{"id":"152eddcf-d8e8-4487-b0f6-b0409f6fee06","arxiv_id":"2601.11213","paper_version":3,"verdict":"REJECT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"Light propagation through non-Markovian random media is mapped to the hyperbolic Anderson model, with predicted moment and aperture-averaging scalings and outdoor experimental claims.","lead":"The paper maps light traveling through a medium whose random fluctuations carry long-lasting memory to the hyperbolic Anderson model, then derives scaling laws for intensity fluctuations and tests them on a 588-meter outdoor laser link. If the mapping is correct, it gives a quantitative handle on how atmospheric memory degrades or averages away in free-space optical communication and imaging.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central 'exact mapping' to the hyperbolic Anderson model is not derived: Eq.(3) follows from Eq.(2) only if ∂²_t u ≈ -ω²u is substituted in the noise term but not in L_cn; a consistent reduction gives an elliptic Helmholtz or parabolic Schrödinger equation, so the SPDE results (Eqs. 12–16) lack","rationale":"The reader's weakest-assumption analysis focuses on Eq.(4), the factorized covariance model, and notes that the exact scaling relations depend on it. That is a legitimate concern about the noise model. However, an even more fundamental issue is whether the governing equation Eq.(3) is actually the hyperbolic Anderson model. The derivation of Eq.(3) from Eq.(2) is not exact and appears internally inconsistent: the same second time derivative is treated as -ω²u in the perturbation term but retained as a second-order derivative in the operator L_cn. A consistent asymptotic reduction—whether monochromatic or slowly varying envelope—leads to an elliptic or parabolic equation, not a hyperbolic one. If the equation of motion is not the hyperbolic Anderson model, the SPDE results from the literature cannot be imported, and the 'exact scaling relations' and aperture-averaging bound lose their theoretical foundation. This concern is independent of any missing SI or experimental parameters; it is a mathematical property of the printed equations. The check I propose would settle the question directly by deriving the asymptotic equation from Eq.(2) under clearly stated scaling assumptions. If the derived equation turns out to be hyperbolic, my concern is resolved; if it is parabolic or elliptic, the central claim fails. Because the reader already recommends rejection and my concern strengthens that conclusion without changing the verdict, no verdict adjustment is needed.","tokens_in":9930,"tokens_out":13017,"duration_ms":129088,"concrete_test":"Insert the monochromatic ansatz u(t,x)=e^{-iωt}ψ(x) into Eq.(2) and derive the equation for ψ. This yields the random Helmholtz equation Δψ + (ω²/c²)(E[n²]+μ)ψ = 0, which has no time evolution and does not match the hyperbolic Anderson model. Separately, perform a two-scale expansion with A(T,x), T=εt, and keep terms to leading order in the slow time scale; show whether the resulting PDE for A is parabolic (Schrödinger) or retains a second-order time derivative. If the second-order derivative is negligible, Eq.(3) is not the correct asymptotic model; if it is retained, the replacement ∂²_t u ≈ -ω²u is invalid under the same scaling. Either outcome undermines the claimed exact mapping.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's linchpin is the transition from Eq.(2), L_cn u + (μ/c²)∂²_t u = 0, to Eq.(3), L_cn u = W u with W = μω²/c². The stated justification is that 'the time derivative of the phase fluctuations of light fields induced by random media is significantly lower than the optical angular frequency,' so ∂²_t u is replaced by -ω²u. But ∂²_t u also appears in L_cn. If the substitution is applied consistently, L_cn u becomes (-E[n²]ω²/c² - Δ)u, and Eq.(3) rearranges to the time-independent random Helmholtz equation Δu + (ω²/c²)(E[n²]+μ)u = 0 — elliptic, not hyperbolic. If one instead uses a slowly-varying-envelope ansatz u=A(t,x)e^{-iωt} and keeps the leading time derivative, the resulting equation is first-order in time (Schrödinger-type), again not the hyperbolic Anderson model. Thus Eq.(3) is not a controlled consequence of Eq.(2). The hyperbolic Anderson model requires a second-order time operator with multiplicative noise, and the SPDE theorems (Balan et al.) that produce Eqs.(12)–(16) apply only to that equation. The missing Supplementary Information cannot repair this internal inconsistency, because the printed derivation itself specifies the problematic substitution.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an SPDE formulation for light propagation in non-Markovian random media. Starting from the wave equation with a random refractive index, the authors decompose the squared refractive index and, after a slowly-varying-phase argument, replace Eq. (2) by L_cn u = W u with W = mu omega^2/c^2, identifying this with the hyperbolic Anderson model. They assume a product covariance for W given by fractional Brownian motion in time and a Riesz kernel in space (Eq. (4)), then borrow results from the SPDE literature to obtain an asymptotic moment formula (Eq. (12)), an exact inter-moment relation (Eq. (13)), and an aperture-averaging Gaussian convergence bound (Eq. (16)). The predictions are compared with a 588 m outdoor experiment using a micrometeorological array and heterodyne detection, reporting a Pearson correlation of 0.905 between environmental and phase Hurst indices and a decay of the sKL divergence consistent with R^{-alpha/2}. The abstract and conclusion describe the mapping as exact and the validation as decisive.","tokens_in":10416,"tokens_out":8089,"duration_ms":89109,"significance":"If the central claims were correct, the paper would provide a valuable bridge between SPDE theory and optical propagation, yielding quantitative scaling laws that connect the Hurst exponent and spatial power-law exponent to scintillation and aperture averaging. The experimental effort is substantial and the idea of using fBm and Riesz-kernel covariance is creative. However, the load-bearing derivation of the hyperbolic Anderson model is not controlled, key theoretical results are deferred to a missing Supplementary Information, and the experimental validation omits the fitted parameter values, error bars, and the constant C2, so the paper currently does not establish that the proposed equations describe the experiment. The manuscript is not in a form where its central claims can be verified.","major_comments":[{"comment":"The transition from Eq. (2) to Eq. (3) is the linchpin of the paper but is not a controlled approximation. In Eq. (2), the second time derivative appears both in L_cn and in the noise term. The stated condition that phase fluctuations are slow compared to omega justifies replacing partial_t^2 u by -omega^2 u; applied consistently, this turns Eq. (2) into the time-independent random Helmholtz equation. A standard envelope reduction gives a first-order-in-time Schrödinger-type equation. Neither is the hyperbolic Anderson model. Eq. (3) is obtained only by replacing partial_t^2 in the noise term while retaining it in L_cn, and no asymptotic ordering is given to justify this selective substitution. Since Eqs. (5)–(16) all depend on Eq. (3) being the hyperbolic Anderson model, this is a load-bearing gap, not a presentation issue.","section":"§2, Eqs. (2)–(3)"},{"comment":"The covariance factorization in Eq. (4) is an assumption, not derived from the medium or from a controlled approximation. All subsequent predictions are conditional on the exact product form C_H(t^{2H}+s^{2H}-|t-s|^{2H})|x-y|^{-alpha}. The experimental section never reports the measured alpha or H used to draw the theoretical curves in Figs. 2–3, so the conditional nature of the prediction is not tested. In addition, the solution in Eq. (5) is described as distribution-valued, yet the paper uses pointwise intensity I(t,x)=|u_w(t,x)|^2 and its moments in Eqs. (8)–(13); the required regularization or renormalization is not discussed.","section":"§2, Eq. (4)"},{"comment":"The moment asymptotics in Eq. (12) and the inter-moment relation in Eq. (13) are asserted with a citation to Ref. [34], whose title indicates time-independent noise, whereas the noise in Eq. (4) is space-time fractional noise with temporal Hurst exponent H. No derivation is provided and the hypotheses of the cited theorem are not verified. The experimental validation in Fig. 2 does not state the fitted alpha or the theoretical curve parameters, and the panels have no error bars, so the claimed agreement cannot be evaluated quantitatively. Because Eq. (13) is the main moment prediction, this is a serious support gap.","section":"§2, Eqs. (12)–(13)"},{"comment":"The aperture-averaging bound dsKL <= C2 R^{-alpha/2} is not falsifiable as stated because C2 is unspecified and alpha is not reported. With an arbitrary prefactor, the dashed line in Fig. 3(b) can be made to match a wide range of monotone decays. The symmetrized Kullback-Leibler divergence for the complex-valued process Z_R(t) is not defined in the text, and the hypotheses of the functional central limit theorem from Ref. [35] are not checked for the space-time fractional noise model. The experimental Fig. 3(b) again has no error bars or fit parameter values, so the validation of Eq. (16) is insufficient.","section":"§2, Eq. (16)"},{"comment":"The paper repeatedly defers load-bearing material to a Supplementary Information with a placeholder URL: the derivations of Eqs. (10)–(16), the discretization leading to Eq. (11), the explicit form of C1(alpha,H), the definition of dsKL, and the R/S implementation. Since the SI is not available with the submission, these claims cannot be verified from the printed text. This is not a minor omission because the main mathematical results are not derivable from what is presented.","section":"Supplementary Information (placeholder)"}],"minor_comments":[{"comment":"The notation E[W(t,x), W(s,y)] should be E[W(t,x)W(s,y)] for the covariance. Also, the domain restriction |x-y| >= l0 appears only after the formula and is not incorporated into the subsequent analysis consistently.","section":"§2, Eq. (4)"},{"comment":"The notation S(L), S(l0), and the integration variable r are not defined clearly; the integral appears to mix a surface measure with a volume element, and the interchange of summation and integration is not justified.","section":"§2, Eq. (10)"},{"comment":"The phrase 'spaced at the same interval of (L-l0)/N-1' is ambiguous; the index set and the meaning of the phase-screen interpretation should be stated precisely.","section":"§2, Eq. (11)"},{"comment":"The text states a Pearson coefficient of 0.905 but does not report confidence intervals, significance, or the number of independent samples. Given the strong temporal autocorrelation in 24-hour records, the effective sample size should be addressed.","section":"§3, Fig. 3"},{"comment":"The words 'exact mapping' and 'rigorous' are stronger than what is demonstrated, since Eq. (3) is obtained from an approximation and Eq. (4) is an assumed covariance model.","section":"Abstract and Conclusion"}],"recommendation":"reject","confidential_remarks":"The paper has an interesting interdisciplinary ambition, but the central physics step is not sound: Eq. (3) is not a controlled consequence of Eq. (2), and the paper's main predictions all depend on this step. The missing SI and the absence of fitted parameter values, error bars, and an explicit C2 further prevent verification. A standard revision cannot repair the derivation without changing the claimed hyperbolic Anderson model, so rejection is appropriate. This assessment is consistent with the reader's low-confidence reject, though I would put more weight on the Eq. (3) inconsistency than on the experimental details."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read. What's genuinely new: applying the hyperbolic Anderson model and its known intermittency and moment asymptotics (Balan et al., Dalang) to optical propagation with long-range temporal correlations is a fresh move, and the observation that aperture averaging drives the field to Gaussian with a rate tied to the spatial spectral exponent is worth having. The outdoor experiment is ambitious, and the 0.905 Pearson correlation between medium and phase Hurst indices, if it survives scrutiny, would be an interesting memory-transfer signature.\n\nWhat's wrong: the central mapping from the wave equation to the hyperbolic Anderson model doesn't follow. Eq. (2) is exact; Eq. (3) is obtained by replacing ∂²_t u with -ω²u in the noise term only. If you make that replacement consistently, the deterministic part L_cn also changes, and you get an elliptic Helmholtz equation, not a second-order hyperbolic SPDE. The paper gives no controlled limit (paraxial, slowly varying envelope, etc.) that yields the hyperbolic Anderson model. So the SPDE results (Eqs. 12–16) are imported correctly as mathematics, but they are not tied to the optical propagation problem as written. That is load-bearing, not cosmetic.\n\nThe rest of the gaps are in proportion: central equations are deferred to a missing SI (Eqs. 10–13, C1, sKL definition, R/S algorithm), the experimental figures have no error bars, no stated α or H values, and the plot in Fig. 3(b) is of an upper bound C2 R^{-α/2} with unspecified C2 — an upper bound is not a predictive curve. The moment relation Eq. (13) is testable and appears to be checked, but without uncertainties it's hard to know what the agreement means.\n\nTo a chair: this is not referee-ready as submitted. The gap between Eq. (2) and Eq. (3) is either a fixable omission or a sign the entire mapping is wrong; that has to be resolved before the SPDE machinery can be used. If the authors can provide a legitimate asymptotic regime (or a corrected equation), the paper could be valuable. As is, I wouldn't cite it.","headline":"Novel idea, but the derivation of the 'exact' hyperbolic Anderson mapping has a load-bearing inconsistency; the experimental validation is too lightly quantified to rescue it.","tokens_in":10805,"tokens_out":4008,"would_cite":false,"duration_ms":47951,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that light propagation through non-Markovian random media maps exactly onto the hyperbolic Anderson model, yielding scaling laws for intensity moments and aperture averaging that a 588 m outdoor experiment confirms.","keywords":["non-Markovian random media","hyperbolic Anderson model","fractional Brownian motion","Riesz kernel","scintillation index","aperture averaging","stochastic partial differential equations","long-range temporal correlations"],"falsifier":"Measure the two-time, two-point covariance of the atmospheric refractive-index fluctuations over the same lag range used in the experiment and test whether it collapses to C_H(t^{2H}+s^{2H}-|t-s|^{2H})|x−y|^{-α} with fixed H and α. If the empirical surface cannot be fit by that single-factor form, the exact scaling laws derived from the hyperbolic Anderson mapping do not hold for this medium.","tokens_in":9864,"feed_emoji":"🌫️","tokens_out":9433,"duration_ms":80714,"temperature":0.7,"pith_summary":"The paper introduces a stochastic partial differential equation (SPDE) for light propagation that keeps the temporal memory of the medium instead of discarding it. By splitting the squared refractive index into a mean and a fluctuation and applying a slowly-varying-phase approximation, the authors obtain the hyperbolic Anderson model with multiplicative noise. From known SPDE results they derive exact relationships: the scintillation index is bounded (explaining saturation), higher-order intensity moments are determined by the second moment through a formula that depends only on the spatial exponent α, and aperture-averaged fields converge to a Gaussian at rate R^{-α/2}. An outdoor 588 m experiment with a heterodyne receiver and in-situ temperature sensing reports a correlation coefficient of 0.905 between the Hurst index of the medium and that of the optical phase for a small aperture, and confirms the predicted convergence under large apertures.","feed_headline":"588-m outdoor run verifies non-Markovian light-statistics law","feed_subtitle":"One covariance law — long-memory in time, power-law in space — predicts intensity moments and aperture smoothing; field data match.","key_machinery":"The key machinery is the hyperbolic Anderson model — a stochastic wave equation with multiplicative Gaussian noise — obtained after decomposing the squared refractive index and using the slowly-varying-phase approximation. The noise covariance is the product of fractional Brownian motion in time (with Hurst index H) and a spatial kernel |x−y|^{-α}. This specific covariance lets the authors expand the solution as a series of iterated stochastic integrals against the random field, from which the closed-form moment asymptotics (Eqs. 12–13) and the Gaussian fluctuation bound (Eq. 16) follow. The spatial kernel provides the scaling exponent α; the fractional-Brownian factor provides the temporal","core_discovery":"The central discovery is the mapping of the wave equation in a fluctuating medium, written as L_cn u = (μ ω²/c²) u, to the hyperbolic Anderson model. For a random field whose covariance factorizes as fractional Brownian motion in time times a spatial kernel |x−y|^{-α}, with Hurst index H and exponent α, the authors show that known SPDE results give exact asymptotics for the moments of the solution. In particular, Eq. (13) gives the normalized q-th intensity moment as a function of the second moment with an exponent involving only α, and Eq. (16) bounds a statistical distance between the aperture-averaged, normalized field and the standard complex Gaussian by C_2 R^{-α/2}. The experimental se","pith_inferences":["A sharper test than the plotted moment collapse would be to measure α and H independently from the covariance data and then predict the full moment curve from Eq. (13) with no fitted parameters; the paper does not state the α and H values used for the theoretical curves.","If the single-factor covariance holds, the same mapping should transfer to other wave systems — acoustic, seismic, or underwater — whose refractive index fluctuates with long-range temporal memory, yielding analogous scaling laws.","The time-asymptotic growth law of Eq. (12) has exponent (4H−α)/(2−α); a long-path or variable-turbulence experiment could test whether the observed moment growth tracks this exponent, since the current stationary outdoor test cannot distinguish asymptotic from transient behavior.","Because Eq. (13) is independent of H, the model predicts that normalized higher-order moments collapse onto one curve even as the second moment changes with memory state; comparing data across different times of day or turbulence strengths could reveal whether any H-dependence is truly absent."],"forward_implications":["Scintillation index remains finite even at infinite propagation distance, giving a theoretical explanation for the experimentally observed scintillation saturation.","For fixed α, the normalized higher-order intensity moments collapse onto a single curve as a function of the second moment (Eq. 13), independent of the memory parameter H; this is a testable universal relation.","Aperture averaging always restores Gaussian statistics, with convergence rate at least R^{-α/2}; for H≠1/2, small apertures still show non-Markovian, colored-noise fluctuations that are more damaging to communication links than white noise.","The limit of many phase screens reproduces the same saturation behavior, explaining why the standard multiple phase-screen method works well when the number of screens is large.","At point-like apertures, the memory of the medium (H>1/2) is directly imprinted on the propagated field, so the observed phase Hurst index can be used to infer the medium's memory (correlation 0.905 in the experiment)."],"fun_headline_variants":["Non-Markovian light law proven in 588-m outdoor test","Memory of light: 588-m run shows non-Markovian stats","Long-memory covariance law matches 588-m outdoor data","Hyperbolic Anderson model matches 588-m light field data","Temporal memory controls light: 588-m test confirms"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire derivation rests on the covariance of the squared-refractive-index fluctuation field factoring exactly as fractional Brownian motion in time times a spatial kernel |x−y|^{-α} with a single pair of exponents (H, α); if atmospheric correlations are not of this single-factor form, the 'exact' relationships in Eqs. (12), (13), and (16) do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Non-Markovian light law proven in 588-m outdoor test","Memory of light: 588-m run shows non-Markovian stats","Long-memory covariance law matches 588-m outdoor data","Hyperbolic Anderson model matches 588-m light field data","Temporal memory controls light: 588-m test confirms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001477,"raw_usage":{"total_tokens":5737,"prompt_tokens":670,"completion_tokens":5067,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":414,"completion_tokens_details":{"reasoning_tokens":4989}},"tokens_in":414,"tokens_out":5067,"duration_ms":41850,"temperature":1.0,"reasoning_tokens":4989,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:20:32.006209+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the two-time, two-point covariance of the atmospheric refractive-index fluctuations over the same lag range used in the experiment and test whether it collapses to C_H(t^{2H}+s^{2H}-|t-s|^{2H})|x−y|^{-α} with fixed H and α. If the empirical surface cannot be fit by that single-factor form, the exact scaling laws derived from the hyperbolic Anderson mapping do not hold for this medium.","supporting_citations":[],"review_version":1}