{"id":"92cf15d8-04c2-49d7-a409-205384edb11a","arxiv_id":"2512.19441","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Imaginary Gaussian multiplicative chaos has Fourier dimension 1 - β² almost surely in the subcritical regime β ∈ (0,1).","lead":"The paper proves that the Fourier dimension of imaginary Gaussian multiplicative chaos on the unit circle equals 1 - β² almost surely for β in (0,1). A smart generalist might read it to see how the high-frequency decay of these random complex distributions is controlled by the parameter β in models from probability and physics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Extension of exact-kernel Coulomb-gas moments and Jack-polynomial asymptotics to covariances perturbed by regular functions lacks explicit error bounds for high-n Fourier decay.","rationale":"The reader correctly isolates the single assumption that bridges the exact-kernel calculations to the stated generality. All other steps (CLT for the exact case, failure to belong to the critical Sobolev space, white-noise limit) are logically downstream of this extension and do not introduce independent gaps.","tokens_in":1864,"tokens_out":435,"duration_ms":24521,"concrete_test":"Fix β=0.5 and a concrete regular perturbation R(θ,φ)=cos(θ−φ). Recompute the second-moment integral E[|M̂(n)|²] for n=10^3…10^4 by direct Monte-Carlo sampling of the perturbed field (or by expanding the Jack polynomials with the modified kernel) and test whether the ratio n^{1−β²} E[|M̂(n)|²] converges to the same explicit constant κ(β) obtained for the pure log kernel; a systematic drift larger than 5% falsifies the extension.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline result (Fourier dimension exactly 1−β² a.s. for the broad class) requires that the leading n^{-(1−β²)} scaling of |M̂(n)|², obtained from partitions with large gaps in the Jack expansion, survives when the covariance is log + R with R sufficiently regular. The abstract derives the moment identities from Coulomb-gas integrals that are specific to the pure log kernel; the perturbation enters the joint law of the field values and therefore perturbs every integral. If the error introduced by R produces a multiplicative factor n^{o(1)} or an additive lower-order term whose exponent differs by more than o(1), the optimal polynomial decay exponent changes. The paper states that the identities “extend,” but supplies no uniform control on the difference of the Pieri coefficients or the gap asymptotics under the perturbed measure, leaving the transfer of the exact exponent unverified.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that the Fourier dimension of imaginary Gaussian multiplicative chaos M_{iβ} = exp(iβ X), where X is a log-correlated Gaussian field, equals 1 - β² almost surely in the subcritical regime β ∈ (0,1). This holds for a broad class of fields whose covariance is the logarithmic kernel plus a sufficiently regular perturbation. For the exact logarithmic kernel on the circle, the paper additionally shows that M_{iβ} fails to belong to the critical Sobolev space H^{-β²/2} almost surely, establishes a central limit theorem in which n^{(1-β²)/2} M̂_{iβ}(n) converges in law to an isotropic complex Gaussian, proves joint convergence for finitely many consecutive coefficients to independent copies, and shows that the rescaled high-frequency content converges in H^s (s < -1/2) to complex white noise with explicit intensity κ(β) = (1/π) Γ(1-β²) sin(π β² / 2). The proof uses Coulomb-gas moment identities and asymptotic analysis of Jack-polynomial expansions, with leading contributions from partitions having large gaps.","tokens_in":2083,"tokens_out":665,"duration_ms":15382,"significance":"If the central claims hold, the work supplies a sharp, explicit description of the high-frequency Fourier decay of imaginary GMC, establishing white-noise behavior after optimal rescaling and furnishing an explicit intensity constant. The combinatorial approach via Jack polynomials and Pieri coefficients, which simplifies for large-gap partitions, is a technical strength that yields parameter-free leading asymptotics and connects GMC regularity to classical orthogonal-polynomial techniques.","major_comments":[{"comment":"Abstract and the section deriving the moment identities: the claim that the Fourier-dimension result extends to covariances differing from the exact logarithmic kernel by a regular function R rests on an unverified transfer of the leading n^{-(1-β²)} scaling. The Coulomb-gas integrals and Jack expansions are derived for the pure log kernel; the perturbation alters the joint law of the field values, and no uniform error bound is supplied showing that the difference in the relevant Pieri coefficients or gap asymptotics remains o(1) in the exponent for the high-n regime. Without such control, the optimal polynomial decay exponent could shift.","section":"Abstract and moment-identity derivation"}],"minor_comments":[{"comment":"The statement of the white-noise convergence in H^s for s < -1/2 would benefit from an explicit reference to the precise Sobolev norm used and a short remark on why the intensity κ(β) is independent of the particular regular perturbation.","section":"White-noise convergence statement"},{"comment":"Notation for the Fourier coefficients M̂_{iβ}(n) is introduced without an immediate reminder of the normalization convention (e.g., whether the circle is equipped with Lebesgue measure normalized to 1); a single clarifying sentence would aid readability.","section":"Notation paragraph"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to extend prior GMC regularity results; verify that the perturbation class is stated with sufficient precision to avoid overlap with existing works on approximate kernels."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and for identifying a point that requires clarification in the extension of our Fourier-dimension result to perturbed kernels. We address this major comment below and will incorporate additional justification in the revised manuscript.","responses":[{"response":"We agree that the explicit Coulomb-gas integrals and Jack-polynomial expansions are derived for the exact logarithmic kernel, and that the manuscript does not currently supply uniform error bounds quantifying the effect of a regular perturbation R on the leading asymptotics. The extension to covariances of the form log + R (with R sufficiently smooth) is justified heuristically by the fact that such an R induces a continuous additive perturbation to the underlying Gaussian field; for the high-frequency moments governing the Fourier dimension, this perturbation contributes only a multiplicative factor whose logarithm remains bounded uniformly in the large-n regime, preserving the exponent 1-β². Nevertheless, to make the argument fully rigorous, we will revise the manuscript by adding a dedicated subsection (or appendix) that derives explicit error estimates. These estimates will show that, for partitions with large gaps, the difference in the relevant Pieri coefficients and gap asymptotics is O(n^{-δ}) for some δ>0 independent of the partition, which is negligible compared with the leading exponential term. With this control, the optimal polynomial decay exponent remains unchanged. We thank the referee for highlighting the need for this quantitative transfer.","revision_made":"yes","referee_comment":"[Abstract and moment-identity derivation] Abstract and the section deriving the moment identities: the claim that the Fourier-dimension result extends to covariances differing from the exact logarithmic kernel by a regular function R rests on an unverified transfer of the leading n^{-(1-β²)} scaling. The Coulomb-gas integrals and Jack expansions are derived for the pure log kernel; the perturbation alters the joint law of the field values, and no uniform error bound is supplied showing that the difference in the relevant Pieri coefficients or gap asymptotics remains o(1) in the exponent for the high-n regime. Without such control, the optimal polynomial decay exponent could shift."}],"tokens_in":1634,"tokens_out":443,"duration_ms":30494,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key result is that for imaginary Gaussian multiplicative chaos with β in (0,1), the Fourier dimension is almost surely 1-β². For the exact log-correlated case it also fails to sit in the critical Sobolev space H^{-β²/2} and the rescaled Fourier coefficients converge in law to an isotropic complex Gaussian, with consecutive ones becoming independent; the whole thing behaves like white noise of intensity κ(β) = (1/π) Γ(1-β²) sin(π β² / 2) in negative Sobolev spaces.","headline":"The paper pins down the Fourier dimension of imaginary GMC to exactly 1-β² a.s. and shows the high-frequency coefficients converge to white noise with explicit intensity.","tokens_in":2558,"tokens_out":194,"would_cite":true,"duration_ms":22891,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Fourier analysis of imaginary GMC via Jack polynomials and Coulomb-gas moments shows no structural overlap with RS forcing chain","alignment":"orthogonal","rationale":"Paper derives Fourier dimension exactly 1−β² a.s. from moment identities (2.2) expanded in Jack polynomials P_λ, reduced via Pieri coefficients and large-gap asymptotics (Lemmas 3.3–3.5) to explicit Beta integrals yielding the exponent. RS theorems (reality_from_one_distinction, Jcost uniqueness via Aczél, phi_fixed_point, 8-tick/D=3 forcing in DimensionForcing.lean, AlexanderDuality) start from bare distinguishability and force reciprocal cost J(x)=½(x+x⁻¹)−1, golden-ratio ladder, and parameter-free constants; none of these appear or are paralleled in the GMC Fourier machinery. Domain (probability/harmonic analysis) lies outside RS structural canon.","tokens_in":59070,"confidence":"high","tokens_out":209,"duration_ms":8920,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Imaginary Gaussian multiplicative chaos has Fourier dimension exactly 1 minus beta squared almost surely for beta between 0 and 1.","keywords":["imaginary gaussian multiplicative chaos","fourier dimension","log-correlated fields","subcritical phase","fourier asymptotics","white noise convergence","sobolev regularity","jack polynomials"],"falsifier":"Numerical computation of the empirical decay rate of the squared Fourier coefficients for large frequencies in a discretized realization of the chaos at a fixed beta such as 0.5, checking whether the observed exponent is statistically consistent with 0.75.","tokens_in":2765,"feed_emoji":"","tokens_out":740,"duration_ms":23628,"temperature":0.7,"pith_summary":"The paper shows that the imaginary Gaussian multiplicative chaos, formally exp of i beta times a log-correlated Gaussian field on the circle, has Fourier coefficients whose squared magnitudes decay polynomially with a precise optimal exponent. That exponent equals 1 minus beta squared almost surely in the subcritical range. This controls the high-frequency regularity of the chaos and determines how its Fourier modes behave at large frequencies. The result extends from the exact logarithmic covariance to a broad class where the covariance differs by a regular function.","feed_headline":"Imaginary Gaussian chaos Fourier dimension equals 1 minus beta squared","feed_subtitle":"High-frequency coefficients of the chaos decay at this exact polynomial rate almost surely, matching the behavior of a scaled white noise.","key_machinery":"Moment identities obtained from Coulomb-gas integrals together with Jack-polynomial expansions, whose asymptotics are controlled by partitions with large gaps where the Pieri coefficients simplify to explicit leading terms.","core_discovery":"In the subcritical phase beta in (0,1), the Fourier dimension of M_i beta, defined via the optimal polynomial decay exponent of the squared Fourier coefficients, equals 1 minus beta squared almost surely. For the exact log-correlated field the chaos fails to lie in the critical Sobolev space H to the power of minus beta squared over 2, the rescaled coefficients converge in law to isotropic complex Gaussians, consecutive coefficients converge jointly to independent copies, and the rescaled and modulated chaos converges in negative Sobolev spaces to a complex white noise with explicit intensity kappa of beta.","pith_inferences":["The white-noise limit for high frequencies suggests that linear statistics of the chaos at fine scales behave like those of Gaussian white noise, which could be tested on other multiplicative chaos constructions.","The explicit intensity formula for the limiting noise may allow direct computation of the variance of integrals against test functions at high frequencies.","The result supplies a concrete rate that can be used to calibrate numerical schemes for sampling imaginary chaos or studying its multifractal spectrum."],"forward_implications":["The chaos lies outside the critical Sobolev space H to the power minus beta squared over 2 almost surely.","The rescaled Fourier coefficients converge in law to independent isotropic complex Gaussians.","The modulated high-frequency content of the chaos converges in negative Sobolev spaces to a complex white noise whose intensity is given explicitly by a Gamma function and sine expression.","The same dimension and white-noise limit hold when the covariance is perturbed by a sufficiently regular function."],"fun_headline_variants":["Imaginary chaos Fourier dimension equals 1 minus beta squared","Fourier dimension of imaginary chaos almost surely 1 minus beta squared","Imaginary Gaussian chaos has Fourier dimension 1 minus beta squared","Chaos Fourier dimension proven equal to 1 minus beta squared"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The covariance of the underlying log-correlated field differs from the exact logarithmic kernel by a sufficiently regular function, and the moment identities plus Jack-polynomial asymptotics extend to this wider class.","fun_headline_variants_meta":{"raw":{"variants":["Imaginary chaos Fourier dimension equals 1 minus beta squared","Fourier dimension of imaginary chaos almost surely 1 minus beta squared","Imaginary Gaussian chaos has Fourier dimension 1 minus beta squared","Chaos Fourier dimension proven equal to 1 minus beta squared"]},"model":"grok-4.3","cost_usd":0.011039,"raw_usage":{"total_tokens":4869,"prompt_tokens":853,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":110390500,"prompt_tokens_details":{"text_tokens":853,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3955,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":853,"tokens_out":61,"duration_ms":23103,"temperature":1.0,"reasoning_tokens":3955,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-16T20:26:59.871190+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical computation of the empirical decay rate of the squared Fourier coefficients for large frequencies in a discretized realization of the chaos at a fixed beta such as 0.5, checking whether the observed exponent is statistically consistent with 0.75.","supporting_citations":[],"review_version":1}