{"id":"28e2ced7-e1e5-46ff-951c-b4e4f860d34b","arxiv_id":"2508.11094","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For the open KPZ equation, the variance of the boundary height grows as a definite power of time for all length-to-time ratios up to t^{2/3}.","lead":"The authors prove matching upper and lower bounds on the height fluctuation variance for the open KPZ equation on a growing interval in the maximal current phase. The result pins down the fluctuation exponent for all interval lengths up to t^{2/3}.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The matching variance bounds hinge on sharp boundary control of the Gibbsian line ensemble for the stationary measure; the supplied full text is corrupted mojibake, so this key input cannot be verified.","rationale":"Only the abstract is readable; the full text is corrupted mojibake, so the proof cannot be checked directly. The central mathematical question is whether the variance of H(0,t) under stationary initial conditions is controlled at the boundary by the line ensemble. Since the process is stationary, the time dependence drops out and the result reduces to a statement about the L-dependence of the stationary measure's boundary variance. The paper's stated strategy is a combination of a periodic-KPZ input (which has no boundary) and line ensemble inputs (which are supposed to handle the boundary). The fragile point is the boundary lower bound: the upper bound may follow from stationarity or from soft arguments, but the matching lower bound requires a sharp one-point estimate at x=0. This is a genuine concern because the abstract explicitly identifies the combination of these techniques as the proof strategy, and the reader's weakest assumption is the same. The proposed check is to reopen the actual source and verify that the cited line ensemble results indeed provide the required boundary one-point lower bound with the correct L-scaling. If they do, the central claim is likely correct; if not, the argument has a gap. Since this check cannot be performed with the corrupted text, the verdict remains UNVERDICTED, unchanged from the reader.","tokens_in":1120,"tokens_out":11138,"duration_ms":121896,"concrete_test":"Repair the corrupted LaTeX source (obtain the original .tex or .pdf from arXiv) and locate the theorem that proves the lower bound for Var H(0,t). Trace the proof back to the cited line ensemble papers: verify that arXiv:2306.05983 or arXiv:2404.13444 contains a lower bound of order L on the boundary height variance of the stationary measure in the maximal current phase, and that the transfer argument from periodic KPZ (arXiv:2111.03650) does not implicitly require L >> t^{2/3} or an additional boundary regularity not present in the cited work. If no such boundary lower-bound statement exists in the cited papers, the matching lower bound is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that for L~t^alpha with stationary initial data, Var H(0,t) has matching upper and lower bounds for alpha in [0,2/3]. If the initial condition is the invariant measure, the open KPZ process is time-stationary, so Var H(0,t) equals the boundary variance of the stationary measure for interval length L. The nontrivial content is therefore a two-sided estimate of the stationary measure's boundary one-point variance, of order L (or t^alpha), in the maximal current phase. The abstract says this is obtained by combining the periodic KPZ techniques of arXiv:2111.03650 with Gibbsian line ensemble methods from the listed prior works. The load-bearing step is that these line ensemble constructions yield sharp (upper and lower, same order) estimates at the boundary point x=0 for the maximal current phase. The periodic KPZ methods have no boundary and cannot by themselves produce boundary control; the line ensemble papers must supply it. If those papers only give bulk bounds, or only upper bounds, or only estimates for fixed L rather than L~t^alpha, the matching lower bound at H(0,t) does not follow. Since the full text is an unreadable encoding artifact, no section or equation can be cited to confirm that this boundary estimate is actually established. The reader's weakest assumption is exactly this, and it is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the open KPZ equation H(x,t) on [0,L] with Neumann boundary conditions in the maximal current phase (parameters u,v >= 0). The announced result is that, for stationary initial conditions and L ~ t^alpha, the variance of the boundary height H(0,t) satisfies matching upper and lower bounds for every alpha in [0,2/3], thereby determining the boundary-height fluctuation exponent in that scaling window. The proof is said to combine the periodic KPZ methods of arXiv:2111.03650 with Gibbsian line ensemble machinery developed for stationary measures in five cited works. The abstract is the only readable part of the submitted file; the full text is an unreadable encoding artifact, so no theorem statements, proofs, or lemmas could be examined.","tokens_in":1373,"tokens_out":5083,"duration_ms":58320,"significance":"If the claimed matching bounds are correct, the result is significant: it would give a two-sided determination of the boundary-height fluctuation exponent for the maximal-current open KPZ equation over the full L ~ t^alpha window, including the KPZ dynamic-scaling endpoint alpha=2/3. The statement is parameter-free in that no fitted parameter is introduced, and it is falsifiable by comparison with exact or numerical variance computations. The announced strategy is plausible and builds naturally on known stationary-measure constructions. However, any assessment of significance is conditional: the current submission does not provide readable mathematical content, so the proof cannot be verified.","major_comments":[{"comment":"The full manuscript as submitted is a single block of unreadable mojibake; no theorem, lemma, definition, or section number is legible. This is not a minor formatting issue: the central claim of matching variance bounds cannot be checked, and no equation or argument can be cited in support. The manuscript must be resubmitted with a readable text before any substantive review is possible.","section":"Full text"},{"comment":"The abstract says the proof combines periodic KPZ techniques from arXiv:2111.03650 with Gibbsian line ensemble methods. Periodic KPZ has no boundary, so the matching lower bound for Var H(0,t) at x=0 must come from the line ensemble estimates for the stationary measure on [0,L]. The submitted text does not state or establish that those estimates are sharp at the boundary for L ~ t^alpha in the maximal current phase. If the cited line ensemble papers supply only bulk bounds, one-sided bounds, or estimates for fixed L, the claimed lower bound does not follow; the manuscript needs an explicit boundary-variance theorem with proof.","section":"Abstract (proof strategy)"},{"comment":"Because the initial condition is the invariant measure, the open KPZ process is time-stationary, so Var H(0,t) equals the variance of the boundary height in the stationary measure on an interval of length L; the t-dependence enters only through L ~ t^alpha. The manuscript should state this reduction explicitly and identify the property of the stationary measure that is responsible for the growth of the boundary variance, since this reduction is the conceptual core of the result.","section":"Abstract (stationary initial conditions)"}],"minor_comments":[{"comment":"The abstract does not describe the constants in the matching upper and lower bounds; the precise theorem should state that the bounds hold with constants independent of t and L uniformly over alpha in [0,2/3].","section":"Abstract"},{"comment":"Clarify whether u and v are fixed positive constants or are allowed to depend on L, since the maximal-current phase and the estimates may be sensitive to their values.","section":"Abstract"},{"comment":"The reference list is garbled in the submitted file; the resubmission should ensure all bibliographic entries and arXiv identifiers are legible and correctly matched to the citations in the abstract.","section":"Full text"}],"recommendation":"uncertain","confidential_remarks":"The recommendation is uncertain solely because the submitted full text is an unreadable encoding artifact; I am not expressing doubt about the mathematics. I recommend asking the authors to resubmit a properly encoded PDF or TeX source, and to include an explicit statement and proof of the boundary variance estimate for the stationary measure under L ~ t^alpha scaling."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The advertised result—matching upper and lower bounds on Var H(0,t) for the open KPZ equation in the maximal current phase, for L ~ t^alpha, alpha in [0,2/3], stationary initial data—is a genuine gap-fill if it holds. The supplied full text is a corrupted mojibake dump; only the abstract and metadata are readable, so the proof is completely uncheckable from what we got.\n\nWhat's genuinely new: prior work treated the periodic KPZ equation (2111.03650) and the Gibbsian structure of stationary measures (the listed line ensemble papers), but the boundary scaling window here hasn't been treated. The abstract states a theorem for a regime that matters for boundary-driven growth. The authors are honest about the ingredients: periodic techniques alone have no boundary, so the line ensemble machinery must supply sharp one-point estimates at x=0. That division of labor is sensible.\n\nThe soft spot is exactly that boundary step, and the stress-test note has it right. If the initial condition is the invariant measure, the process is time-stationary, so Var H(0,t) equals the stationary measure's boundary variance for interval length L. The nontrivial content is therefore a two-sided estimate of that stationary variance, of order L, for all alpha in the window. The abstract doesn't state a lemma that guarantees the line ensemble gives matching lower and upper bounds at the boundary in the maximal current phase. That could be in the full text, and I'd guess the authors have it, but from the abstract alone it's an assertion. The other cited works may only give bulk bounds or fixed-L estimates; the whole theorem rests on this.\n\nThe citation pattern looks normal: several works by overlapping authors, but that's standard for a construction built over a series of papers, and the cited results are real.\n\nWhere does this leave us? The claim is plausible and the abstract is well written. The math can't be checked, and that's a practical problem, not necessarily a flaw in the work. If the editorial office can get a readable version, this deserves a serious referee. Without it, desk rejection isn't fair either—ask for a corrected source first. For a reading group, I'd bring it up as a topic, but I wouldn't ask anyone to read the corrupted text. I wouldn't cite it in my own work until the proof is verified.\n\nMy recommendation: engage with it, but only with a clean copy.","headline":"The abstract advertises a real gap-fill for the open KPZ maximal current phase, but the supplied full text is a mojibake dump, so the proof is uncheckable and the boundary control step is the key unknown.","tokens_in":1799,"tokens_out":2496,"would_cite":false,"duration_ms":25541,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60K35","82B23"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the open KPZ boundary height variance is of order L for L~t^alpha, alpha at most 2/3.","keywords":["open KPZ equation","maximal current phase","fluctuation exponents","boundary height variance","stationary initial conditions","Gibbsian line ensemble","Brownian Gibbs property","KPZ universality"],"falsifier":"Run a high-resolution simulation of the open KPZ equation (or an exactly solvable lattice model in the same universality class) with stationary initial data and take $L=t^{1/3}$, measuring $\\operatorname{Var} H(0,t)$ over a wide range of $t$. The paper's claim predicts this quantity stays comparable to $t^{1/3}$ (linear in $L$) with constants independent of $t$; a clear drift toward the $t^{2/3}$ rate, or a value independent of $L$, would contradict the result.","tokens_in":965,"feed_emoji":"📈","tokens_out":23786,"duration_ms":277618,"temperature":0.7,"pith_summary":"The paper treats the open KPZ equation, a one-dimensional random-growth model on an interval $[0,L]$ with boundary noise in the maximal-current phase. With stationary initial data and interval length growing as $L\\sim t^\\alpha$, it proves matching upper and lower bounds on the variance of the height at the boundary, $\\operatorname{Var} H(0,t)$, for every $\\alpha\\in[0,2/3]$. The matching bounds show the variance is of order $L=t^\\alpha$ up to constants, so the boundary fluctuation exponent in this scaling window is exactly $\\alpha$. The interest is that the open boundary and its stationary measure are non-Gaussian and not translation invariant; the bounds establish that, on short enough intervals, the boundary height is nonetheless governed by the same linear-in-length variance as the equilibrium Brownian-like profile.","feed_headline":"Open KPZ boundary height variance grows linearly with interval length","feed_subtitle":"Matching bounds fix the boundary fluctuation exponent for all length growth rates up to the two-thirds power.","key_machinery":"The load-bearing object is the Gibbsian line ensemble representation of the stationary measure of the open KPZ equation in the maximal-current phase. The stationary height profile is realized as the top curve of an ordered ensemble of random curves with a Brownian Gibbs resampling property and boundary weights set by $u$ and $v$. The proof adapts the periodic-KPZ strategy to this ensemble and compares the boundary point $H(0,t)$ with the equilibrium Brownian profile of length $L$; the Brownian Gibbs property supplies the control of increments needed to turn ensemble-level estimates into matching upper and lower variance bounds.","core_discovery":"The central claim is the two-sided estimate $$c\\, L \\le \\operatorname{Var} H(0,t) \\le C\\, L$$ for $L\\sim t^\\alpha$ and $\\alpha\\in[0,2/3]$, with $c,C>0$ independent of $L$ and $t$; equivalently, $c\\, t^\\alpha \\le \\operatorname{Var} H(0,t) \\le C\\, t^\\alpha$. Throughout this range the interval length is at most the KPZ correlation length scale, so the system sits in the equilibrated regime and the length, rather than the elapsed time, sets the fluctuation scale. This determines the fluctuation exponent $\\gamma(\\alpha)=\\alpha$ for the boundary height in the maximal-current phase over the full stated length-time window.","pith_inferences":["A natural next target is the complementary regime $\\alpha>2/3$, where the interval is longer than the correlation length and one expects the variance to cross over to the $t^{2/3}$ time scaling; the present bounds give the matching endpoint at $\\alpha=2/3$.","The mechanism suggests that an exactly solvable lattice discretization of the open KPZ in the maximal-current phase will show the same linear-in-$L$ boundary variance, so Monte Carlo data there would be a direct universality test.","Because the line ensemble controls the whole profile, the same two-sided bounds likely extend to interior points $H(x,t)$ away from the boundary, with constants depending on $x$ but the same linear growth in $L$."],"forward_implications":["In the entire window $\\alpha\\in[0,2/3]$, the boundary height variance is of order $L$, not of order $t^{2/3}$; the length scale wins because the system has equilibrated.","For stationary initial data on a fixed interval ($\\alpha=0$), the variance remains of order $L$ as $t\\to\\infty$, a quantitative saturation statement.","The upper and lower bounds match up to constants, so the fluctuation exponent is sharp with no logarithmic corrections in this scaling window.","The proof transfers the periodic-KPZ variance mechanism to open boundaries, showing that the maximal-current stationary line ensemble carries the same variance behavior as the periodic geometry."],"supporting_citations":[],"fun_headline_variants":["Open KPZ boundary variance scales linearly with L","Maximal-current KPZ: height variance scales as interval length","Boundary height variance in open KPZ grows like L","Open KPZ: variance exponent equals length growth exponent","Open KPZ fluctuation exponent: boundary variance linear"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the line ensemble representation of the stationary measure gives sharp, boundary-regular control of the top curve; if the Brownian Gibbs construction fails to control the boundary value $H(0,t)$ at these parameters, the matching variance bounds do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Open KPZ boundary variance scales linearly with L","Maximal-current KPZ: height variance scales as interval length","Boundary height variance in open KPZ grows like L","Open KPZ: variance exponent equals length growth exponent","Open KPZ fluctuation exponent: boundary variance linear"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000893,"raw_usage":{"total_tokens":3799,"prompt_tokens":842,"completion_tokens":2957,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":458,"completion_tokens_details":{"reasoning_tokens":2880}},"tokens_in":458,"tokens_out":2957,"duration_ms":22898,"temperature":1.0,"reasoning_tokens":2880,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:26:58.178599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a high-resolution simulation of the open KPZ equation (or an exactly solvable lattice model in the same universality class) with stationary initial data and take $L=t^{1/3}$, measuring $\\operatorname{Var} H(0,t)$ over a wide range of $t$. The paper's claim predicts this quantity stays comparable to $t^{1/3}$ (linear in $L$) with constants independent of $t$; a clear drift toward the $t^{2/3}$ rate, or a value independent of $L$, would contradict the result.","supporting_citations":[],"review_version":2}