{"id":"13f559fc-a4e2-4b86-8832-f405d4ace736","arxiv_id":"2509.03328","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Corner-flip interfaces above a hard wall on the half-line converge, under diffusive scaling, to the reflected stochastic heat equation whose invariant measure is the 3D Bessel law.","lead":"This mathematics paper proves that a lattice model of a random interface pushed against a hard wall converges, after rescaling, to a reflected stochastic heat equation on a half-line. It extends known segment results to an unbounded domain and identifies the 3-dimensional Bessel process as the stationary law of the continuum equation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.5 applies Dynkin's formula to a non-cylindrical functional without a domain or truncation argument; the Fourier-increment bound (3.7) and therefore the tightness of (h^ε) are unsupported as written.","rationale":"I confirm the reader's weakest assumption is the single most load-bearing step: Lemma 3.5 is the sole source of time-increment decay in Fourier space, and the entire tightness chain for h^ε (Lemma 3.6, Proposition 3.4, Lemma 3.7, Theorem 3.1) plus the η^ε tightness via (5.4) depends on it. The paper never justifies Dynkin's formula for the non-cylindrical f_ζ on the infinite lattice, and although the generator action is in fact absolutely convergent for each fixed ε (the crude ε^{-3/2} bound I cite is a uniform-in-ε estimate), the forward/backward decomposition (3.8)-(3.9) still requires f_ζ to be in the domain of the closed generator or to be approached by an approximation argument, neither of which is supplied. I re-derived the surrounding estimates and found them consistent: the bracket bounds (3.10)-(3.11), the double-BDG inequality (3.12), the time-linearization Lemma 3.6, and the interpolation parameters in Lemma 3.7 all check out, and the noise bracket in Section 4 converges to t∥φ∥²_{L²} as claimed. I did not find the 'exponent mismatch in the Hölder chain of Proposition 3.4' mentioned in the reader's report; the exponents 3/8, 3p/4, and κ = (3/4)(1−θ) re-derive cleanly. I did find a separate, smaller slip in the proof of Theorem 1.2: with F_{t,φ} defined as in the text (with −V_t(φ)) and the discrete equation (5.2) containing −√2W^ε_t(φ), one obtains F_{t,φ}(h^ε,W^ε,η^ε)+R^ε_t(φ) = (√2−1)W^ε_t(φ), not 0; the displayed equality (6.2) holds only if the noise component is √2W^ε or if F uses −√2V_t(φ). This is a normalization typo in the identification step, not a structural flaw: the direct limit of (5.2) yields (1.5) with √2W, and the uniqueness theorem of Hambly-Kalsi [8] cited by the paper then characterizes the limit externally, with Corollary 1.3 following from the invariance principle [3] — so no circularity is present. The central theorem appears sound, but the written proof needs the Dynkin-domain repair plus the √2 fix, consistent with the reader's CONDITIONAL verdict.","tokens_in":25368,"tokens_out":62525,"duration_ms":479798,"concrete_test":"Settle the concern by writing the missing truncation argument for Lemma 3.5. For F_ζ^N(h) = c_{ζ,ε}Σ_{εn≤N}e^{-iζεn}e^{-ρεn}h(εn) (bounded cylindrical), verify: (i) F_ζ^N(h^ε_t) − F_ζ^N(h^ε_s) → ĝ^ε_t(ζ) − ĝ^ε_s(ζ) in L^m(Ω), uniformly in ε ∈ (0,1] and t ∈ [0,T]; (ii) the forward and backward Dynkin martingales for F_ζ^N converge in L^m to limits whose quadratic variations obey the uniform bounds (3.10)-(3.11); (iii) the drift residuals between truncations vanish in L^m, so the Lyons-Zheng identity passes to the limit and (3.7) holds. If step (ii) or (iii) fails, the H^{-s0} moment estimates of Proposition 3.4 and hence the tightness of (h^ε) have no proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the proof of Lemma 3.5 (Section 3.2). The observable f_ζ(h) = ⟨h, e^{-ρx-iζx}⟩, effectively F_ζ(h) = c_{ζ,ε} Σ_{x∈εN*} e^{-iζx}e^{-ρx} h(x), is non-cylindrical on the infinite lattice, yet equations (3.8)-(3.9) apply Dynkin's formula to it, for the forward and backward processes, without showing f_ζ belongs to the domain of the generator (or any truncation/approximation argument). This is not a cosmetic issue: writing L^εF_ζ(h) = c_{ζ,ε}ε^{-2}Σ_x e^{-(ρ+iζ)x}Δ^εh(x)1{...}, the per-site bound gives a crude ε^{-3/2} divergence, so the decomposition is not justified by an absolutely convergent series, and the Lyons-Zheng subtraction producing the bound (3.7) has no rigorous premise. Since (3.7) feeds Lemma 3.6, Proposition 3.4, Lemma 3.7, and Theorem 3.1, the tightness of (h^ε), the joint tightness in Theorem 1.2, and the tightness of (η^ε) via (5.4) all rest on this step. The gap is probably repairable — the bracket estimates (3.10)-(3.11) are uniform in spatial truncation, so a truncation argument should close it — but as written the central proof is missing this premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stationary fluctuations of a corner-flip interface model on the half-line lattice N above a hard wall, with pinning at the origin. Under diffusive scaling, it claims joint convergence in law of the rescaled interface h^ε, the discrete noise W^ε, and the discrete reflection measure η^ε to (u, W, η), where W is a cylindrical Wiener process and (u, η) is the solution of the reflected stochastic heat equation on [0,∞) with Dirichlet condition at 0, starting from a 3-dimensional Bessel initial condition independent of W. Corollary 1.3 concludes that the 3D Bessel law is invariant for this SPDE. The strategy is modular: Section 3 proves tightness of the interface via weighted Sobolev/Hölder estimates and a Lyons–Zheng decomposition; Section 4 proves convergence of the discrete noise to white noise via martingale brackets and Rebolledo’s theorem; Section 5 proves tightness of the reflection measure from the semi-discrete PDE; Section 6 identifies any joint limit point as the unique strong solution of the SPDE. The proof relies on an external invariance principle for the conditioned random walk and on external strong uniqueness for the reflected SPDE.","tokens_in":25611,"tokens_out":18955,"duration_ms":166410,"significance":"The result, if completed, is significant: it provides the first scaling limit of a discrete interface above a hard wall on an infinite spatial domain, connecting it to a Nualart–Pardoux type reflected stochastic heat equation on the half-line, and it gives a dynamical proof of invariance of the 3-dimensional Bessel law. The paper is well structured and contains several reusable quantitative estimates (Lemmas 2.4, 3.3, 3.5–3.8, and the bracket computations in Section 4). The identification of limit points is not circular: it uses external strong uniqueness and an external invariance principle, and the self-reference to prior work is limited to a technical segment result. The main concerns are localized technical gaps in the tightness proof of the interface (Section 3.2), both of which appear repairable; they do not undermine the overall strategy.","major_comments":[{"comment":"Dynkin's formula is applied to the non-cylindrical functional f_ζ(h) = ⟨h, e^{-ρx - iζx}⟩ on the infinite lattice, and to its backward-time counterpart, without showing that f_ζ belongs to the domain of the generator of the Feller process or supplying a cylindrical approximation argument. The generator L was defined only on cylindrical functions, and for this f_ζ one must justify passing from finite sums over sites to the infinite sum defining Lf_ζ and the associated martingale. This is not a cosmetic point: the bound (3.7) on Fourier increments, and hence Lemma 3.6, Proposition 3.4, Lemma 3.7, Theorem 3.1, and ultimately the tightness of η^ε via (5.4), all rest on this step. The gap is likely repairable: truncating f_ζ to finitely many sites gives a cylindrical function, and the estimates (3.10)–(3.11) are uniform in the truncation, so a dominated-convergence or uniform-tightness argument should close it; the manuscript should provide that argument explicitly.","section":"Section 3.2, Lemma 3.5, Eqs. (3.8)–(3.9)"},{"comment":"The displayed moment chain E∏_{j=1}^p |A_j|^2 ≤ ∏_{j=1}^p (E|A_j|^{2j+1})^{1/(2j)} ≤ ∏_{j=1}^p (c_{2j+1}(t-s)^{3/8})^2 is not justified. The first inequality is not a valid Hölder or Cauchy-Schwarz estimate (it already fails in general for p=2), and the second does not follow from Lemma 3.6: Lemma 3.6 controls the L^{2j+1} norm, i.e. (E|A|^{2j+1})^{1/(2j+1)}, not the displayed 1/(2j)-th power. A correct route is to apply Hölder with a common exponent p, obtaining (E|A_j|^{2p})^{1/p}, and then use Lemma 3.6 with m=2p; the claimed (t-s)^{3p/4} bound can be recovered that way. As written, however, the proof of the crucial estimate (3.6) is invalid, and (3.6) is used in Lemma 3.7 and Theorem 3.1.","section":"Section 3.2, proof of Proposition 3.4"}],"minor_comments":[{"comment":"The displayed normalization of W^ε appears inconsistent: equation (1.9) and the bracket computations in Section 4 correspond to a coefficient ε/√2, whereas the displayed equations suggest ε√2. Please check and correct the displayed coefficient if it is not a typesetting artifact.","section":"Equations (1.8) and (4.1)"},{"comment":"The statement that h^ε is equal to √ε on the support of η^ε is not correct in general: a site with h^ε=0 and Δ^ε h^ε = -2√ε satisfies the reflection condition, so h^ε can vanish on the support. The inequality h^ε ≤ √ε on the support is sufficient for the argument, since it still gives F(h^ε,η^ε) ≤ √ε ∫ xψ dη^ε.","section":"Section 6, proof of item (v), after (6.5)"},{"comment":"In the sentence 'for all φ∈S′([0,∞))' the space should be S([0,∞)) (Schwartz functions), not the space of distributions S′; the same correction applies in the surrounding argument where φ is used as a test function.","section":"Section 4, proof of Theorem 4.1"},{"comment":"The proof applies Dynkin's formula to a complex-valued function without comment; this is harmless after splitting into real and imaginary parts, but a brief remark would improve readability.","section":"Section 3.2, Lemma 3.5"}],"recommendation":"major_revision","confidential_remarks":"The two major comments are both localized in Section 3.2 and appear fixable without changing the architecture of the proof. The rest of the paper, especially the noise convergence and the limit identification, is in good shape. I therefore recommend major revision rather than rejection; the central claim is plausible and the gaps are technical rather than conceptual."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe headline: this paper proves the scaling limit of a pinned corner-flip interface above a hard wall on the half-line to the reflected stochastic heat equation, with the 3D Bessel law as invariant. That closes the natural infinite-volume case left open by Etheridge and Labbé's segment result. The new work is in the infinite-volume estimates: Sobolev-norm bounds uniform in the lattice spacing, and control of the reflection measure through the semimartingale structure. The identification step uses external uniqueness from Hambly and Kalsi, so there is no circularity, and the citation pattern is clean.\n\nThe paper is clearly written and the strategy is standard for this literature, but two proof details are off. In Lemma 3.5, Dynkin's formula is applied to a non-cylindrical functional on the infinite lattice. For fixed ε the expression for the generator makes sense: the Laplacian is uniformly bounded and the exponential weight makes the sum converge, so I don't think there is a divergence issue at fixed ε. What is missing is a short domain or truncation argument showing f_ζ is in the domain of the Markov generator. That is a minor omission, easily repaired because the bracket estimates are uniform in spatial cutoffs.\n\nIn Proposition 3.4, the Hölder chain as printed does not follow from Lemma 3.6: replacing (E|·|^{2j+1})^{1/(2j)} by (c(t-s)^{3/8})^2 is not justified by the lemma. The standard argument with m=2p gives the same conclusion, so the exponent chain needs a fix, not a new idea.\n\nNeither gap falsifies the main theorem. The tightness route is sound in outline and the repairable steps are localized. I would send this to a serious referee, expecting a minor revision. It is not a desk reject.","headline":"Half-line hard-wall interface convergence to reflected SHE is novel and essentially correct; two proof details need fixing before acceptance.","tokens_in":26246,"tokens_out":5823,"would_cite":true,"duration_ms":51974,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60K35","60F17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that stationary fluctuations of a hard-wall corner-flip interface, after diffusive rescaling, converge in law to the reflected stochastic heat equation on the half-line, with the 3D Bessel law as an invariant measure.","keywords":["reflected stochastic heat equation","hard wall","corner flip dynamics","random interface","3-dimensional Bessel process","invariant measure","scaling limit","half-line"],"falsifier":"A decisive check is to apply the generator to that infinite weighted sum explicitly and see whether the result is integrable under the stationary measure; if it diverges for some frequency, the time-increment bound breaks and the tightness proof collapses. A numerical surrogate: simulate the stationary interface at small $\\epsilon$ and estimate the worst-case Fourier increment in time; any growth faster than $(t-s)^{1/2}+\\epsilon^{3/4}$ contradicts the paper's core estimate.","tokens_in":25063,"feed_emoji":"🧱","tokens_out":15790,"duration_ms":128826,"temperature":0.7,"pith_summary":"Consider a height function on the nonnegative integers, starting at zero, whose neighboring heights always differ by one, and which flips local corners as long as the flip does not push it below zero, with an extra pinning at the origin. This paper proves that, starting from the stationary law of this process and rescaling space by $\\epsilon$, time by $\\epsilon^2$, and heights by $\\sqrt{\\epsilon}$, the interface fluctuations converge in law to the solution of the reflected stochastic heat equation on the half-line. The convergence is joint: the discrete noise and the discrete reflection term produced by the wall also converge, to space-time white noise and to the continuum reflection measure. As a corollary, the law of the 3-dimensional Bessel process starting from zero is invariant for the limiting SPDE. The result matters because it establishes a dynamical invariance principle through the wall constraint: the discrete exclusion effect becomes a reflection measure that pushes the continuum solution upward exactly when it touches zero.","feed_headline":"Hard-wall random interface scales to a reflected heat equation","feed_subtitle":"Stationary fluctuations converge to the reflected heat equation; the Bessel-3 law is invariant for it.","key_machinery":"The load-bearing identity is the rescaled semimartingale decomposition, which rewrites the discrete dynamics as $\\langle h^\\epsilon_t, \\varphi \\rangle_\\epsilon = \\langle h^\\epsilon_0, \\varphi \\rangle_\\epsilon + \\int_0^t \\epsilon^{-2} \\langle \\Delta^\\epsilon h^\\epsilon_s, \\varphi \\rangle_\\epsilon\\, ds + \\sqrt{2}\\, W^\\epsilon_t(\\varphi) + \\int \\varphi\\, d\\eta^\\epsilon$, with the discrete noise $W^\\epsilon$ and discrete reflection measure $\\eta^\\epsilon$ defined explicitly from the corner-flip rates. This is the exact discrete mirror of the weak form of the reflected SPDE. Around this identity, three tools carry the proof: static moment estimates for the conditioned random walk that gives the invariant measure $\\pi$, whose scaling limit is the 3-dimensional Bessel process; the Lyons-Zheng decomposition, which uses reversibility under $\\pi$ to reduce time-increment bounds on the Fourier transform of $h^\\epsilon$ to martingale moment estimates; and martingale convergence criteria applied to $W^\\epsilon$, whose bracket process is shown to converge to $t\\,\\|\\varphi\\|^2_{L^2}$.","core_discovery":"The central claim is Theorem 1.2: for an interface $h$ started from its stationary measure $\\pi$ and then rescaled, the triple $(h^\\epsilon, W^\\epsilon, \\eta^\\epsilon)$ converges in law as $\\epsilon \\to 0$ to $(u, W, \\eta)$, where $W$ is a cylindrical Wiener process and $(u, \\eta)$ is the solution of the reflected stochastic heat equation $\\partial_t u = \\partial^2_{xx} u + \\sqrt{2}\\,\\dot W + \\eta$ on $[0,\\infty)$, with Dirichlet condition $u(t,0)=0$, $u \\ge 0$, $\\eta \\ge 0$, and $\\int u\\, d\\eta = 0$, starting from an independent 3-dimensional Bessel-distributed initial condition. The proof shows that any limit point of the rescaled semimartingale equation satisfies the weak formulation of this SPDE, and then invokes strong uniqueness for the continuum equation to identify the limit. Corollary 1.3 states that the law of the 3-dimensional Bessel process starting from zero is invariant for the reflected SPDE.","pith_inferences":["A natural extension the authors leave implicit: varying the pinning strength or adding a slope at infinity should produce the same reflected SPDE with different boundary conditions, and the invariant measure should arise from a different conditioning transform of the simple random walk.","The explicit joint convergence of noise and reflection measure suggests that occupation-time and current fluctuations in the discrete model could be studied through the continuum pair $(u, \\eta)$, with $\\eta$ acting as a local time at the wall.","One could test universality numerically by running the stationary dynamics at small $\\epsilon$ and checking that the one-time spatial law matches the Bessel-3 law while the reflection measure sits on rare zero-set times; if another discrete interface model with the same constraint converges to the same SPDE, that would corroborate a universality conjecture the paper does not state."],"forward_implications":["If the theorem is right, the reflected stochastic heat equation on the half-line is the exact scaling limit of the stationary corner-flip interface with a hard wall and pinning at the origin.","The joint convergence gives a continuum description of the wall: the reflection measure $\\eta$ is supported on the zero set of $u$, so in the limit the interface touches zero only at exceptional times but is still pushed upward by that measure.","Corollary 1.3 provides an explicit invariant measure: the Bessel-3 law is stationary for the reflected SPDE, giving a concrete starting point for studying long-time behavior and correlations.","The convergence of the discrete noise identifies the noise strength in the limiting SPDE, fixing the coefficient $\\sqrt{2}$ in the equation as the correct fluctuation scale."],"supporting_citations":[{"why":"Supplies the double martingale moment technique used in the dynamical estimates.","marker":"[1]"},{"why":"Supplies the invariance principle for the random walk conditioned to stay nonnegative, giving convergence of the stationary laws to the 3D Bessel law.","marker":"[3]"},{"why":"Established the segment-case scaling limit to a Nualart-Pardoux reflected SPDE, the strategy this paper extends to the half-line.","marker":"[6]"},{"why":"Provides strong existence and uniqueness for the reflected SPDE on the half-line, used to identify the limit point.","marker":"[8]"},{"why":"Provides the Lyons-Zheng decomposition used to reduce tightness of the interface to martingale moment bounds.","marker":"[15]"},{"why":"Introduced the Nualart-Pardoux reflected SPDE whose half-line analogue is the target equation.","marker":"[17]"}],"fun_headline_variants":["Reflected heat equation limit from hard-wall interface","Half-line SPDE emerges from corner flip dynamics","Bessel-3 law invariant for reflected heat equation","Interface scaling limit: reflected heat equation on half-line"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's most delicate premise is that a standard calculus identity (Dynkin's formula) can be applied to an infinite weighted sum of the interface heights, with no approximation argument showing that the sum lies in the domain of the generator; if that identity fails, the central bound on time increments of the interface, and hence the compactness of the rescaled interfaces, loses its main support.","fun_headline_variants_meta":{"raw":{"variants":["Reflected heat equation limit from hard-wall interface","Half-line SPDE emerges from corner flip dynamics","Bessel-3 law invariant for reflected heat equation","Interface scaling limit: reflected heat equation on half-line"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1581,"prompt_tokens":860,"completion_tokens":721,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":660}},"tokens_in":476,"tokens_out":721,"duration_ms":6604,"temperature":1.0,"reasoning_tokens":660,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:31:28.930221+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is to apply the generator to that infinite weighted sum explicitly and see whether the result is integrable under the stationary measure; if it diverges for some frequency, the time-increment bound breaks and the tightness proof collapses. A numerical surrogate: simulate the stationary interface at small $\\epsilon$ and estimate the worst-case Fourier increment in time; any growth faster than $(t-s)^{1/2}+\\epsilon^{3/4}$ contradicts the paper's core estimate.","supporting_citations":[{"cited_title":"Bertini and G","cited_arxiv_id":null,"evidence_quote":"Supplies the double martingale moment technique used in the dynamical estimates."},{"cited_title":"Bryn-Jones and R","cited_arxiv_id":null,"evidence_quote":"Supplies the invariance principle for the random walk conditioned to stay nonnegative, giving convergence of the stationary laws to the 3D Bessel law."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established the segment-case scaling limit to a Nualart-Pardoux reflected SPDE, the strategy this paper extends to the half-line."},{"cited_title":"Hambly and J","cited_arxiv_id":null,"evidence_quote":"Provides strong existence and uniqueness for the reflected SPDE on the half-line, used to identify the limit point."},{"cited_title":"Lyons and W","cited_arxiv_id":null,"evidence_quote":"Provides the Lyons-Zheng decomposition used to reduce tightness of the interface to martingale moment bounds."},{"cited_title":"Nualart and E","cited_arxiv_id":null,"evidence_quote":"Introduced the Nualart-Pardoux reflected SPDE whose half-line analogue is the target equation."}],"review_version":2}