ tanh(πτ))/(4π²(g−1)L)‖f‖², exactly the value predicted by the random wave model, up to explicit error terms. The proof reduces the statistic to a sum over geodesic loops, identifies the dominant diagonal contribution, and shows that the remaining off-diagonal terms are negligible. A new geometric tool, the notion of length-minimising geodesic loops, is introduced, and it is proven that such loops are always simple and intersect trivially, which controls the off-diagonal terms. The paper also shows that if eigenfunctions were independent Gaussians, the variance would agree with the theorem up to the stated errors.","feed_headline":"Local Weyl law variance matches random-wave prediction","feed_subtitle":"On large random surfaces, the pointwise eigenfunction count has variance τ tanh(πτ)/(4π²(g−1)L)||f||².","key_machinery":"Three pieces carry the argument. First, a weight function H(ℓ) encodes the space-averaged diagonal contribution of closed geodesics of length ℓ, turning the diagonal term into a weighted sum over the length spectrum; an integration formula for Weil–Petersson volumes over moduli space then reduces the expectation to a single integral of H against a volume ratio. Second, an exact integral identity involving the complete elliptic integral K and the cosine transform evaluates that integral, producing the constant τ tanh(πτ). Third, length-minimising loops are geodesic loops that are shortest outside a given subgroup of the fundamental group; the paper proves every such loop is simple and that su","core_discovery":"Theorem 1.1 states that as the genus g tends to infinity, the variance over Weil–Petersson random surfaces of the smoothed eigenfunction count N(X,z) equals (1/(4π(g−1)))·(τ tanh(πτ)/(πL))‖f‖²_{L²(R)} plus error terms of size O_{f,τ}(1/(L²g)) and O_{f,L,τ}(1/g²). Equivalently, the fluctuations of the local Weyl law are those of a Gaussian random wave field. The proof splits the second moment of the oscillatory part into diagonal and off-diagonal pairs of geodesic loops; the diagonal pairs give the main term through an exact integral involving the complete elliptic integral of the first kind, and the off-diagonal pairs are shown to be O(1/g²) using new results on length-minimising loops.","pith_inferences":["The simplicity theorem for length-minimising loops likely extends to any negatively curved manifold, since its proof uses only the arc-loop-arc decomposition and length minimality; if so, the off-diagonal control would generalise.","A natural extension would be to let L grow with g, e.g. L ~ c log g; the current error O(1/(L²g)) suggests a window where the random wave prediction remains valid, beyond which the variance may change.","The proof leaves a potential gap: the stationary-phase/uniform-asymptotics step in Lemma 4.18 lacks an explicit uniform bound for the remainder integral; if that integral is not O(1/L²), the leading constant could acquire a logarithmic L-dependence.","The statement of Lemma 4.4 appears misprinted; the proof suggests the intended identity, but the main constant depends directly on it, so a corrected statement is needed before relying on the constant."],"forward_implications":["The pointwise local Weyl law on large random hyperbolic surfaces is asymptotically deterministic: its variance decays like 1/g, so the smoothed eigenfunction count concentrates on its mean.","If eigenfunctions are replaced by independent Gaussian variables of the same variance, the variance of the statistic agrees with the theorem up to the stated error terms, confirming the Gaussian wave picture for this statistic.","The length-minimising loop theorems supply a general method to control statistics that depend on pairs of short geodesic loops, which appear in other spectral and geometric questions.","The explicit constant gives a precise target for numerical simulation of eigenfunctions on random surfaces of large genus.","The exploring loops construction yields a sequence of simple, trivially intersecting loops that weakly fill the surface, providing a new topological decomposition."],"fun_headline_variants":["Random surfaces confirm Berry's random wave model","Exact variance for local Weyl law on large random surfaces","Length-minimizing loops sharpen eigenfunction count variance","Weyl law variance tied to random wave prediction","New proof links geodesic loops to Weyl law variance"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The theorem stands on the interchange in Lemma 4.18, where the product of two test functions is replaced by its value at ℓ with an error term that is then integrated against a kernel over all y,v ≥ 0; the paper does not supply a uniform-in-L bound for this error integral, and if that integral contributes at order 1/L rather than 1/L², the stated leading constant would acquire an uncontrolled L-dependent correction.","fun_headline_variants_meta":{"raw":{"variants":["Random surfaces confirm Berry's random wave model","Exact variance for local Weyl law on large random surfaces","Length-minimizing loops sharpen eigenfunction count variance","Weyl law variance tied to random wave prediction","New proof links geodesic loops to Weyl law variance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000162,"raw_usage":{"total_tokens":1093,"prompt_tokens":781,"completion_tokens":312,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":235}},"tokens_in":525,"tokens_out":312,"duration_ms":3768,"temperature":1.0,"reasoning_tokens":235,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T23:08:06.137187+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the remainder integral in Lemma 4.18 with the explicit error O_f((y+v)/L) and test numerically whether the y,v integral converges to O(1/L²); if it instead saturates at O(1/L), the leading constant is not established.","supporting_citations":[],"review_version":1}