{"id":"95b2033b-1f74-48a9-8e6d-beed68ec8264","arxiv_id":"2608.11297","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Rényi entanglement and non-hydrodynamic correlators in noisy symmetric systems follow from the geometry of k-commutant manifolds, with singularities from frozen states producing diffusive √t growth and e^{-√t} decay in 1D.","lead":"This paper shows that the slow dynamics of noisy quantum systems with conserved charge or spin is controlled by the shape of a special space of states, including its singular points. It explains previously known slow growth of entanglement and makes new predictions for systems with non-Abelian symmetries.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"SU(2) logarithmic-entanglement prediction rests on an unverified equal-superposition postulate for a continuum of degenerate TDVP trajectories; the paper provides no microscopic derivation or numerical check for the SU(2) effective model.","rationale":"The reader's weakest assumption groups the TDVP formalism and the two singularity/superposition prescriptions. I agree that the overall TDVP-on-singular-manifold program is the soft spot, but the pressure-equalization condition (Eq. 87) is not load-bearing for the headline exponents because the crossing points are fixed by symmetry in the infinite-system domain-wall and single-particle setups; it only affects finite-size saturation. In contrast, the equal-superposition prescription is the entire origin of the SU(2) log-t entanglement claim, which is a central advertised result and has no numerical support in the SU(2) setting. The Heisenberg and U(1) tests are encouraging but involve respectively a smooth manifold and a discrete two-trajectory superposition; neither validates the continuum-superposition step on a singular manifold. Since this is a testable prediction rather than an internal inconsistency, it does not warrant rejection; it strengthens the case for a conditional verdict pending either a simulation of the SU(2) effective model or an analytic derivation of the superposition measure from the replica path integral. The reader already reached CONDITIONAL, and my concern refines the same weakness without moving the verdict, so UNCHANGED is appropriate.","tokens_in":57853,"tokens_out":7733,"duration_ms":77878,"concrete_test":"Perform high-precision MPS imaginary-time evolution of the microscopic SU(2) Brownian effective Hamiltonian P^(2) (derived from the generator \\vec S_i·\\vec S_j, Eq. (167)) for L≈20–30 starting from the half-chain domain wall |A:\\bar A⟩; measure the half-chain second Rényi entropy of the evolved replicated state as a function of κt. If it grows logarithmically with the coefficient predicted by Eq. (187) before finite-size saturation, the equal-superposition continuum prescription is supported; if it saturates or grows at a different rate, the SU(2) central claim would be undermined. As a cross-check, repeat the semiclassical ansatz with a non-uniform measure on S^2 and verify whether the log prefactor changes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central U(1) results—√t Rényi growth and e^{-√κt} non-hydrodynamic correlator decay—are driven by single TDVP trajectories and are numerically supported in the esη and U(1) models; the two-trajectory superposition in U(1) is further corroborated by the log-2 entropy offset in Fig. 7c. The least secure element of the central claim is the equal-superposition prescription for a continuous family of degenerate trajectories, which is the sole basis for the SU(2) prediction of logarithmic entanglement growth within the void [Eq. (187), Appendix E]. Sec. IV B 4 introduces this prescription as a postulate ('we posit') and it is calibrated only in the Heisenberg model, where the manifold is smooth, and in the two-trajectory U(1) case. For SU(2) the manifold intersection is (S^2)^4, so the family is a continuum; the measure dμ(n) is not derived from the microscopic Brownian model, and no SU(2) effective-Hamiltonian simulation is presented to check even the existence of the cat state. If the true evolved replicated state is not an equal-weight coherent superposition—for example, if fluctuation determinants around different saddles are n-dependent or the saddles decohere—the log t growth could be replaced by saturation or a different rate. This is a load-bearing gap because the abstract's non-Abelian claim depends on it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a geometric framework for the averaged imaginary-time dynamics of replica Hamiltonians that arise in noisy Brownian circuits with continuous symmetries. The authors show that the ground-state manifolds of the effective replica Hamiltonians are controlled by the geometry of k-commutants, and that in interacting systems these manifolds are unions of smooth branches (e.g., (S^2)^k for U(1)) that intersect at \"frozen\" states, producing singularities. Using the time-dependent variational principle (TDVP) on these manifolds, they derive diffusive equations of motion for low-energy field configurations, with additional prescriptions for branch-switching singularities (Dirichlet boundaries plus pressure equalization) and for equal superpositions of degenerate semiclassical trajectories. The framework is first calibrated on the ferromagnetic Heisenberg model and on a newly introduced three-state \"esη\" toy model, then applied to U(1)-symmetric models, where it yields sub-ballistic Rényi entanglement growth (~√t in 1D) and stretched-exponential decay of non-hydrodynamic correlators (~e^{-√κt} in 1D), and to SU(2)-symmetric models, where a continuous family of degenerate trajectories is predicted to produce logarithmic entanglement growth within the void. The paper also contrasts these interacting cases with free-fermion systems, whose commutant manifolds are smooth, and discusses higher replica numbers, higher dimensions, and extensions to other entropic quantities.","tokens_in":58296,"tokens_out":4968,"duration_ms":85136,"significance":"If the central claims hold, this paper provides a unified geometric explanation for several previously disparate phenomena: the diffusive growth of Rényi entropies in charge-conserving systems, the stretched-exponential decay of non-hydrodynamic correlators, the role of void states in rigorous bounds, and the difference between Abelian and non-Abelian symmetry structures. The TDVP equations are derived carefully and the analytic solutions are compared with MPS numerics in the Heisenberg and esη models, with partial checks in the U(1) sector; the predictions are concrete and falsifiable. The paper also clearly credits prior work on void arguments and on membrane pictures, and it is honest about the status of several key ingredients. The main significance risk is that the non-Abelian logarithmic-growth claim rests on an equal-superposition prescription for a continuum of trajectories that is not derived from the microscopic model and is not numerically checked in the SU(2) effective Hamiltonian.","major_comments":[{"comment":"The SU(2) logarithmic-entanglement prediction rests entirely on the equal-superposition prescription for a continuum of degenerate TDVP trajectories. The paper itself labels this a postulate in Sec. IV B 4, and it is calibrated only for the smooth Heisenberg domain wall and the discrete two-trajectory U(1) case. For SU(2), the degenerate family is parametrized by (S^2)^4, the measure dμ(n) in Eq. (181) is not derived, and no simulation of the SU(2) effective model is presented to check even the existence of the cat state. If the true replicated state is not an equal-weight coherent superposition---for example, if fluctuation determinants around different saddles are n-dependent or the saddles decohere---the log t growth could be replaced by saturation or a different rate. Because the abstract's non-Abelian claim depends on this, the gap is load-bearing and needs either a microscopic derivation or a numerical test.","section":"Sec. VII B 2-3, Eqs. (181)-(187)"},{"comment":"The exact determination of the ferromagnetic ground-state spaces and the manifold identifications, including Eq. (49) for U(1) and Eq. (168) for SU(2), is asserted to be proven in the in-preparation Ref. [75], with the text stating at one point \"This is rigorously proven in Ref. [75].\" These manifolds are the foundation of the TDVP analysis: if the ferromagnetic ground-state structure or the k-design property fails, the whole geometric picture changes. Since Ref. [75] is not available to the reader, the exactness of these central mathematical inputs cannot be checked from the manuscript alone. The authors should either include self-contained proofs, state the precise conditions under which the k-design/ferromagnetism holds, or explicitly mark these statements as conjectures for the purposes of this paper.","section":"Sec. III C and VI A 2, Eq. (128)"},{"comment":"The predicted log(κt) growth is computed for the entanglement of the variational state |A:\\bar A(t)⟩ itself, whereas the physical annealed Rényi entropy is defined through the replicated boundary-state overlap in Eq. (26), whose U(1) analog is evaluated in Eq. (147). The paper does not show how logarithmic entanglement growth of the variational state translates into logarithmic growth of the overlap observable that defines the Rényi entropy. Without such a mapping, the SU(2) logarithmic-growth claim is not quantitatively connected to the quantity originally defined; the authors should either spell out this connection or restrict the claim to the variational state.","section":"Sec. VII B 3 and Appendix E, Eq. (187)"}],"minor_comments":[{"comment":"The numerical comparison for the U(1) purity prediction is restricted to short times, and the text notes that the predictions with the cusped weight function r(x,t) and the Gaussian weight function ř(x,t) are almost indistinguishable for the tested initial states. This limitation should be stated more prominently, since the cusp is a distinctive qualitative prediction of the present framework.","section":"Sec. VI B 4 and Fig. 8"},{"comment":"The abstract mentions Haar-random circuits as part of the framework's scope, but the body explicitly treats Brownian models and only argues that the results are expected to generalize to other circuits forming symmetric k-designs. A more precise wording in the abstract would avoid overstating the scope.","section":"Abstract and Sec. VIII"},{"comment":"The t^{1/3} growth of the sink region is reported numerically but no derivation is given. Since this exponent is used as evidence that fluctuations beyond TDVP are subleading, a short analytic argument or a reference to a derivation would strengthen the claim.","section":"Sec. V D 2 and VI B 2"},{"comment":"The notation |σ_p⟩ with p∈{0,1} is introduced in Eq. (126), but the meaning of the subscript p and its relation to the local parity sector defined in Eq. (125) could be stated more explicitly for readers not familiar with the replica basis.","section":"Sec. VI A 2, Eq. (126)"},{"comment":"The discussion of the breakdown of the entanglement membrane picture is clear, but the relationship between the present cusped weight function and the large-q extrapolation of Ref. [31] would benefit from a more explicit statement that the two predictions also differ in their functional form, not only in their derivation.","section":"Eq. (150) and Sec. VI B 3"}],"recommendation":"major_revision","confidential_remarks":"This is a well-executed paper with a sound central derivation for the U(1) sector and a clearly posed but unverified SU(2) prediction. The heavy reliance on the in-preparation Ref. [75] for exact ground-state manifold statements should be flagged as a dual-submission/availability issue: if Ref. [75] is not accessible, the exactness claims lack support. I recommend major revision rather than rejection because the U(1) results and the TDVP machinery are credible and locally supported by numerics, while the SU(2) logarithmic-growth claim is the main load-bearing gap; the abstract's non-Abelian claim should be deferred until the equal-superposition prescription is derived or numerically checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a serious paper that deserves referee time. It builds a geometric framework—k-commutant manifolds, with singularities produced by frozen states—that unifies the known U(1) results (√t Rényi growth, e^{-√t} non-hydrodynamic correlator decay) and makes a genuinely new prediction for non-Abelian symmetries: log t entanglement from a continuum of degenerate TDVP trajectories. The U(1) core is in good shape; the non-Abelian headline rests on unverified ground.\n\nWhat's new and good. The singular manifold picture is a real conceptual step. The esη toy model is well chosen and the calibration against MPS data is careful—void grows diffusively, sink subdiffusively. The TDVP equations, pressure-equalization condition, and normalization conventions are worked out in gory detail. The exponents are not fit; they come from solving diffusion equations on the manifold. The cusped weight function r(x,t), as opposed to the Gaussian of Ref. [31], is a concrete, testable difference, even if current numerics cannot yet discriminate it.\n\nWhere it's soft, in order. (1) The foundational input—ferromagnetism of P^(2), the exact ground-state manifolds, the symmetric 2-design verification—is deferred to the companion Ref. [75]. That is acceptable if that paper delivers, but it makes this paper non-self-contained on its central objects. (2) The SU(2) log t prediction depends on the equal-superposition postulate for a continuum of trajectories (Sec. IV B 4 and Eq. (181)). The measure dμ(n) is not derived, and there is no numerical test of the resulting cat state in the SU(2) effective model, whose local Hilbert space is 16-dimensional. The U(1) two-trajectory case is supported by the log(2) entropy offset in Fig. 7c, but that is a single finite-size check; the continuous case could easily fail if fluctuation determinants are n-dependent or the saddles decohere. The paper honestly labels this as a postulate, but the abstract's non-Abelian claim leans entirely on it. (3) The singularity prescriptions (Dirichlet conditions, pressure equalization) are posited and calibrated on the toy model; their extension to U(1) is plausible and partly tested, to SU(2) much less so. Minor: the d≥3 η-particle dynamics introduces a UV cutoff b that is a free parameter, though the resulting exponential decay is probably robust.\n\nWho it's for. Random-circuit and replica-theory people, and anyone working on entanglement growth in symmetric systems. It deserves a serious referee. My recommendation: send it to review, and ask the authors to state precisely which statements are proven in [75] (ideally with that paper available), and to either derive the superposition measure in a simplified non-Abelian setting or test it numerically on a minimal model—or, until that is done, to scale back the abstract's claim. The U(1) framework and the toy model stand on their own.","headline":"A serious geometric-framework paper with a solid U(1) core and an intriguing but unverified SU(2) log-t prediction that rides on a continuous equal-superposition postulate.","tokens_in":58678,"tokens_out":5772,"would_cite":true,"duration_ms":46310,"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":"Late-time dynamics of noisy many-body systems with continuous symmetries — Rényi entanglement growth and non-hydrodynamic correlator decay — is controlled by the quantum geometry of k-commutant manifolds and their singularities.","keywords":["Brownian circuits","k-commutant manifolds","replica Hamiltonians","Rényi entanglement entropy","non-hydrodynamic correlators","continuous symmetries","time-dependent variational principle","void states"],"falsifier":"Simulate the $k=2$ effective Hamiltonian of the $U(1)$ Brownian model (Eqs. 124–129) and measure the spatial profile of the purity weight function of an evolved half-chain domain wall at long times: the paper predicts a purity decay $e^{-\\sqrt{\\kappa t}}$ with the error-function weight $r(x,t)=\\tan(\\pi/4\\,(1-|\\mathrm{erf}(x/\\sqrt{4\\kappa t})|))$ and a cusp at the interface, whereas the alternative large-$q$ membrane picture (Ref. [31], Eq. 150) predicts a Gaussian weight at the same exponent; observing a Gaussian profile, or a one-dimensional purity exponent different from $1/2$, would falsify the singularity-driven void mechanism.","tokens_in":57687,"feed_emoji":"⚛️","tokens_out":21683,"duration_ms":163030,"temperature":0.7,"pith_summary":"The paper claims that in noisy (Brownian) quantum circuits carrying a continuous global symmetry, the average late-time dynamics of Rényi entanglement entropies and of non-hydrodynamic correlation functions is set by the quantum geometry of $k$-commutant manifolds — the ground-state manifolds of the effective replica Hamiltonians that describe the noise-averaged evolution. In generic interacting systems these manifolds are fixed by the symmetry group alone and contain singularities, generated by frozen states, where different replica-symmetry branches touch. A domain wall of replicas (the object that computes entanglement) or a non-hydrodynamic local operator relaxes by driving the field through such a singularity, nucleating a diffusively melting void; the void produces sub-ballistic $S_2\\sim\\sqrt{\\kappa t}$ Rényi entropy growth in one dimension and stretched-exponential decay $e^{-\\sqrt{\\kappa t}}$ of non-hydrodynamic correlators, with exponential decay in $d\\ge 3$ and $e^{-\\kappa t/\\log t}$ in $d=2$. For non-Abelian symmetries the singular set is itself a continuous manifold, so a continuum of degenerate semiclassical trajectories contributes and the entanglement inside the void grows logarithmically. The authors calibrate these geometric rules on exactly solvable toy models and check them against tensor-network simulations of the effective models.","feed_headline":"Entanglement grows as √t in symmetric noisy circuits","feed_subtitle":"Rényi entropies grow sub-ballistically and non-hydrodynamic correlators decay as e^{−√κt}, both set by k-commutant geometry.","key_machinery":"The $k$-commutant manifold: the set of fully polarized product states spanning the ground-state space of the effective replica Hamiltonian $P^{(k)}$, equivalently the symmetry algebra of $k$ replicas of the noisy circuit. In interacting systems it is a complex projective variety whose singularities sit at replicated frozen states, and its Fubini-Study metric turns the time-dependent variational principle into a diffusive heat equation $\\partial_t v^\\gamma = 2J(\\nabla^2 v^\\gamma + \\Gamma^\\gamma_{\\alpha\\beta}\\nabla v^\\alpha\\cdot\\nabla v^\\beta)$ for the semiclassical field. The machinery is completed by two posited rules: branch-switching points act as Dirichlet boundaries that drift to equalize the field pressure $|\\partial_x\\mathbf{n}(x^-)|^2 = |\\partial_x\\mathbf{n}(x^+)|^2$, and when several lowest-energy trajectories exist the evolved state is their equal superposition. These ingredients convert observable dynamics into pure geometry: entanglement growth and correlator decay are fixed by how fields on the manifold traverse its singularities.","core_discovery":"The central claim is that the late-time physics of symmetric Brownian circuits — for both entanglement and correlation observables — reduces to the imaginary-time dynamics of smooth field configurations on ferromagnetic ground-state manifolds of the effective replica Hamiltonians $P^{(k)}$, the $k$-commutant manifolds. For interacting systems that form symmetric $k$-designs these manifolds are determined by the symmetry alone: for $U(1)$ the $k=2$ manifold is two copies of $S^2\\times S^2$ touching at the two frozen states, and for $SU(2)$ it is two copies of $\\mathrm{CP}^3\\times \\mathrm{CP}^3$ intersecting on a continuous $(S^2)^4$ submanifold. The authors show that the low-energy excitations obey TDVP equations that reduce to a heat equation along the manifold's Fubini-Study metric, and they add two calibrated prescriptions: singularities act as Dirichlet boundaries whose position drifts to equalize field pressure, and equally energetic trajectories superpose with equal weights. Applying these rules, one-dimensional Rényi entanglement grows as $\\sqrt{\\kappa t}$ because the domain wall melts through the singularity into a void, and the product of suppressed local weights inside the void makes the purity decay as $e^{-\\sqrt{\\kappa t}}$; non-hydrodynamic operators generate voids of the same structure, giving the same stretched-exponential decay of their squared correlators, with exponential decay in $d\\ge 3$ and $e^{-\\kappa t/\\log t}$ in $d=2$. For non-Abelian symmetries the void is threaded by a continuous family of degenerate trajectories, giving logarithmic entanglement growth within the void and a dynamical distinction between Abelian and non-Abelian continuous symmetries.","pith_inferences":["If the manifold geometry is the controlling datum, the same TDVP machinery should give the exponents for observables the paper lists but does not compute — symmetry-resolved entropies, entanglement asymmetry, and OTOCs — by dressing the $\\eta$-branch states with charged operators, with no new replica calculation required.","The numerically observed $t^{1/3}$ growth of the sink region attributes the leading correction to quantum fluctuations of the singularity; checking this exponent at longer times in the full $U(1)$ model would test whether the fluctuation picture survives beyond the $\\mathrm{es}\\eta$ toy model.","The Abelian/non-Abelian difference in trajectory degeneracy implies a measurable fingerprint: $U(1)$ voids should plateau at $\\log 2$ entanglement from the two-branch cat state, while $SU(2)$ voids should show persistent logarithmic growth, a contrast that is sharper than the shared $\\sqrt{t}$ entropy exponent.","Multipole-conserving and fragmented systems are a natural next testbed: the paper expects their commutant manifolds to be non-ferromagnetic, but if an analogous geometric structure exists, a void-like mechanism might survive with modified exponents, linking the geometric picture to constrained dynamics."],"forward_implications":["In one-dimensional interacting noisy $U(1)$-symmetric systems, the annealed second Rényi entropy of a half-chain domain wall grows as $\\sqrt{\\kappa t}$ with the explicit weight function $r(x,t)$ of Eq. (147), and the entanglement-membrane picture of sharp domain walls is replaced by diffusively melting voids.","Squared non-hydrodynamic autocorrelators decay as $e^{-\\sqrt{\\kappa t}}$ in $d=1$, as $e^{-\\kappa t}$ in $d\\ge 3$, and as $e^{-\\kappa t/\\log t}$ in $d=2$, while hydrodynamic correlators decay algebraically ($\\sim 1/(\\kappa t)$ in $d=1$) because replicas decouple within a single branch.","For $SU(2)$ and more generally any non-Abelian continuous symmetry with multiplicity-free irreps, the intersection of the $e$- and $\\eta$-branches is a continuous manifold, so the melting domain wall passes through a continuum of degenerate solutions and the void acquires logarithmic entanglement growth, unlike the finite two-branch 'cat' entanglement of the $U(1)$ case.","Because the $k$-commutant manifolds depend only on the symmetry whenever the generators form a symmetric $k$-design, the same exponents and weight functions apply to any isotropic interacting noisy model with the same symmetry — including Haar-random circuits — independent of microscopic details of the noise.","Extrapolating to $k\\ge 3$, the paper conjectures that the two shortest trajectories through the replicated frozen states still dominate, so all Rényi entropies grow diffusively as $\\sqrt{\\kappa t}$, while the von Neumann limit remains open and subtle."],"supporting_citations":[{"why":"establishes the mapping from commutant algebras of symmetric generators to ground states of local superoperators, giving the k=1 ferromagnetic description the paper extends to k≥2.","marker":"[45]"},{"why":"supplies the replica formulation in which Rényi entanglement is a domain wall between ground states in the effective Hamiltonian, the starting point for the TDVP computation.","marker":"[23]"},{"why":"develops the ground-state-manifold (commutant manifold) framework and its geometry for free-fermion models, the conceptual template and point of comparison used throughout.","marker":"[56]"},{"why":"the deferred companion work the paper cites for rigorous proofs that the U(1), SU(2) and higher-spin effective Hamiltonians are ferromagnetic with the stated manifold forms.","marker":"[75]"},{"why":"the earlier derivation of sub-ballistic Rényi growth in U(1) circuits via void bounds, whose weight function the paper refines and contrasts with its own.","marker":"[31]"},{"why":"the void-based argument for subexponential decay of local correlations, recovered here as the stretched-exponential signature of singularity crossing.","marker":"[57]"},{"why":"the stochastic-model analysis of non-hydrodynamic correlators whose t^{1/3} subdiffusion benchmarks the fluctuations around the semiclassical trajectories.","marker":"[58]"},{"why":"provides the TDVP equations that, on the manifold metric, become the diffusive heat equation driving the semiclassical dynamics.","marker":"[59]"}],"fun_headline_variants":["√t entanglement growth in symmetric noisy circuits","Geometry sets √t entanglement and anomalous correlator decay","Noisy symmetric circuits: k-commutant geometry rules","Voids in k-commutants drive √t entanglement and stretched-exponential decay","Abelian vs non-Abelian symmetries split noisy-circuit dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the semiclassical TDVP dynamics on the $k$-commutant manifold — including the posited singularity rules (Dirichlet pinning with pressure equalization) and equal-weight superposition of degenerate trajectories — captures the leading late-time imaginary-time evolution of the replicated boundary states; these prescriptions are calibrated on the $\\mathrm{es}\\eta$ toy model and then applied to the $U(1)$ and $SU(2)$ effective Hamiltonians without a derivation from the microscopic models, so if quantum fluctuations around the trajectories are not subleading, the predicted exponents and weight functions fail.","fun_headline_variants_meta":{"raw":{"variants":["√t entanglement growth in symmetric noisy circuits","Geometry sets √t entanglement and anomalous correlator decay","Noisy symmetric circuits: k-commutant geometry rules","Voids in k-commutants drive √t entanglement and stretched-exponential decay","Abelian vs non-Abelian symmetries split noisy-circuit dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00136,"raw_usage":{"total_tokens":5658,"prompt_tokens":1227,"completion_tokens":4431,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":843,"completion_tokens_details":{"reasoning_tokens":4346}},"tokens_in":843,"tokens_out":4431,"duration_ms":29531,"temperature":1.0,"reasoning_tokens":4346,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:13:11.075281+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the $k=2$ effective Hamiltonian of the $U(1)$ Brownian model (Eqs. 124–129) and measure the spatial profile of the purity weight function of an evolved half-chain domain wall at long times: the paper predicts a purity decay $e^{-\\sqrt{\\kappa t}}$ with the error-function weight $r(x,t)=\\tan(\\pi/4\\,(1-|\\mathrm{erf}(x/\\sqrt{4\\kappa t})|))$ and a cusp at the interface, whereas the alternative large-$q$ membrane picture (Ref. [31], Eq. 150) predicts a Gaussian weight at the same exponent; observing a Gaussian profile, or a one-dimensional purity exponent different from $1/2$, would falsify the singularity-driven void mechanism.","supporting_citations":[{"cited_title":"McCulloch, J","cited_arxiv_id":null,"evidence_quote":"the void-based argument for subexponential decay of local correlations, recovered here as the stretched-exponential signature of singularity crossing."}],"review_version":1}