{"id":"ec78e2fd-a6e8-4cf5-9d9d-a086467af3b8","arxiv_id":"2607.28167","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Temporal increments of the nonlinear stochastic fractional heat equation with rough spatial noise satisfy Khinchin's and Chung's laws of the iterated logarithm with explicit constants.","lead":"What did the paper find: exact formulas for the largest and smallest time-fluctuations of solutions to a fractional heat equation driven by rough spatial noise. Why read: it sharpens earlier regularity bounds into two-sided laws of the iterated logarithm, the standard precision tool for random path behavior.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The LIL constants in Cor. 3.1 rest on Eq. (1.5), but the stated κ does not match the linear solution's temporal covariance when α≠2; a direct covariance check at α=1.3,H=0.45 shows a mismatch by roughly a factor of √3.5.","rationale":"The reader correctly identified Eq. (1.5) as the most load-bearing unproved premise, but did not test its constant. My covariance calculation suggests a concrete numerical contradiction: the κ required by (1.5) is not Γ(2rH)/Γ(rH) unless α=2. The rest of the paper's technical estimates are less suspect: the proof of Theorem 3.1 is detailed, the exponent bookkeeping is plausible, and the L^p-to-a.s. step is deferred but standard in the cited literature. However, the exact LIL constants are the central claimed output, and they are obtained by combining Theorem 3.1 with (1.5) and Lemma 5.3. If the constant in (1.5) is wrong, Corollary 3.1 is wrong even though the scaling exponent rH/2 and the qualitative LIL structure may be correct. The one concrete test of rederiving the covariance will settle the issue. I therefore recommend REJECT rather than CONDITIONAL: the paper's headline constants appear to be in error, not merely insufficiently justified. If the covariance check surprisingly matches the paper, I would return to CONDITIONAL on the remaining a.s. sup lemma, but the current balance of evidence points to a real numerical error.","tokens_in":16264,"tokens_out":51622,"duration_ms":687734,"concrete_test":"Compute the temporal covariance of v(t,x) directly from the spectral representation: R_v(t,s)=c_{1,1}(2/α)Γ(1−rH)/(2rH)[(t+s)^{rH}−|t−s|^{rH}]. Then (a) verify this formula by evaluating the integral ∫_0^{min(t,s)}∫_R e^{-(t+s−2u)|ξ|^α} c_{1,1}|ξ|^{1−2H}dξ du; (b) compare the leading coefficient of E[(v(t+ε,x)−v(t,x))²] (namely c_{1,1}(2/α)Γ(1−rH)/rH) with κ²=Γ(2rH)/Γ(rH). Use a concrete valid parameter pair, e.g. α=1.3,H=0.45, and also α=2,H=0.25, where the two should coincide. If the coefficients differ for α≠2, Eq. (1.5) and the LIL constants must be revised.","verdict_should_be":"REJECT","load_bearing_attack":"The central transfer from Theorem 3.1 to Corollary 3.1 is Eq. (1.5): for fixed x, t ↦ κ B^{rH/2}_t − v(t,x) is C∞ on (0,∞), with κ² = Γ(2rH)/Γ(rH), rH = (α+2H−2)/α. This equation is not merely unproved; it appears numerically inconsistent with the covariance of the linear solution. From the spectral representation of v, R_v(t,s) = c_{1,1}(2/α)Γ(1−rH)/(2rH) [(t+s)^{rH} − |t−s|^{rH}]. Therefore the leading coefficient of E[(v(t+ε,x)−v(t,x))²] is c_{1,1}(2/α)Γ(1−rH)/rH. For this to be compatible with (1.5), κ² must equal that coefficient. But the paper's κ² = Γ(2rH)/Γ(rH). Taking a permitted point, α=1.3 and H=0.45, gives rH≈0.1538, c_{1,1}=Γ(1.9)sin(0.45π)/(2π)≈0.1512, and the required κ²≈1.68, while Γ(2rH)/Γ(rH)≈0.48. The same discrepancy appears at the origin, where rκ² in Lemma 5.3 should be the variance coefficient of v(h) but is instead about 0.27 versus the computed 0.95. If this calculation is correct, the Khinchin/Chung constants in Corollary 3.1 are wrong by a factor ≈√(1.68/0.48)≈1.87 for α≠2. The L^p approximation theorem may still hold, but it cannot yield the stated exact constants through (1.5).","agreement_with_reader":"partial"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Theorem 3.1 is worth a referee. Corollary 3.1, as stated, is not.\n\nThe genuinely new piece is the sharp L^p approximation of temporal increments for the SFHE with α∈(1,2) and H∈((3−α)/4,1/2). That regime was not covered by [32] or [27], and the proof has real content—Proposition 2.2 gives a new kernel estimate, and the exponent bookkeeping in Section 4 mostly holds up. If the paper only claimed (3.1)–(3.2), I would be fairly comfortable.\n\nThe problem is the transfer from Theorem 3.1 to Corollary 3.1. Equation (1.5) is asserted, not proved, and the stress-test is right: from the spectral covariance used in the paper,\n\nE[(v(t+ε,x)−v(t,x))²] ~ [2 c_{1,1} Γ((2−2H)/α)/(α rH)] ε^{rH}.\n\nFor α=2 this coefficient reduces to Γ(2rH)/Γ(rH) by the usual gamma identity, which is presumably why the formula was imported from [27]. For α≠2 it does not. At α=1.3, H=0.45 the coefficient is about 1.68, while Γ(2rH)/Γ(rH) is about 0.48. So Eq (1.5) cannot hold with the stated κ, and the Khinchin/Chung constants in (3.3) and (3.4) are off by a factor around √3.5. This is not a fillable gap; it is a wrong constant in the main advertised result.\n\nThe rest of the soft spots are proportionate. Lemma 5.2 and Lemma 5.3 are borrowed from [27] by “adapting,” and Corollary 3.1’s proof is one sentence. That would be acceptable if the borrowed result had the right constant, but here it doesn’t. The self-citation itself is not a problem.\n\nBottom line: Theorem 3.1 deserves a serious referee; Corollary 3.1 needs the constant corrected and the decomposition (1.5) either proved or replaced. I would not cite the LIL results as they stand, but I would send the paper to review and require the authors to fix the linear analysis before publication.","headline":"The temporal approximation theorem is a real extension, but the LIL constants in Corollary 3.1 rest on a decomposition that contradicts the linear solution's covariance when α≠2.","tokens_in":17247,"tokens_out":37201,"would_cite":false,"duration_ms":289559,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60G17","60G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the temporal process of the nonlinear stochastic fractional heat equation driven by rough spatial noise behaves, at small scales, exactly like a fractional Brownian motion with index (α+2H−2)/(2α), yielding explicit Kh","keywords":["stochastic fractional heat equation","fractional Brownian motion","law of the iterated logarithm","Khinchin's LIL","Chung's LIL","rough spatial noise","fractional Laplacian","temporal increments"],"falsifier":"Compute the exact leading-order variance of the temporal increment of the linear solution $v(t,x)$ in (1.4) for small $\\varepsilon$: if $\\operatorname{Var}(v(t+\\varepsilon,x)-v(t,x))$ does not equal $\\kappa^2 \\varepsilon^{2\\tilde H/2}$ plus smaller-order terms, with $\\kappa$ as in (1.6), then the decomposition (1.5) fails. Alternatively, simulate the linear equation at a fixed $x$ and estimate the Khinchin limsup; any deviation from $\\kappa$ would falsify the paper's core mechanism.","tokens_in":16098,"feed_emoji":"🕒","tokens_out":4346,"duration_ms":177626,"temperature":0.7,"texified_at":"2026-08-05T21:52:20.166340+00:00","pith_summary":"The paper establishes that, for the nonlinear stochastic fractional heat equation with space-rough Gaussian noise, the temporal increment at a fixed point equals the coefficient $\\sigma(u(t,x))$ times the linear solution's increment, plus a remainder of order $\\varepsilon^{(\\alpha+2H-2)/(2\\alpha)+\\eta}$. From this approximation it derives sharp Khinchin and Chung laws of the iterated logarithm for the temporal process $t \\mapsto u(t,x)$, with explicit constants. The result matters because it shows the local temporal oscillation of the solution is governed by a single scaling exponent, and that the nonlinearity enters only through the value of $\\sigma$ at the point $(t,x)$. If correct, it provides a complete first-order description of the temporal sample paths in the regime of rough spatial dependence and fractional diffusion.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":6603,"prompt_tokens":834,"completion_tokens":5769,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":834,"completion_tokens_details":{"reasoning_tokens":4957}},"feed_headline":"Heat equation with rough noise has fBm temporal LIL","feed_subtitle":"Khinchin and Chung constants reduce to one scaling exponent and the value of σ at (t,x).","key_machinery":"The load-bearing object is the Gaussian decomposition (1.5): for fixed $x$, the linear solution's temporal process can be written as $\\kappa B^{\\tilde H/2}_t$ minus an infinitely differentiable remainder. This transfers the Khinchin and Chung LIL constants from a fractional Brownian motion to the linear solution and then, via the approximation theorem, to the nonlinear solution. The approximation theorem itself is proved by decomposing the temporal increment into J0 (initial condition), J1 (historical stochastic convolution), and J2 (short-time stochastic convolution), using a time splitting $s \\in [0, t-\\varepsilon^\\theta]$ and $[t-\\varepsilon^\\theta, t]$, fractional heat-kernel estimates, and a BDG-type inequality for the rough spati","core_discovery":"The central claim is Theorem 3.1: for every $p \\ge 1$ and $\\eta$ in the stated interval, the $L^p$ norm of the approximation error — the difference between the temporal increment of the nonlinear solution and the deterministic initial-value term plus $\\sigma(u(t,x))$ times the linear temporal increment — is bounded by $c_{3,1} \\varepsilon^{(\\alpha+2H-2)/(2\\alpha)+\\eta}$. Using this and the asserted Gaussian decomposition of the linear solution, $t \\mapsto \\kappa B^{\\tilde H/2}_t - v(t,x)$ with infinitely differentiable remainder, the paper derives Corollary 3.1: with probability one, the Khinchin limsup equals $\\kappa |\\sigma(u(t,x))|$ and the Chung liminf equals $\\kappa \\lambda_{\\tilde H/2}/\\tilde H |\\sigma(u(t,x))|$, where $\\tilde H = (\\alpha+2H-2)/\\alpha$. Thus the local temporal process","pith_inferences":["The paper treats the Gaussian decomposition (1.5) as an input without proof; if this decomposition were only Hölder rather than C∞, or if κ were miscomputed, the LIL constants would change — so verifying this decomposition directly for the linear equation would be a decisive test of the whole argument.","Because the constant κ depends only on \\tilde H and not on α and H separately, the local temporal law may be universal across the allowed parameter region; this could be tested numerically by simulating the linear equation for different (α,H) pairs and checking the same κ.","The dependence on σ(u(t,x)) suggests that at points where σ(u(t,x)) vanishes, the stated LIL degenerates and a different normalization might be needed; the paper does not explore this regime.","The same approximation method could plausibly yield a functional LIL or a Chung-type law for the entire temporal path, since the smooth remainder in (1.5) is negligible at the LIL scale."],"forward_implications":["If Theorem 3.1 and Corollary 3.1 are correct, the temporal process at each fixed point has an exact first-order asymptotic: its normalized oscillations converge to those of a fractional Brownian motion with index (α+2H−2)/(2α), and the LIL constants are known explicitly in terms of κ, the small-ball constant λ, and σ(u(t,x)).","The same scaling exponent \\tilde H/2 controls both the limsup and liminf rates, so the paper yields a consistent self-similarity index for the temporal process.","At t=0, the Khinchin and Chung constants change by the factor 2^{(1−\\tilde H)/2} (via \\tilde κ), provided the initial condition is sufficiently Hölder continuous, extending the results to the origin.","The a.s. sup bound in Lemma 5.2, if justified, upgrades the L^p approximation error to a uniform-in-ε almost sure statement, which is what allows the LIL to be stated with probability one."],"fun_headline_variants":["Heat equation with rough noise: exact temporal LIL constants","Stochastic heat with fractional noise: temporal LIL proven","Heat equation's time path: Khinchin and Chung constants","Rough noise heat equation: temporal LIL with fBm scaling","Exact temporal LIL constants for stochastic heat equation"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The Gaussian decomposition (1.5) — that the linear temporal process is a scaled fractional Brownian motion plus an infinitely differentiable remainder — is asserted without proof or citation, and it is what transfers the fBm LIL constants to the nonlinear solution; Lemma 5.2's a.s. sup bound is also stated only by 'adapting' a cited work.","fun_headline_variants_meta":{"raw":{"variants":["Heat equation with rough noise: exact temporal LIL constants","Stochastic heat with fractional noise: temporal LIL proven","Heat equation's time path: Khinchin and Chung constants","Rough noise heat equation: temporal LIL with fBm scaling","Exact temporal LIL constants for stochastic heat equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000592,"raw_usage":{"total_tokens":2593,"prompt_tokens":705,"completion_tokens":1888,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":1804}},"tokens_in":449,"tokens_out":1888,"duration_ms":13444,"temperature":1.0,"reasoning_tokens":1804,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T03:18:47.689557+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact leading-order variance of the temporal increment of the linear solution $v(t,x)$ in (1.4) for small $\\varepsilon$: if $\\operatorname{Var}(v(t+\\varepsilon,x)-v(t,x))$ does not equal $\\kappa^2 \\varepsilon^{2\\tilde H/2}$ plus smaller-order terms, with $\\kappa$ as in (1.6), then the decomposition (1.5) fails. Alternatively, simulate the linear equation at a fixed $x$ and estimate the Khinchin limsup; any deviation from $\\kappa$ would falsify the paper's core mechanism.","supporting_citations":[],"review_version":2}