{"id":"223dd5eb-bb08-42f8-a30e-f03f11103674","arxiv_id":"2512.13148","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A field-level Breuer-Major CLT is proposed via Wiener chaos, with applications to powers of the discrete GFF, but the odd-power GFF limit uses an incorrect normalization.","lead":"The paper states a lattice version of the Breuer-Major central limit theorem for nonlinear functionals of stationary Gaussian fields, proved with the fourth-moment theorem, and applies it to powers of the discrete Gaussian free field. Its main applications claim even powers become white noise and odd powers become a continuous Gaussian free field, but the odd-power theorem contains a scaling error that invalidates the stated result.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's scaling is off: Eq. (19) treats N^{-d} Σ G(i,j) as converging to the continuum Green covariance, but for G(i,j)~|i-j|^{2-d} this term is O(N^2); hence the odd-power GFF limit as stated cannot hold.","rationale":"The reader's weakest assumption exactly identifies the decisive flaw: the Riemann-sum identity in Eq. (19) is false for the discrete GFF because the double sum of the Green's function grows like N^{d+2} after the N^{-d} prefactor, leaving O(N^2). This invalidates Theorem 3, the paper's central advertised application. The even-power part may be recoverable from known results, and Theorem 1 may be repairable with a corrected C_m, but as stated the main claim of an odd-power continuous GFF limit does not follow. The reader's REJECT verdict is appropriate; my independent reading reaches the same conclusion, so the verdict should remain unchanged.","tokens_in":11376,"tokens_out":8649,"duration_ms":72238,"concrete_test":"Test the Eq. (19) identity analytically and numerically: take d=3, f≡1, and compute L_N = N^{-d} Σ_{i,j∈B_N} G(i,j) for N=64,128,256 using the known discrete Green's function. If L_N/N^2 converges to a positive constant rather than to 0, the asserted convergence to ∫∫ G_cont fails. Alternatively, re-derive the scaling: substitute G(i,j) = N^{2-d}|i/N - j/N|^{2-d} + lower order; then N^{-d} Σ f(i/N)f(j/N)G(i,j) = N^2 ∫∫ f(x)f(y)|x-y|^{2-d}dxdy + o(N^2), contradicting Eq. (19).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing flaw is in Section 1.3.2, Eq. (19), which claims that the linear term of the odd-power field has variance c_1^2 N^{-d} Σ_{i,j∈B_N} f(i/N)f(j/N)G(i,j) and that this converges to c_1^2 ∫∫ f(x)f(y)G_cont(x,y)dxdy. This is the Riemann-sum identity that carries Theorem 3. For the discrete GFF on Z^d, d≥3, G(i,j) ~ |i-j|^{2-d}. Taking f≡1, N^{-d} Σ_{i,j∈B_N} G(i,j) ≈ N^{-d} · (N^{2d} · N^{2-d}) = N^2 ∫∫ |x-y|^{2-d} dxdy, not O(1). More generally, N^{-d} Σ f(i/N)f(j/N)G(i,j) ≈ N^2 ∫∫ f(x)f(y)|x-y|^{2-d}dxdy, so the variance of the linear term diverges quadratically. The limit object in Theorem 3, a finite continuous GFF with covariance ∫∫ f G_cont g, would require an additional factor N^{-1} — i.e., normalization N^{-d/2-1} rather than N^{-d/2}. Because Theorem 3 is the paper's central advertised application and its proof rests directly on Eq. (19), the central claim is not supported. A separate normalization mismatch also appears in Theorem 1, where C_m uses Σ|ρ|^q while the variance is Σρ^q, but the odd-power scaling error is already decisive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a field-level Breuer–Major CLT for non-linear functionals of stationary Gaussian lattice fields. The proof represents H(X_j) via its Hermite expansion, proves a CLT for each Wiener chaos using the Nualart–Peccati fourth-moment theorem, and adds a tightness argument in Sobolev spaces H^{-α}(D). The main applications are to powers of the discrete Gaussian free field: even powers are claimed to converge to Gaussian white noise, while odd powers are claimed to converge to a continuous Gaussian free field with explicit covariance (17). The odd-power claim is the advertised central novelty. As written, the proof of that claim rests on a false Riemann-sum identity; in addition, the normalization in Theorem 1 is inconsistent with the variance computed in Proposition 1, and Lemma 4 contains a false eigenvalue asymptotic.","tokens_in":11782,"tokens_out":9719,"duration_ms":94667,"significance":"If the field-level formulation were correct, it would be a natural extension of the classical Breuer–Major theorem and would provide a unified proof of some known GFF fluctuation results, particularly for even powers. The exposition of the standard chaos machinery is clear, and the proof of Proposition 1 is a mostly standard contraction argument. However, the odd-power GFF theorem is the paper's principal new application and it is false as stated because of the scaling error in Eq. (19). The normalization mismatch in Theorem 1 also affects the correctness of the main theorem for signed covariances. The contribution as it stands is not established; substantial correction of the statements and proofs would be required.","major_comments":[{"comment":"The variance computation in Eq. (19) is not a Riemann-sum approximation. For the discrete GFF Green function on Z^d, G(i,j) ≍ |i-j|^{2-d}. Taking f≡1 gives N^{-d} ∑_{i,j∈B_N} G(i,j) ≍ N^{-d} · N^{d+2} = N^2, and for general f the expression grows like N^2 ∫∫ f(x)f(y)|x-y|^{2-d} dxdy. Hence the first term in (19) diverges, and no limit with covariance (17) is obtained at normalization N^{-d/2}; the required normalization for the linear term is N^{-d/2-1}. Since this identity is the structural premise of Theorem 3, the theorem as stated is false. Separately, the coefficient in Eq. (18), c_1 = E[X_o^{2p+2}], equals G(o,o)^{p+1}(2p+1)!!, not G(o,o)^p(2p+1)!!.","section":"§1.3.2, Eq. (19) / Theorem 3"},{"comment":"The normalization constant in Theorem 1 is inconsistent with the variance computation in Proposition 1. Theorem 1 defines C_m = ∑_{q≥m} q! c_q^2 (∑_u |ρ(u)|^q), but Proposition 1, Eq. (33), gives the limit variance q! c_q^2 (∑_u ρ(u)^q) ∫ f^2. Unless ρ≥0 pointwise, the normalized field has variance (∑ ρ^q)/(∑ |ρ|^q) ∫ f^2, not the claimed white-noise covariance ∫ f^2. The theorem needs an explicit nonnegativity assumption or a C_m built from ∑ ρ^q (with a separate positivity/convergence condition). As stated, the normalization cannot produce the claimed limit for signed covariances.","section":"Theorem 1 / Proposition 1"},{"comment":"The statement lim_{k→∞} λ_k/k^2 = 1 is false for the Dirichlet Laplacian on D ⊂ R^d with d≥2; Weyl's law gives λ_k ≍ k^{2/d}. The sup-bound (54) and the subsequent summability in (59) remain true for α>d/2 once the correct eigenvalue growth is used, so the tightness argument is repairable, but the displayed asymptotic is incorrect and must be fixed.","section":"Lemma 4 (tightness)"}],"minor_comments":[{"comment":"The sentence 'for f≡1, the last expression implies that ∑ ρ(u)^q > 0 for every N>0' is not justified. Positivity of the variance gives nonnegativity only of the weighted double sum, and the conclusion about the infinite sum is at best a limiting statement.","section":"Proposition 1, after Eq. (35)"},{"comment":"Absolute values on ρ should be introduced explicitly before bounding by Eq. (54); as written the step from ρ(j-ℓ)^q to |ρ(j-ℓ)|^q is implicit.","section":"Eqs. (57)–(59)"},{"comment":"Typographical issues: 'forth moment theorem' should be 'fourth'; 'reminder' should be 'remainder'. The proof of Theorem 3 is labelled a sketch, but the convergence of finite-dimensional distributions of L_N + R_N to the claimed GFF is asserted rather than demonstrated.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper advertises the odd-power GFF result as its main contribution, and that result is false as stated due to the scaling error in Eq. (19). The mismatch between C_m in Theorem 1 and the variance computed in Proposition 1 is a second, independent correctness issue. Even though the field-level Breuer–Major framework may be salvageable after corrections, the current manuscript's central advertised claims are not supported, and the required changes go beyond local revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead this one with a careful eye on the scaling. The paper's main claim, Theorem 3, says odd powers of the discrete GFF, normalized by N^{d/2}, converge to a continuous GFF with Green's covariance. That cannot be right. The linear term in the variance at Eq. (19) involves N^{-d} Σ f(i/N)f(j/N)G(i,j). For the GFF Green's function G(i,j) ~ |i-j|^{2-d}, that sum behaves like N^2, not O(1). So the normalization would need an extra factor N^{-1}; as stated, the variance diverges. The stress-test note is correct: the Riemann-sum identity in Section 1.3.2 is simply false for the discrete Green's function. This is not a minor gap; it is the load-bearing step that carries Theorem 3.\n\nCredit where it's due. The first part of the paper, the field-level formulation of Breuer-Major with tightness in Sobolev spaces, is a clean and honest repackaging of known tools. The proof via the Nualart-Peccati fourth-moment theorem is standard but competently executed. The even-power application (white noise for even powers of the GFF for d≥5) is a correct and useful observation, though it follows quickly from known results like [CHRR23] and the original Breuer-Major theorem.\n\nThere are two other soft spots worth flagging. Theorem 1 defines C_m with |rho(u)|^q but the limiting variance in Proposition 1 uses rho(u)^q. If the covariance is signed, those don't match, and the theorem as stated is false; the paper should either assume rho≥0 or replace the variance with the absolute-value version. And Lemma 4 states lim λ_k/k^2 = 1, which is the one-dimensional Weyl law; in d dimensions the eigenvalues grow like k^{2/d}. That's a careless misstatement, though the subsequent bound may still hold for the intended α > d/2, so it might not break the tightness argument.\n\nOverall, the paper does a decent job of reviewing and repackaging known material, but its advertised new application is incorrect, and the general theorem has an unaddressed sign issue. A serious referee would catch these, and the paper deserves that scrutiny, but as it stands it's not publishable without a substantial revision that either fixes the odd-power normalization or drops that claim entirely.\n\nRecommendation: reject in current form, but encourage the authors to resubmit with the odd-power section corrected or removed, and with the C_m sign issue resolved.\n\nWould I bring it to reading group? Maybe, as a cautionary example of how a scaling error can look like a Riemann sum. Would I cite it? Not now.\n\nBest,\n[Your name]","headline":"The field-level Breuer-Major packaging is tidy, but the advertised odd-power GFF limit is off by a factor of N in the normalization, so the central application fails.","tokens_in":12260,"tokens_out":2493,"would_cite":false,"duration_ms":23513,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G60","60G15","60F05","60F17","33C45"],"pacs":[],"model":"deepseek-v4-flash","headline":"A field-level central limit theorem turns nonlinear functionals of stationary Gaussian lattice fields into Gaussian white noise, with an application to powers of the discrete Gaussian free field.","keywords":["Wiener chaos expansion","Hermite rank","central limit theorem","Gaussian white noise","discrete Gaussian free field","continuous Gaussian free field","fourth moment theorem","Sobolev spaces"],"falsifier":"Compute the left side of equation (19) for f ≡ 1 and H(x) = x^3 on a d = 3 box: using G(i,j) ~ c|i-j|^{2-d}, the leading term c_1^2 N^{-d} sum_{i,j in B_N} G(i,j) diverges like N^2, so the variance of N^{-d/2} sum_{j in B_N} X_j^3 grows with N; observing this growth for increasing N refutes the claimed convergence to a continuous Gaussian free field.","tokens_in":11220,"feed_emoji":"🎲","tokens_out":11179,"duration_ms":92879,"temperature":0.7,"pith_summary":"The paper sets out to prove a field-level central limit theorem for nonlinear functionals of stationary Gaussian fields on the integer lattice, generalizing the classical scalar CLT for such functionals. Its main theorem states that if the function H has Hermite rank m and the field's covariance is absolutely summable at power m, then the normalized field N^{d/2} Phi_N converges in the Sobolev space H^{-alpha}(D) to Gaussian white noise on the unit cube D. The proof works directly on the field: each Hermite component of H(X_j) is written as a multiple stochastic integral, the fourth moment theorem forces each component to a Gaussian limit, and a separate tightness estimate in H^{-alpha}(D) upgrades convergence of observables to convergence of the field. The paper applies this machinery to powers of the discrete Gaussian free field: even powers have Hermite rank 2 and converge to white noise once the covariance is square-summable (d >= 5, or d >= 2 for gradients), while odd powers, whose linear Hermite term cannot be removed, are claimed to converge to a continuous Gaussian free field with explicit Green-function covariance. The odd-power result rests on a Riemann-sum approximation of the discrete Green's function.","feed_headline":"A white-noise limit theorem for nonlinear Gaussian lattice fields","feed_subtitle":"Even powers of the discrete Gaussian free field converge to white noise; odd powers are claimed to reach a continuous free field.","key_machinery":"The mechanism is the Wiener chaos decomposition: H(X_j) is expanded into Hermite polynomials H_q(X_j), and each term is represented as a multiple stochastic integral I_q(u_j^{⊗q}) with respect to the isonormal process W over the Hilbert space with inner product rho(j-k). The proof then invokes the fourth moment theorem, which says that for a sequence of variables living in a single Wiener chaos, convergence to a Gaussian is equivalent to the vanishing of the contractions f_N ⊗_r f_N; the paper checks this for the normalized kernels s_{N,q}(f) by splitting the covariance sum into small and large scales and applying a standard norm inequality. Tightness in H^{-alpha}(D) is obtained by expandin","core_discovery":"On the paper's own terms, the central discovery is that a white-noise limit can be obtained simultaneously for all chaos components of a nonlinear Gaussian lattice field, not just for scalar sums. Concretely, for a stationary unit-variance Gaussian field X on Z^d and a function H with Hermite expansion starting at order m, the condition sum_u |rho(u)|^m < infinity implies N^{d/2} Phi_N converges in law to W in H^{-alpha}(D) for every alpha > d/2, where W is Gaussian white noise on D = [-1/2,1/2]^d. The paper further claims that for the discrete Gaussian free field, H(x) = x^{2p} falls under this theorem in d >= 5, whereas H(x) = x^{2p+1} escapes it because the covariance is not summable, and","pith_inferences":["A direct computation of the leading term in the paper's key variance identity for odd powers and f ≡ 1, using the discrete Green's function G(i,j) ~ |i-j|^{2-d}, gives growth of order N^2 rather than convergence; if this is correct, the odd-power claim needs a different normalization or a different limiting object.","The same contraction technique could yield quantitative rates of convergence for the even-power white-noise limit by tracking how fast the fourth-moment gap decays.","For d = 3 and d = 4, even powers fall outside the theorem because sum_u |rho(u)|^2 diverges; whether a different scaling produces a Gaussian or non-Gaussian limit is left open."],"forward_implications":["The field-level statement is stronger than a scalar CLT: for any finite collection of test functions, the vector (⟨Phi_N,f_i⟩) converges to a Gaussian vector with covariance diagonal in Hermite order, and disjoint-box indicators give independent standard normal limits.","Even powers of the discrete Gaussian free field, normalized by N^{d/2}, converge to white noise for d >= 5; the gradient version extends the result to every d >= 2.","Odd powers of the discrete Gaussian free field are claimed to have a continuum Gaussian free field limit whose covariance kernel is the continuous Green's function; the linear Hermite term dominates and the higher-order Hermite terms vanish in the limit.","The tightness bound in H^{-alpha}(D) shows the convergence holds as random distributions, so the limit statement applies to nonlinear functionals evaluated against Sobolev test functions."],"fun_headline_variants":["Even powers of discrete GFF converge to white noise","Wiener chaos yields new CLTs for lattice Gaussian fields","GFF powers: even to white noise, odd to continuous field","Nonlinear Gaussian lattice fields: a chaos-based CLT","Discrete GFF: even powers white noise, odd powers continuous"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The odd-power theorem depends on the assumption that the Riemann sum c_1^2 N^{-d} sum_{i,j in B_N} f(i/N)f(j/N) G(i,j) converges to c_1^2 int int f(x) G_cont(x,y) g(y) dx dy; for the discrete Gaussian free field Green's function in d >= 3 the sum grows like N^2, so this premise is not met.","fun_headline_variants_meta":{"raw":{"variants":["Even powers of discrete GFF converge to white noise","Wiener chaos yields new CLTs for lattice Gaussian fields","GFF powers: even to white noise, odd to continuous field","Nonlinear Gaussian lattice fields: a chaos-based CLT","Discrete GFF: even powers white noise, odd powers continuous"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000181,"raw_usage":{"total_tokens":1120,"prompt_tokens":698,"completion_tokens":422,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":442,"tokens_out":422,"duration_ms":5176,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:30:41.157002+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the left side of equation (19) for f ≡ 1 and H(x) = x^3 on a d = 3 box: using G(i,j) ~ c|i-j|^{2-d}, the leading term c_1^2 N^{-d} sum_{i,j in B_N} G(i,j) diverges like N^2, so the variance of N^{-d/2} sum_{j in B_N} X_j^3 grows with N; observing this growth for increasing N refutes the claimed convergence to a continuous Gaussian free field.","supporting_citations":[],"review_version":1}