{"id":"efcdc3b0-516d-4b39-bfd8-8f767eab8703","arxiv_id":"2608.08480","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The scaled cumulant generating function for annealed current fluctuations across a disk in the two-dimensional SSEP is obtained in closed form via a nonisospectral integrable reduction.","lead":"This paper derives an exact formula for the probabilities of rare net particle losses from a disk in a two-dimensional interacting particle gas. If correct, it is a rare exact result for current fluctuations in a curved finite region and introduces a new integrability method.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim depends on unreleased scattering algebra: identities (29) and (35) are deferred to the Supplemental Material and the paper concedes missing Jost/analyticity control, so Eq. (6) is not established in the visible text.","rationale":"We read the paper in good faith. The central claim is precise, and the visible part contains several independent consistency checks that make Eq. (6) credible: the first cumulant matches the heat-content calculation, the fluctuation symmetry follows from ω(λ)=ω(A−λ), the small and large r0 limits match the bulk and boundary-layer regimes, and the spectral measure has the right total weight and channel form. However, none of these checks validates the scattering derivation. The steps leading to (29) and (35) are delegated to Supplemental Material, and the paper itself states that a rigorous analysis would require control of Jost solutions, analytic continuation, and contour-crossing zeros. The reader's weakest_assumption correctly targets these premises; our pass agrees. We therefore do not move the verdict: CONDITIONAL remains appropriate, requiring release of the supplement or independent verification of (29) and (35). We would not reject the paper, because the claim is internally consistent and passes visible checks.","tokens_in":9929,"tokens_out":29181,"duration_ms":314447,"concrete_test":"Use the same iterative scheme as Fig. 3 to solve the radial MFT saddle problem (9)-(12) for r0=2 and at least two physical λ values, e.g. (ρ1,ρ2,λ)=(0.5,0.2,1) and (0.7,0.1,2). From the converged endpoint fields, build u(x,0), v(x,1), compute bv_>(z), bu_>(z), γ0(z), γ1(z) by quadrature on a grid of real z, and test identity (29) pointwise. Then compute the on-shell action S(λ) by quadrature and verify the trace identity (35) and the resulting Eq. (6). If (29) or (35) fails at any tested λ, the central claim is invalid; if both pass, the missing rigor becomes a formal rather than substantive gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The result is plausible: Eq. (6) reproduces the mean current, the fluctuation symmetry, the small-r0 support limit, and the large-r0 boundary-layer asymptotics, and Fig. 3 checks endpoint densities for one parameter set. But the derivation of Eq. (6) is not contained in the paper. After the Lax pair (16)-(17) and triangular endpoint matrices (26), the scalar factorization identity (29) is obtained by 'direct evaluation' and 'standard decay and analyticity assumptions' with all details in [83]; the same holds for the trace identity (35), which connects the first spectral coefficient to the derivative of the SCGF. If the half-line Jost solutions do not have the claimed regularity, or if the endpoint constants do not satisfy αβ = η0ω, then (29) fails, and neither the scalar factorization (32) nor the trace identity (35) follows. The paper's Discussion explicitly concedes that rigorous control of Jost solutions, analytic continuation, and contour-crossing zeros is missing. Since the visible text does not even contain the derivation of (29) from (26)-(28), the closed form (6) is unsupported as written; it is a plausible conjecture conditional on the supplement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers annealed current fluctuations in the two-dimensional symmetric simple exclusion process across a circular passive counting boundary. The initial state is a Bernoulli product measure with density rho1 inside a disk of radius R and rho2 outside, and the observable is the net particle loss from the disk over a finite time. The authors formulate the problem in macroscopic fluctuation theory, argue via convexity and rotational averaging that the optimal history can be taken radially symmetric, and then reduce the radial MFT equations to a nonisospectral Lax pair on the half line. Their main result is the closed-form scaled cumulant generating function mu_ann^2(lambda; r0) reproduced in Eq. (6), expressed as a one-dimensional integral over the combination omega(lambda). From this formula they derive the rate function, cumulants, fluctuation symmetry, a spectral channel representation, and a reconstruction of endpoint densities. The central technical route is a half-line scattering construction leading to a scalar factorization problem and a trace identity that relates the first spectral coefficient to the derivative of the SCGF. Several load-bearing steps are not shown in the main text and are deferred to Supplemental Material [83], including the radial optimality proof, the endpoint algebra, the normalization of the trace identity, and the final resummation.","tokens_in":10346,"tokens_out":3479,"duration_ms":41872,"significance":"If correct, Eq. (6) is a significant exact result: it provides the full annealed large-deviation statistics for a finite, curved counting region in a genuinely two-dimensional interacting diffusive system, going beyond the one-interface one-dimensional solutions. The formula is parameter-free in that all dependence on the initial densities and lambda enters through the single function omega(lambda), and it is checked against the mean current, the fluctuation symmetry, small- and large-radius asymptotics, and a pointwise endpoint-density comparison. The nonisospectral Lax pair and the reduction of the scattering problem to a scalar factorization are conceptually interesting and may be useful beyond this particular observable. However, the visible text does not contain the derivations of the key identities needed to establish Eq. (6); the essential steps are delegated to a Supplemental Material that is not provided in the manuscript. The result is therefore presently a plausible and well-tested conjecture rather than a demonstrated theorem, and the manuscript itself concedes the absence of rigorous control of the Jost solutions and analytic continuation.","major_comments":[{"comment":"Equation (29) is the hinge of the whole derivation: it reduces the matrix half-line scattering problem to a scalar factorization problem. Yet the text obtains it by \"direct evaluation\" of the endpoint frames and by \"standard decay and analyticity assumptions,\" with all details confined to [83]. In particular, the relation alpha*beta = eta0*omega and the elimination of the origin-side transforms are not demonstrated. Without a derivation of (29) from (26)-(28), the scalar factorization (32), the Cauchy formula (33), and hence the spectral reconstruction of the SCGF are unsupported. This is a load-bearing gap that must be closed in the visible manuscript or in an actually available supplement.","section":"Half-Line Scattering and Scalar Factorization, Eq. (29)"},{"comment":"The trace identity (35) is normalized using the deterministic step rho1=1, rho2=0 and then asserted to hold for general densities because \"the annealed endpoint algebra depends on (rho1, rho2, lambda) only through omega.\" This transfer is nontrivial: C1 in (34) is defined through the normalized scalar factor Phi+, which depends on the gauge frames and the Jost normalization, and it is not evident that the same constant of proportionality survives for arbitrary rho1, rho2. A derivation showing that the ratio C1/omega is independent of the endpoint densities, or an explicit computation for general rho1, rho2, is needed. As written, the extension from one representative case to the full statement of Eq. (6) is an assumption rather than a demonstrated identity.","section":"Spectral Reconstruction of the SCGF, Eq. (35)"},{"comment":"The manuscript explicitly concedes that \"a rigorous analysis of the present construction would also require control of the half-line Jost solutions, analytic continuation, and possible contour-crossing zeros.\" These are not peripheral technicalities: the scalar factorization (32)-(33) relies on analytic continuation into complementary spectral domains, and the principal-value regularization in (34)-(35) is justified only if the contour deformation and zero-crossing structure are controlled. The paper should either supply sufficient conditions under which these manipulations are valid or clearly state the precise regularity and zero-avoidance assumptions under which Eq. (6) is claimed to hold. Without this, the central formula remains conditional on unverified analytic behavior.","section":"Discussion and Outlook"},{"comment":"The reduction of the full two-dimensional variational problem to radially symmetric histories is a central premise, but the proof is deferred entirely to [83]. The convexity/Jensen argument sketched in the text is plausible for the dynamical cost, but the endpoint terms F0 and the constraint that QT is preserved under rotational averaging require explicit treatment. Since the radial reduction is what makes the rest of the construction possible, the proof should be included in the main text or in an available supplement rather than referenced as an unpublished document.","section":"MFT and Radial Optimality, first paragraph"}],"minor_comments":[{"comment":"Reference [83] is listed as \"(2026), see Supplemental Material\" with no further information. If the supplement is not included with the posted arXiv version, a reader cannot check any of the deferred derivations. The authors should make the supplement available or state explicitly where it can be obtained.","section":"References, [83]"},{"comment":"The factor 1/(2*pi*i) in front of a principal-value real integral is notationally unusual; the reader must infer that this is a contour integral representation with the contour taken along the real axis. A brief clarifying remark would avoid confusion.","section":"Equation (34)"},{"comment":"The endpoint-density comparison is shown for a single parameter set and a single value of lambda. Given that this comparison is the main direct test of the scattering solution at the field level, one more parameter set or a short statement of how many cases were tested in [83] would strengthen confidence in the reconstruction formulas.","section":"Figure 3"},{"comment":"The symbol B(r) is used in Eqs. (11)-(12) as the indicator of the disk, but earlier in Eq. (2) the same letter denotes the particle number in the ball. This reuse of notation is slightly confusing and should be fixed by using, for example, chi(r) for the indicator.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"This is a nicely written and intellectually attractive Letter, and the result may well be correct. My major concern is structural rather than technical: the visible paper is a sequence of assertions whose proof is entirely in the Supplemental Material, and that supplement is not provided. For a Letter this is acceptable if the supplement is actually available and the claims are checkable, but in the current posting [83] is just a placeholder. I would urge the editor not to send the paper back for scientific review of Eq. (6) until the authors make the supplement available, since the central derivation cannot be verified otherwise. The paper's own discussion candidly admits the missing analytic control, which supports a conditional rather than definitive assessment at this stage."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: if the supplement is as good as the Letter claims, this is a new result — the first exact annealed full counting statistics for a finite curved region in d>1, with a nonisospectral Lax pair that is genuinely different from the 1D isospectral MFT constructions. But the case is not closed in the visible text. The main formula (6) is a one-dimensional integral that reproduces the mean current, the fluctuation symmetry, the small- and large-r0 limits, and a single numerical endpoint-density check. Good signs. The defect is that the derivation of Eq. (6) lives almost entirely in the Supplemental Material: the endpoint algebra αβ=η0ω, the scalar factorization (29), the trace identity (35), and the Bessel resummation are all 'in [83]'. A reader cannot verify the central claim from the Letter alone.\n\nWhat I like. The rotational averaging argument for radial optimality is clean. The change of variable x=r^2/2 that flattens the measure and puts the MFT equations in Lax form is a nice observation, and the time-dependent k(t)=1/(z+4it) is a recognizable nonisospectral twist. The channel representation (36)-(37) is a useful by-product. The citation pattern is appropriate: the 1D MFT integrability references are the right ones, and the paper correctly identifies the finite-curved-region problem as open. The paper is honest about what is missing: the Discussion explicitly concedes control of Jost solutions, analytic continuation, and contour-crossing zeros. That counts for something.\n\nThe soft spots, in proportion. Load-bearing steps are deferred, and the paper tells you they are not rigorous. The trace-identity normalization is argued for the step ρ1=1, ρ2=0 and then extended to general densities 'since the endpoint algebra depends only on ω'; that is plausible but still asserted. The numerical check is one parameter set at the field level; reassuring but not a proof. None of this is circular — the ω-dependence of the endpoint algebra is the real input, and the mean current fixes one normalization constant — so the formula is not fitted to the target. It is a well-motivated conjecture whose proof is out of view.\n\nBottom line: if you work on MFT or exact large deviations, this is worth your time and worth citing with a 'supplement pending' caveat. It deserves a serious referee, but the referee must be given the supplement and should insist that the main algebraic steps appear in the paper. I would send it to peer review rather than desk-reject. Bring it to the group; the incompleteness is a good thing to argue about.\n\nBest,","headline":"If the supplement holds up, this is the first exact full counting statistics for a finite curved region in d>1, but the visible text does not demonstrate the central identity.","tokens_in":10710,"tokens_out":4897,"would_cite":true,"duration_ms":45256,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims an exact closed form for the annealed scaled cumulant generating function of the current through a circular boundary in the two-dimensional symmetric simple exclusion process, valid for any fixed disk radius, with the…","keywords":["symmetric simple exclusion process","macroscopic fluctuation theory","large deviations","current fluctuations","scaled cumulant generating function","nonisospectral Lax pair","inverse scattering","full counting statistics"],"falsifier":"Directly solve the radial MFT boundary-value problem (9)-(12) numerically for a fixed $r_0$ and several $\\lambda$, evaluate the on-shell action, and compare with Eq. (6); a pointwise discrepancy beyond numerical precision at moderate $\\lambda$, away from contour-crossing zeros, would falsify the closed form, as would a mismatch in the second or fourth current cumulant measured in a large-$T$ lattice SSEP simulation.","tokens_in":9729,"feed_emoji":"📊","tokens_out":11068,"duration_ms":103728,"temperature":0.7,"pith_summary":"The paper claims a complete exact solution to a previously open problem: the annealed large-deviation statistics of the net particle current through a circular counting boundary in the two-dimensional symmetric simple exclusion process, starting from two different Bernoulli densities inside and outside the disk. The main result is the closed one-dimensional integral $\\mu^{\\rm ann}_2(\\lambda;r_0)=4r_0^2\\int_0^1\\sqrt{1-y^2}\\,\\log(1+\\omega(\\lambda)e^{-r_0^2y^2})dy$, where $\\omega(\\lambda)$ encodes the densities and the counting field. The derivation combines a convexity argument showing the optimal fluctuation may be chosen radially symmetric, a change of variables $x=r^2/2$ that flattens the radial measure, a nonisospectral Lax pair on the half-line, and a scalar factorization problem that reduces the variational problem to spectral data depending only on $\\omega$ and $r_0$. If correct, this gives the full scaled cumulant generating function, all current cumulants, the rate function, and a fluctuation symmetry for arbitrary fixed disk radius in the diffusive limit. The interest is that this is an exact, geometry-dependent result for a finite curved counting region in two dimensions, where previously exact results were largely confined to one-dimensional or planar-interface settings.","feed_headline":"One integral gives exact current statistics for a disk in 2D SSEP","feed_subtitle":"A closed one-dimensional integral predicts the probabilities of rare particle losses through a circle, for any disk size.","key_machinery":"The central machinery is the nonisospectral Lax pair (16)-(17), whose time-dependent spectral parameter $k(t;z)=1/(z+4it)$ compensates the explicit factor of $x$ in the radial MFT equations, together with the area coordinate $x=r^2/2$ that flattens the radial measure and produces a linear-coefficient divergence form. After a diagonalizing gauge, the spatial problem takes Volterra form with off-diagonal potentials $u,v$; at the two temporal endpoints the annealed conditions make the half-line transfer matrix factor into triangular matrices, leading to the scalar factorization $\\Phi_-(s)\\Phi_+(s)=1+\\omega K(s)$ on the real line, where $K(z)=\\exp(-4r_0^2/(z(z+4i)))$. The first large-$z$ coefficient of $\\log\\Phi_+$ yields the trace identity $\\partial_\\omega F_2(\\omega;r_0)=\\frac{1}{2\\omega}\\mathrm{PV}\\int_\\mathbb{R}\\log\\frac{1+\\omega}{1+\\omega K(s)}ds$; expanding in $\\omega$, evaluating radial moments, and resumming produces the closed integral (6). The radial convexity reduction guarantees that the radial minimum equals the full two-dimensional minimum.","core_discovery":"On its own terms, the paper establishes that the annealed scaled cumulant generating function for the net outward particle current through a disk of radius $R=r_0\\sqrt{T}$ in the two-dimensional SSEP is exactly $F_2(\\omega(\\lambda);r_0)=4r_0^2\\int_0^1\\sqrt{1-y^2}\\log(1+\\omega e^{-r_0^2y^2})dy$, with $\\omega(\\lambda)=\\rho_1(1-\\rho_2)(e^\\lambda-1)+\\rho_2(1-\\rho_1)(e^{-\\lambda}-1)$. From this closed form the paper obtains the rate function by Legendre transform, all current cumulants by differentiation, the fluctuation symmetry $\\mu(\\lambda)=\\mu(A-\\lambda)$ with $A=\\log[\\rho_1(1-\\rho_2)/(\\rho_2(1-\\rho_1))]$, and a channel representation $F_2(\\omega;r_0)=\\int_0^1\\log(1+\\omega\\theta)\\,\\nu_2(d\\theta)$ in which the counting geometry is encoded in a measure $\\nu_2$ of total weight $\\pi r_0^2$. The paper further shows that optimal endpoint density profiles reconstructed from the spectral factors agree pointwise with a direct numerical solution of the macroscopic fluctuation theory equations.","pith_inferences":["Inference: The same nonisospectral closure may extend to other radially symmetric diffusive lattice gases whose MFT equations share the structure (14), provided the convexity reduction still holds; the exact form of the kernel $K(z)$ would change but the factorization strategy would not.","Inference: The channel measure $\\nu_2$ suggests a testable finite-system conjecture: the occupation generating polynomial for a disk on the lattice may admit a Poisson-binomial factorization whose spectral measure converges to $\\nu_2$ in the diffusive limit, which could be checked numerically for moderate $T$.","Inference: Because the simultaneous flattening of the radial measure and the linear divergence form occurs only in two dimensions, the exact one-integral formula is unlikely to have a direct analogue in higher dimensions; approximate variational or spectral approaches there would need a different closure.","Inference: The triangular endpoint factorization suggests that annular or multi-interface counting regions could be treated by composing additional endpoint shears, though the scalar factorization would become matrix-valued and the closed form would likely be replaced by a system of coupled equations."],"forward_implications":["The annealed rate function $I(q)$ for the disk current is obtained by Legendre transform of Eq. (6) and is non-Gaussian for finite $r_0$. All current cumulants are derivatives of Eq. (6), and the mean matches an independent heat-content calculation based on ordinary diffusive evolution.","The SCGF obeys the fluctuation relation $\\mu^{\\rm ann}_2(\\lambda;r_0)=\\mu^{\\rm ann}_2(A-\\lambda;r_0)$, so the rate function satisfies $I(q)-I(-q)=Aq$, with the minimum of the SCGF at $\\lambda=A/2$ independent of $r_0$.","The large-$|\\lambda|$ slopes of Eq. (6) approach $\\pm\\pi r_0^2$, in agreement with the full annealed support of the disk occupation.","The full counting statistics admit a spectral-channel form $F_2(\\omega;r_0)=\\int_0^1\\log(1+\\omega\\theta)\\,\\nu_2(d\\theta)$, encoding the radial geometry in a measure $\\nu_2$ of total weight $\\pi r_0^2$, which suggests a microscopic Poisson-binomial interpretation in the diffusive limit."],"supporting_citations":[{"why":"Supplies the macroscopic fluctuation theory variational action and saddle-point equations from which the calculation starts.","marker":"[15]"},{"why":"Gives the one-dimensional SSEP current fluctuation solution whose saddle structure and boundary conditions are adapted to the radial problem here.","marker":"[41]"},{"why":"Provides the inverse-scattering formulation for time-dependent MFT that the nonisospectral half-line construction extends.","marker":"[48]"},{"why":"Companion detailed inverse-scattering treatment of the one-dimensional SSEP current, supplying the spectral reconstruction and trace-coefficient method used to extract the SCGF.","marker":"[49]"},{"why":"Self-referenced Supplemental Material containing the radial optimality proof, endpoint gauge algebra, trace normalization, moment evaluation, and resummation steps leading to Eq. (6).","marker":"[83]"}],"fun_headline_variants":["Exact current statistics from a single integral","One integral solves 2D SSEP current statistics","Nonisospectral integrability yields exact current","Closed form for disk current in 2D SSEP","2D exclusion process: exact current fluctuations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the half-line Jost solutions of the nonisospectral Lax pair satisfy the standard decay and analyticity assumptions and that the endpoint transfer matrices factor into the stated triangular forms with $\\alpha\\beta=\\eta_0\\omega$; if either fails, the scalar factorization, the trace identity, and hence the closed form are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Exact current statistics from a single integral","One integral solves 2D SSEP current statistics","Nonisospectral integrability yields exact current","Closed form for disk current in 2D SSEP","2D exclusion process: exact current fluctuations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1591,"prompt_tokens":959,"completion_tokens":632,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":560}},"tokens_in":575,"tokens_out":632,"duration_ms":6275,"temperature":1.0,"reasoning_tokens":560,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:36:06.685018+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly solve the radial MFT boundary-value problem (9)-(12) numerically for a fixed $r_0$ and several $\\lambda$, evaluate the on-shell action, and compare with Eq. (6); a pointwise discrepancy beyond numerical precision at moderate $\\lambda$, away from contour-crossing zeros, would falsify the closed form, as would a mismatch in the second or fourth current cumulant measured in a large-$T$ lattice SSEP simulation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the inverse-scattering formulation for time-dependent MFT that the nonisospectral half-line construction extends."},{"cited_title":"Carinci, C","cited_arxiv_id":null,"evidence_quote":"Self-referenced Supplemental Material containing the radial optimality proof, endpoint gauge algebra, trace normalization, moment evaluation, and resummation steps leading to Eq. (6)."}],"review_version":1}