Papers

We use microlocal tools to define distinguished incoming and outgoing vacuum states of the quantum theory of massive Klein–Gordon fields in non-trapping asymptotically flat spacetimes, including radiative perturbations of Minkowski space. The asymptotic vacuum states satisfy the Hadamard condition, and the incoming and outgoing particle representations are unitarily equivalent, or in other words the quantum S-matrix is well-defined. These results are based on a description of scattering of classical fields which keeps track of the asymptotic distinction between positive- and negative-frequency solutions of the Klein–Gordon equation. Our emphasis is on developing the theory without recourse to explicit asymptotics for classical waves.

We establish quantitative hypocoercivity for singular stochastic damped wave equations arising in stochastic quantization. Building on the finite-dimensional argument of Cao, Lu, and Wang [Arch. Rat. Mech. Anal. 2023], we derive a dimension-free criterion that replaces uniform semiconvexity by a log-Sobolev inequality together with exponential lower-tail bounds for the bottom of the Hessian spectrum. We verify this criterion for the two-dimensional $\Phi^4_2$ and sine-Gordon models and for the three-dimensional $\Phi^4_3$ model, through estimates on the lowest eigenvalue of associated singular random Schrödinger operators. Combined with the continuum limit established in a companion paper, this yields exponential $L^2$ convergence to equilibrium for the $\Phi^4_2$ dynamics and for sine-Gordon with $\beta^2<2\pi$. For $\Phi^4_2$, we verify that in a weak-coupling, small-mass regime, optimizing the damping yields a kinetic relaxation rate of at least order $m$. This yields a genuine acceleration over the order-$m^2$ relaxation rate of the overdamped dynamics. The lattice estimates are uniform in the discretization for sine-Gordon throughout $\beta^2<4\pi$, and for $\Phi^4_3$ at fixed volume, including regimes beyond those for which the continuum dynamics is vpresently constructed.

Replacing the circle by a manifold leads to current Lie algebras and their central and derivation extensions. We separate the coefficient algebra, harmonic analysis and geometric data. Differentials modulo exact forms describe central extensions; divergence-free vector fields give explicit cocycles, whose compatible derivations satisfy a stabilizer equation. Compact examples include the three-sphere and its current–Virasoro extension. We also explain the analytic qualifications for SL(2,R) and its symmetric space and the role of transported multiplication in soft geometries. The recent harmonic constructions of Campoamor-Stursberg, Marrani and Rausch de Traubenberg are compared with the cohomological work of Coquereaux, Frappat, Ragoucy and Sorba and Castellani's loop geometry and infinite-dimensional free differential algebras. A coefficient-algebra construction connects currents to higher brackets and clarifies the additional requirements of a dynamical theory. For Lie algebras of order F, central extensions obey coupled binary and symmetric F-ary conditions, with explicit examples and obstructions.

We prove that the simulated annealing algorithm efficiently samples from the Sherrington-Kirkpatrick model at inverse temperature $\beta < 1$ to an arbitrary inverse polynomial accuracy in total variation.

We prove Yau's number-theoretic conjecture on positive integral points in real right simplices in every dimension $n\ge3$, and apply the resulting sharp polynomial bound to quantum harmonic oscillators. The lattice-point bound holds for every choice of positive real intercepts for which the count is nonzero, with equality precisely when all intercepts equal the same integer. For oscillators in dimensions $n\ge3$, we obtain explicit upper bounds for eigenvalue counts and spectral remainders for arbitrary positive frequencies at every threshold at or above the lowest eigenvalue. For frequencies with rational ratios in dimensions $n\ge2$, an additional lattice coordinate and interpolation give bounds for energy sums, or first Riesz means, at every threshold. When all frequencies are equal (the isotropic case), the oscillator attains these bounds: the counting bound is exact at its energy levels, and interpolation on its natural energy grid recovers the energy sum at every threshold. In three dimensions, the exact formula displays the periodic contribution associated with the resonance discussed by Fefferman and Seco. The proof builds on the slicing method of Yau and Zhang. Arithmetic and geometric consequences include explicit bounds for smooth numbers and lattice-point discrepancies.

We study certain simple affine vertex algebras $L_{-1-h/6}(\mathfrak{g})$ from rank-one Argyres-Douglas theories, where $\mathfrak{g}$ is a non-simply-laced simple Lie algebra and $h$ is its Coxeter number. For $L_{-3}(F_4)$, we determine the maximal ideal of $V^{-3}(F_4)$, Zhu's algebra, and the associated variety. We also classify the irreducible highest-weight $L_{-3}(F_4)$-modules and determine the characters of its irreducible ordinary modules. The associated variety is $\overline{\mathbb{O}}_{\widetilde A_1}$, and the level $-3$ is totally collapsing for the orbit $\mathbb{O}_{\widetilde A_1}$. The Poisson coinvariants of the $C_2$-algebra of $L_{-3}(F_4)$ are isomorphic to $\mathbb{C}[\ell]/(\ell^5)$, where $\ell$ is induced by the conformal vector. Consequently, the characters of ordinary $L_{-3}(F_4)$-modules satisfy a fifth-order modular linear differential equation. We also give modular linear differential equations of orders five and eleven satisfied by the characters of ordinary modules for $L_{-2}(B_3)$ and $L_{-2}(G_2)$ respectively.

In this paper, we study planar compressible cavity flows fast-decaying vorticity and establish a Schauder estimate in large scale. As an application, we obtain an a priori asymptotic growth rate estimate of the cavity, that $s^{\frac{1}{2}-\delta}\lesssim h(s)\lesssim s^{\frac{1}{2}+\delta}$. This is by far the most accurate asymptotic estimate for compressible cavity flows and is probably the first paper that takes vorticity into account in the asymptotic analysis.

We prove that for $k=4$ the quantum uniformity measures are norms, whereas for every $k\geq5$ the triangle inequality fails. This gives a complete answer to an open question of Bu, Gu, and Jaffe. The $k=4$ case is proved using the Cauchy–Schwarz inequality for the quantum uniformity pairing $\langle\cdot,\cdot\rangle_{Q^4}$ introduced here, together with the triangle inequality in a suitable finite-dimensional Bochner $L^8$ space. For $k\geq5$, we construct systems of linear equations over a finite field and derive a contradiction from the structure theory of norming linear systems.

This paper concerns the Helmholtz scattering problem for open waveguide networks, extending the analysis from [arXiv:2302.04353, arXiv:2310.05816, arXiv:2401.04674] to plane-wave incident data. We first reduce the problem to a Fredholm second-kind integral equation on an infinite interface. This produces an integral representation of the solution to the original scattering problem, which is then used to obtain detailed asymptotic estimates of the solution at infinity. To leading order and away from the waveguides, the scattered field is itself a plane wave. Its amplitude and phase shift exhibit a jump across a critical polar angle depending on the direction of the incident plane-wave. Our estimates also imply that the scattered field is outgoing in a weak sense, allowing us to establish a uniqueness result. We conclude by calculating the waveguide-mode contributions to the scattered field in a perturbative regime.

We study Yang–Baxter deformations of two-dimensional sigma models on compact symmetric spaces $\mathrm{SU}(N)/\mathrm{SO}(N)$, with particular emphasis on their ultraviolet behavior. We claim that the ultraviolet fixed point is described by the $\mathrm{SO}(N,\mathbb C)/\mathrm{SO}(N,\mathbb R)$ non-compact WZW model together with $N-1$ compact free bosons. We determine the leading correction to this fixed point and identify it with the perturbation generated by the corresponding screening currents. We also extend our construction to the broader class of symmetric spaces of maximal rank.

We develop a Sobolev theory for the kinetic Fokker-Planck equation subject to a singular Morrey-form $L^\infty+M^q_K$ force field which can produce local blow-ups due to strong attraction to a submanifold. Our approach is based on the fractional representation for the Duhamel series of the equation and on establishing kinetic Adams-type estimates. Accordingly, we construct weak solutions of Brownian-driven kinetic SDEs in a critical Morrey set-up, obtain kinetic Krylov-type estimates and a strongly continuous Feller propagator, and establish uniqueness criteria for a large class of weak solutions.

We identify the $(\infty,1)$-category presented by the dagger Bergner, dagger Joyal, and dagger Rezk models of Parts I and II with that of univalent flagged dagger $(\infty,1)$-categories. We first construct a realization–nerve adjunction between reversal simplicial anima and Hermitian flags. We prove that its nerve restricts to an equivalence between flagged dagger $(\infty,1)$-categories and dagger Segal anima, under which univalence corresponds to unitary completeness. We then construct the adjunctions for univalentization and Hermitian completion and identify univalentization with unitary completion.

This paper develops a Willis-type thermal homogenization framework for two-dimensional particulate composites under harmonic heat transfer and establishes a unique effective constitutive relation without introducing residual fields. The local thermal fields are obtained using the dual-reciprocity inclusion-based boundary element method, in which two eigen-fields are introduced to simulate the mismatches in thermal conductivity and heat capacity. Instead of retrieving effective kernels from several independent macroscopic solutions generated by residual fields, the complete constitutive operator is constructed upon relations between eigen-fields and generalized macroscopic responses. The effective constitutive relations are validated against a one-dimensional analytical solution and an extended Fourier-series homogenization scheme. The Fourier formulae further establish the opposite-wavenumber reciprocity relations between two coupling tensors and reflection-symmetry parity relations, which explain when transverse coupling components vanish and the sign reversal of off-diagonal conductivity components under reflection. Numerical results further demonstrate that the de-phased average is required to extract physically consistent effective thermal properties under the Bloch conditions, and reveal their nonlocal frequency/wavenumber-dependent behavior. The framework provides a general computational method for Willis-type homogenization of particulate composites and is extendable to three-dimensional, multiphysical problems.

Global synchronization has been largely investigated in static higher-order networks and little attention has been brought to directed, temporal higher-order networks. We hereby make a step in this direction by studying the emergence of global synchronization in periodically fast switching systems with directed higher-order interactions. By working in the framework of the Master Stability Function, we show that the linearized dynamics includes two time-dependent Jacobian matrices, both evolving on the timescale of the synchronous solution, together with a generalized Laplacian evolving on a fast switching timescale, resulting from the hypergraph time evolution. To address this two-timescale structure, we establish a fast-switching stability criterion for directed balanced higher-order structures. The latter condition allows to decouple the parallel and transverse perturbations to the synchronization manifold. By relying on the average theory for dynamical systems, we are able to prove exponential stability of global synchronization in the fast switching higher-order network. The proposed theory is then illustrated by using a higher-order Stuart–Landau model, which provides a natural benchmark for the theory. Directionality can move Laplacian eigenvalues into instability regions off the real axis, hence destabilizing synchronization; our results nevertheless reveal that temporal switching can counteract this destabilizing effect and restore global synchronization when the directed structures alternate sufficiently rapidly, even though the instantaneous directed higher-order structures are individually unstable.

We prove sharp arithmetic localization for one-dimensional quasiperiodic potentials with jump discontinuities, beyond uniform transversality. Our results include every non-constant $C^1$ monotone sampling function, even with arbitrarily large flat regions, and also certain non-monotone potentials. A key new mechanism is a symbolic unique-continuation principle: first-return words to the region where $f'>0$ have no common Dirichlet resonance, forcing eigenfunction mass to transfer away from flat or negatively sloped regions and recovering quantitative monotonicity of finite-volume eigenvalue branches.

Spatial sudden death of entanglement refers to the disappearance of quantum correlations between subsystems after a certain distance called the entanglement length. Specifically, given a density matrix $\rho_{ABC}$ on a tripartite system $ABC$ with $B$ spatially separating $A$ and $C$, the entanglement length is the distance $\ell(A,C)$ after which the reduced density matrix $\rho_{AC}$, given by tracing out the $B$ subsystem, is separable. We show in the case of gapped ground states in any dimension that the entanglement length is finite on finite subsystems. Restricting to one-dimensional systems, we prove that normal matrix product states have a finite entanglement length independent of $A$ and $C$, and that a finite pure state with such uniform entanglement length has a matrix product state representation with bond dimension independent of $A$ and $C$. Consequently, in one dimension, entanglement length distinguishes matrix product states from all other pure states.

We derive a Fredholm-Pfaffian formula for the one-time multi-point transition probabilities of half-space polynuclear growth with any nonnegative boundary nucleation rate and general deterministic upper semi-continuous initial profiles. We prove the formula by showing that it satisfies an integral Kolmogorov equation with a unique bounded solution. We also discuss the formal 1:2:3 scaling limit of the kernel and its relation to the half-space KPZ fixed point.

The Dotsenko-Fateev equation is a third-order linear ordinary differential equation studied in two-dimensional conformal field theory. We study the confluent Dotsenko-Fateev system, a rank-three system of linear ordinary differential equations obtained by confluence from a first-order system equivalent to the Dotsenko-Fateev equation. It has an irregular singular point at infinity, and its solutions admit double integral representations of confluent Selberg type. Using these representations, we construct sectorial fundamental solution matrices around infinity, establish their asymptotic expansions, and determine the associated Stokes matrices explicitly. We further study three aspects of the confluent Dotsenko-Fateev system: its equivalence to the degree-two polynomial reduction of the Schrödinger equation for the quantum Painlevé equation of type V, its realization as a restriction of a confluent version of the linear Pfaffian system associated with Appell's hypergeometric function $F_4$, and its reduction to the trivial rank-one equation via integral transforms and a quadratic change of the independent variable.

For $H^1(\mathbb{R}^2)$-orthonormal families $\{u_n\}_{n=1}^N$ with one-body density $\rho=\sum_{n=1}^N |u_n|^2$, we prove the whole-space endpoint Moser–Trudinger inequality $$ \int_{\mathbb{R}^2}\frac{e^{4\pi\rho(x)}-1-4\pi\rho(x)}{\rho(x)}\,dx \le C\left(N-1+\int_{\mathbb{R}^2}\rho(x)\,dx\right), $$ where the integrand on the left-hand side is defined to be zero on $\{x: \rho(x)=0\}$. We develop a heat-semigroup refinement of Rumin's density method to treat both the endpoint coefficient $4\pi$ and the whole-space setting. Our inequality is sharp in two respects: the optimality of the coefficient $4\pi$ and the linear growth in the rank $N$. Moreover, through a semiclassical analysis of the density generated by normalized Dirichlet eigenfunctions, we establish the collective asymptotic behaviors of the quantities appearing in the inequality in various regimes. These semiclassical behaviors show that the rank-dependent term $N-1$ on the right-hand side is indispensable.

Can the Alexander-Orbach conjecture hold on hierarchical lattices? We construct Gromov-Hausdorff-Prokhorov scaling limits of critical percolation clusters on hierarchical lattices in the chemical metric and determine the typical spectral dimensions of the associated diffusions. These dimensions are dense in $(1,2)$; however, outside a narrow region in dimension space, we prove that the Alexander-Orbach conjecture fails on every hierarchical lattice under consideration.

The purpose of this note is twofold. First, we revisit the classical regularity theory of harmonic maps in the setting of maps defined on a lattice $\mathbb{Z}^d$ instead of $\mathbb{R}^d$ or some other manifold. The target space of our harmonic maps will be spheres $\mathbb{S}^{N-1}, N\geq 2$. A harmonic map $\sigma : \mathbb{Z}^d \to \mathbb{S}^{N-1}$ is a map which in any finite subgraph $\Lambda$ minimizes the internal Dirichlet energy $\frac 1 2 \sum_{i \sim j \in \Lambda} \|\sigma_i -\sigma_j\|^2$ given the boundary condition $\sigma_{|\partial \Lambda}$. One key difference between our setting and the continuum is that our main Hölder regularity statement in a large domain $\Lambda\subset \mathbb{Z}^{d=2}$ does not require any regularity assumption on the map $\sigma$ restricted to the boundary of the domain. For example, when $d=2$ and $N=3$, the Hölder regularity induced in the bulk of the domain holds uniformly in all boundary configurations $\sigma \in (\mathbb{S}^{2})^{\partial \Lambda}$ (this is in contrast with the continuum which requires extensions in suitable Sobolev spaces). The fact we obtain results uniformly in the boundary conditions allows us to extend our results for positive temperatures. Namely, as $T \searrow 0$, we obtain a local regularity theory for all infinite volume Spin $O(N)$ Gibbs measures, at least when $d=2$ and $N\geq 3$. We also discuss implications of our analysis in dimensions $d\geq 3$.

Polyakov predicted in 1975 ([Pol75) that the spin $O(3)$-model (or classical Heisenberg model) should have exponential decay of correlations at all positive temperatures on $\mathbb{Z}^2$. Our main result is as follows: if we replace the Euclidean scalar product \begin{align*} \langle \sigma_i , \sigma_j\rangle = \sigma_i^x \sigma_j^x + \sigma_i^y \sigma_j^y + \sigma_i^z \sigma_j^z \end{align*} by the $\lambda$-modified scalar product \begin{align*} \langle \sigma_i , \sigma_j\rangle_\lambda := \sigma_i^x \sigma_j^x + \sigma_i^y \sigma_j^y + \lambda \sigma_i^z \sigma_j^z \end{align*} we show that for all $\lambda\in(-1,1)$, the $\mathbb{S}^2$ valued spin system with Gibbs measure $\exp(\beta \sum_{i\sim j} \langle \sigma_i,\sigma_j\rangle_\lambda)$ has power-law decay of correlations if $\beta$ is chosen large enough. This is the first evidence that arbitrary small smooth modifications of the classical Heisenberg model produce a BKT phase in $2d$. It was proved in the 1970s and 80s that there exist arbitrary small smooth modifications of the classical Heisenberg model which produce instead an Ising phase. In these work, they considered the regime $\lambda>1$ while in this paper, we consider the regime $\lambda<1$ which cannot be analyzed via reflection positivity and chessboard estimate. The two main ingredients used in our proof are the stability of the BKT phase under percolation type disorder ([DG25]) and weak regularity type properties of the groundstates or $\mathbb{S}^2$-valued harmonic maps associated to our $\lambda$-energy functional. This analytic step is interesting on its own. Finally, the exponential and power-law decay regime should be separated by a curve $\lambda \mapsto \beta_c(\lambda)$. We prove a rigorous lower bound in the case of the $\lambda$-anisotropic hemisphere model $\mathbb{S}^2_+$ which matches with RG flow predictions.

Since Brill's work of 1959, the conformal factor of finite-amplitude Brill waves, the axisymmetric and time-symmetric vacuum initial data on $\mathbb{R}^3$ that describe pure gravitational radiation, has been obtained only numerically, up to the most recent studies. Here an explicit one-parameter family of such data is constructed: the first explicit real-analytic, non-trivial, asymptotically flat vacuum initial data with trivial topology and a single end. The free metric function is a closed-form combination of elementary functions and a dilogarithm. The data have the toroidal conformal symmetry of Beig and Husa, and a change of variable of Natanzon type reduces every mode equation to the Gauss hypergeometric equation, so that the conformal factor is a series with explicit Gamma-function coefficients. The quantities that have so far required numerical work are thereby obtained exactly. A ground-state identity gives a sharp existence criterion: complete, asymptotically flat data exist precisely for $k<k_c=0.8048016772\ldots$, and the explicit series is identified with the unique variational solution. The ADM mass is an absolutely convergent digamma series, with $m/a=\pi k^2/10+O(k^3)$ for small amplitudes and an explicit residue at its critical divergence. Sufficiently strong waves contain trapped surfaces: it is proved that for $k$ close to $k_c$ the rescaled conformal factor converges to an explicit function and explicit tori and spheres become outer trapped, so that the data contain an apparent horizon. A numerical search, restricted to equatorially symmetric star-shaped surfaces, finds marginally trapped (minimal) spheres, which appear in a fold near $k=0.7670$. No thin-torus or weak-field approximation is used.

This paper studies the topological entropy of real-analytic Liouville integrable Hamiltonian systems. We prove that the Hamiltonian flow on a compact connected regular energy hypersurface has zero topological entropy whenever a complete family of commuting real-analytic first integrals is generically independent in a connected neighborhood of that hypersurface. The proof uses finite normal expansions along minimal analytic carriers and an ergodic version of Ziglin's leading-term argument to obtain independent invariant normal polynomials, whose Poisson geometry makes all tangent Lyapunov exponents of the carriers vanish. In particular, we prove that a real-analytic Liouville integrable geodesic flow on a closed real-analytic Riemannian manifold has zero topological entropy, giving a positive answer to the conjecture proposed by Paternain in 1992.

Stable bound states living on a discrete one-dimensional N-point lattice are assumed generated by a certain two-parametric PT symmetric potential $V=V(A,B)$. At any $N$, the coordinates $A=A^{(EP)}$ and $B=B^{(EP)}$ of the Kato's exceptional point (marking the phase-transition instant of the loss of stability) are defined, exactly, via roots of a certain elementary quartic polynomial.

In this paper we construct a one-parameter family of periodic Airy line ensembles $\mathcal{A}^{\mathrm{per},R}$ for $R>0$ as weak scaling limits of Borodin's periodic Schur processes. We establish several properties of $\mathcal{A}^{\mathrm{per},R}$, including a determinantal point process structure with an explicit correlation kernel. We also show that, after a parabolic shift and normalization, $\mathcal{A}^{\mathrm{per},R}$ satisfies, on its domain $[-R/2,R/2]$, the same Brownian Gibbs property enjoyed by the usual Airy line ensemble. A main technical contribution of the paper is a general tightness criterion for determinantal line ensembles, which extends the argument used by Johansson in [Comm. Math. Phys. 242 (2003), 277-329] to prove a functional limit theorem for the discrete polynuclear growth model.

The focusing coupled Kundu-Eckhaus (CKE) equation extends the coupled nonlinear Schrödinger (CNLS) equation by adding quintic and derivative nonlinearities. We study the Cauchy problem for the focusing CKE equation with Schwartz initial data under the assumption of no discrete spectrum and no spectral singularities on the real axis. Based on the gauge equivalence between the focusing CKE and CNLS equations, we analyze the associated $3\times3$ matrix Riemann-Hilbert problem by the Deift-Zhou nonlinear steepest descent method. By establishing joint error estimates for the CNLS solution and the gauge phase, we obtain a uniform long-time asymptotic formula for the dispersive radiation of the CKE equation with an $O(t^{-1}\log t)$ remainder in bounded velocity cones.

One main aim of our paper is to formulate noncommutative versions of the Lebesgue dominated convergence theorem in the setting of quantum integration theory on von Neumann algebras. Our starting point is a reformulation of classical dominated convergence in terms of weak* convergence. Using our recent sequential-continuity theorem, we obtain dominated convergence formulations for sequentially normal states and, under domination by a fixed bounded integrable majorant, for sequentially normal weights. Integrable, possibly unbounded, majorants are treated first in the tracial setting, then for arbitrary von Neumann algebras using Haagerup's L1 space, and finally using normal semifinite weights and their GNS representations. For any faithful normal semifinite weight $\phi$, weak* convergence of norm-bounded sequences is characterized by `integrating against' bounded positive integrable majorants. We also study noncommutative almost uniform and local almost uniform convergence. In particular, for norm-bounded sequences, local $\phi$-bilaterally almost uniform convergence implies weak* convergence and L1 convergence to 0 of $h^{1/2}(x_n-x)h^{1/2}$ for every positive Haagerup L1 density h, without assuming sigma-finiteness. We also prove a dominated convergence theorem in the predual: fixed positive order domination and Petz's almost uniform convergence imply convergence in the predual norm. We establish implications and equivalences among these convergence notions, connecting dominated convergence with tracial and nontracial integration.

We present a self-contained, algorithmic construction of almost invariant subspaces [Panati et al. (2003), De Nittis and Panati (2010), Freund and Teufel (2016)] for Hofstadter-like Hamiltonians $H_b $ acting on $\ell^2(\mathbb{Z}^2,\mathbb{C}^n)$. More precisely, for rational values of $b_0/(2\pi)$ we describe an iterative procedure that takes a “locally gapped" projection $P_{0}$ (possibly topologically non-trivial) which commutes with $H_{b_0}$, into a projection $P_{\varepsilon}$ that commutes with $H_{b_0+\varepsilon}$ up to an error of order $\varepsilon^\infty$. Our construction insures that the Chern number of $P_{\varepsilon}$ equals the Chern number of $P_0$ as long as $P_\varepsilon$ exists. Moreover, its integrated density of states varies linearly with $\varepsilon$ (the “Středa formula"). We also show that any spatially localized Parseval frame of $P_{0}$ can be mapped into to a localized Parseval frame of $P_{\varepsilon}$. We choose to work with a discrete model in order to keep the technicalities to a minimum level, with no need for advanced pseudo-differential calculus. Nevertheless, we are confident that most of our results can be extended to (unbounded) magnetic (pseudo)differential operators, but the technical burden would be much heavier. This is left for future work.

An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.