Papers

Let $p$ be a prime, $d\ge 2$, $H\in [1,p)$, and $\ln\ln p = o(\ln H)$. We prove that $$ \min_{1\le n \le H} n \left\|\frac{a_1 n}{p}\right\|\ldots \left\|\frac{a_d n}{p}\right\| \approx \frac{1}{(\ln p)^{d-1} \ln H} $$ for "almost all" $a \in ({\Bbb Z} / p{\Bbb Z})^d$.

We establish spacetime counterparts of fill-in inequalities for spin initial data sets satisfying the dominant energy condition in dimensions three through seven, controlling the minimum and the integral of the norm of the spacetime mean curvature vector by intrinsic boundary data. The proof relies on a Reilly–MIT estimate for complete spin manifolds with compact boundary. As further consequences, under suitable nonnegativity assumptions, Gromov's hyperspherical-radius fill-in inequality and Bär's recent total mean-curvature estimate extend to complete, possibly noncompact spin manifolds with compact boundary.

We consider the stochastic heat equation $\partial_t u(t,x) = \Delta u(t,x) + \dot{W}_\alpha(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_\alpha$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-\Delta)^{-\alpha}$ with $\alpha>0$. We prove that a unique pointwise defined mild solution exists if and only if $\alpha>d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<\alpha<d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.

We determine the exact first-order asymptotics of the expected optimal cost in two-dimensional random bipartite matching, for every finite power cost $q \ge 1$, on the flat torus. In the endpoint case $q=1$, this answers a question by Talagrand, in the periodic case. The argument involves the closely related asymptotics of the energy of the solution of the $p$-Poisson equation with a regularized white-noise source. In the limit of vanishing regularization parameter, we identify this energy as the solution of a Variational Martingale Problem on a limiting Gaussian filtration, whose value is characterized by a parabolic Monge-Ampère flow.

This paper gives a categorical interpretation of Baumeler & Wolf's logically consistent non-causal circuits, connecting them to postselected quantum teleportation. Looped feedback is represented by a trace in the category of non-negative matrices, and it is shown that the traced process is stochastic precisely when the induced loop transition matrix has trace $1$ for every external input, a condition shown to be equivalent to a unique fixed point for the loop for each input to a deterministic circuit. A classical non-causal circuit is represented by a measure-and-prepare quantum channel with an internal register utilising a maximally entangled Bell-state with postselection. The main result is that classical logical consistency is equivalent to the postselected Bell outcome having, for loop dimension $d$, probability exactly $1/d^2$ for each classical input distribution. The subsequent normalised conditional output then agrees exactly with the classical categorical trace. This work identifies a class of quantum Bell-postselection constructions whose conditional evolution maintains linear dependence on classical input distributions.

We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator on $\mathrm{L}^1_{\sigma,n}(\Omega)$ does not generate a $\mathrm{C}_0$-semigroup, even though the resolvent problem is uniquely solvable. In stark contrast, its realization on ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$ generates a compact, analytic $\mathrm{C}_0$-semigroup, which leaves $\mathrm{L}^1_{\sigma,n}(\Omega)$ invariant. The key point is that these two realizations, which are canonically identified for $1<p<\infty$ through the Helmholtz decomposition, cease to be equivalent at the endpoint $p=1$. This leads to genuinely different functional analytic properties. Our result provides the first positive generation theorem for the Stokes operator with no-slip boundary conditions in a pure $\mathrm{L}^1$-setting on a bounded domain and settles a problem that had remained open for nearly fifty years; see, e.g., \cite{Koz:01,DHP:01}. In this sense, it completes the theory of the Stokes semigroup across the full scale of solenoidal Lebesgue spaces on (smooth) bounded domains. As an intermediate step, some results on the space of Radon measures are obtained. The proof combines the sun-dual construction with a precise analysis of the failure of the Helmholtz decomposition in $\mathrm{L}^1$, the celebrated result of Abe and Giga \cite{AG:12} on the Stokes semigroup on $\mathrm{C}_{\sigma,0}(\Omega)$ and the regularity theory refinements for the Stokes operator recently developed by Breit and the second author \cite{BG:25}.

Let \(\{Z_n\}_{n\geq0}\) be a supercritical branching random walk with deterministic rooted \(b\)-ary tree, \(b\geq2\), and symmetric displacements satisfying \(\lim_{x\to+\infty}x^{-\alpha}\log\Pp(X>x)=-\lambda \) with \(\alpha,\lambda>0\). Set $\overline Z_n(\cdot):=Z_n(\cdot)/Z_n(\mathbb{R}),~n\geq0.$ For a finite union \(A\) of intervals, we establish the following full large-deviation principle: for every Borel set \(\Gamma\subset[0,1]\), \begin{align*} -\inf_{q\in\Gamma^\circ}Q_A(q) &\leq \liminf_{n\to\infty}n^{-\alpha/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\in\Gamma\right) &\leq \limsup_{n\to\infty}n^{-\alpha/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\in\Gamma\right) &\leq -\inf_{q\in\overline\Gamma}Q_A(q), \end{align*} where $Q_A$ is a good rate function on $[0,1]$. Meanwhile, we obtain large deviation probabilities for $\{\overline Z_n(\sqrt n\,A)\}_{n\geq1}.$ This strengthens the nonmatching upper and lower bounds obtained by Chen and He [Probab. Theory Related Fields 175 (2019) 255-307] for regular trees with Weibull displacements. Our method combines a large-deviation principle for rescaled displacement tree fields and exponential equivalence.

Delcourt and Postle proved that every $K_t$-minor-free graph is $O(t\log\log t)$-colorable and reduced the Linear Hadwiger Conjecture to coloring $K_t$-minor-free graphs on $O(t\log^4 t)$ vertices. In this paper, we use the star contraction technique to improve their bound for small graphs and use their reduction to extend this improvement to all $K_t$-minor-free graphs. Thus we improve their $O(t\log\log t)$ bound to $O(t\sqrt{\log\log t})$.

Hydronic radiator systems are among the most widely used heating systems in buildings. In the literature, radiator heat outputs are often assumed to be independent of one another and arbitrarily adjustable in time when designing controllers. However, in practice, radiators are supplied by one or multiple common heat sources and are hydraulically coupled through the water circulation system. Maintaining occupant comfort in multi-zone buildings therefore requires not only an appropriate supply temperature but also proper hydraulic balancing to distribute the available water flow according to the heating demand of each zone. To this end, we develop a grey-box thermal model that captures the hydraulic interactions among radiators and the effects of radiator valves, circulation pumps, and heat sources. In a real building, we show that accounting for hydraulic interactions reduces the root mean square error (RMSE) between the measured and modeled zone temperatures by approximately 11% compared with a model that neglects these interactions. Additionally, we integrate the developed model into a model predictive control (MPC) framework for dynamic hydraulic balancing that jointly optimizes valve openings and the supply temperature to maintain thermal comfort in each zone while reducing energy consumption. Through both real-world experiments and numerical case studies, we demonstrate that, compared with existing MPC formulations that neglect hydraulic interactions or do not control radiator valves, the proposed MPC reduces comfort-range violations by at least 27% while requiring a similar or even lower cumulative supply temperature.

Let $h\geq 3$ be an integer and $0<\alpha<1/h$. In this paper, we prove that there exists a minimal asymptotic basis of order $h$ with asymptotic density $\alpha$. This solves an open problem posed by Erdős and Nathanson in 1988.

We prove that on the metric graph of $\mathbb{Z}^d$ for $d\in \{3,4,5\}$, clusters of the critical loop soup (or equivalently, clusters of the critical Gaussian free field level-set) admit a scaling limit. To the best of our knowledge, this presents the first scaling limit result for a percolation model on a three-dimensional lattice.

We consider the exclusion process with energy, a model with mass and energy conservation, and we put it in a boundary-driven setting. We are interested in its hydrodynamic and hydrostatic limits, together with the associated dynamical and stationary fluctuations. We provide a new proof for the derivation of dynamical fluctuations that does not require explicit estimates on the two-point correlation functions. While the hydrodynamic limit is given by two uncoupled heat equations, the dynamical fluctuations are described by a system of coupled generalized Ornstein-Uhlenbeck processes.

Direction-dependent instrumental polarisation is a major systematic limitation in high-fidelity single-dish radio polarimetry, yet a unified characterisation of beam leakage and the conditioning of polarisation recovery is lacking for the Ghana Radio Astronomy Observatory (GRAO) 32-m telescope. We aim to establish a quantitative, direction-dependent polarimetric beam model at 5.0 and 6.7 GHz. High-resolution GRASP electromagnetic simulations of the nominal telescope configuration are used to derive the full complex Jones response and corresponding Mueller matrices, from which beam shape, beam squint, instrumental Stokes leakage, and intrinsic cross-polarisation ratio (IXR) are evaluated across the primary beam. The Stokes $I$ beams have half-power beam widths of $396.4$ and $295.3$ arcsec at 5.0 and 6.7 GHz, respectively, with main-beam efficiencies of $56.4$ and $55.6$ per cent. At 5.0 GHz, the circular-polarisation beam squint is $9.96$ arcsec ($2.5$ per cent of the HPBW), whereas no statistically significant squint is detected at 6.7 GHz. Leakage from Stokes $I$ into linear polarisation remains below $0.1$ per cent within the half-power beam but increases substantially towards the sidelobes. IXR reaches approximately $80$ dB on axis and decreases with angular offset. These results establish the intrinsic electromagnetic polarimetric response of the GRAO 32-m telescope and provide a quantitative baseline for direction-dependent calibration and subsequent observational validation.

Fix two positive braid words $\beta_+,\beta_-$, and let $\text{Conf}(\beta_+,\beta_-)$ be the corresponding (type A) double Bott-Samelson variety. Let $\beta$ be a double braid word containing $\beta_+,\beta_-$ as the top, bottom words. We consider the open cluster torus $T(C_\beta)$ associated to a triangulation $C_\beta$ in $\text{Conf}(\beta_+,\beta_-)$ from arXiv:1904.07992, and we identify these with weighted plabic fences, where the usual local moves on planar bipartite graphs naturally correspond to change of torus coordinates in double Bott-Samelson variety. Moreover, $T(C_\beta)$ can be parametrized explicitly by certain matrix products, and also has monomial coordinates given by the cluster variables, which are matrix minors. We interpret these minors as a generalized notion of perfect matchings on weighted plabic fences, which may not be reduced plabic graphs. Using this interpretation, we show that the cluster variables are given by "generalized minimal matchings", extending the minimal matchings from arXiv:1606.08383. Lastly, we derive a Chamber Ansatz formula for double Bott-Samelson varieties via face alternating products from dimer theory. In general, plabic fences are not reduced plabic graphs, yet we are able to extend and apply the standard tools such as local moves on plabic graphs, trips, and minimal matchings to them.

We consider Thouless pumping of two-dimensional quadratic solitons composed from coherently interacting fundamental frequency (FF) and second harmonic (SH) components propagating in a periodic $\chi^{(2)}$ material. The pumping is induced by two mutually sliding two-dimensional lattices defined by shallow transverse and longitudinal periodic refractive index modulation. Focusing on solitons in the semi-infinite gap, we find three distinct pumping scenarios: the absence of transport for small-amplitude solitons, non-quantized transport in a transient regime at intermediate amplitudes, and stable quantized transport for solitons with relatively large amplitudes. Different directions of the sliding velocity were investigated resulting in different trajectories of soliton center. The transition to the regime of quantized transport is found to depend strongly on phase mismatch between FF and SH waves that also influences stability properties of two-dimensional $\chi^{(2)}$ lattice solitons. We also show that pumping with longitudinal periods in the x and y directions, whose ratio approximates an irrational number, induces quantized transport whose direction rapidly converges, with increase of the accuracy of the approximation, to a limiting pumping direction determined by this irrational number and corresponding to truly incommensurate longitudinal periods.

We give a new equivalent formulation of Verdier's octahedral axiom. An initial square in a pre-triangulated category determines an obstruction in a quotient of ordinary Hom groups. Its vanishing is equivalent to the existence of a good completion, and $\mathrm{TR4}$ is equivalent to this vanishing for the squares associated with composable morphisms. As a consequence, if two pre-triangulations have the same underlying additive category and one is triangulated, then the other satisfies $\mathrm{TR4}$ precisely when their relative inverse Heller comparison admits an exact lift along every short exact sequence in the Freyd category. As applications, we prove that the exotic pre-triangulated category of Díaz Cabrera–Muro is in fact a triangulated category over every algebraically closed field. We construct scalar families for type-$A_{3a-1}$ preprojective algebras over arbitrary fields and for a range of Dynkin preprojective algebras in characteristic two; in each family the zero parameter is the unique triangulated member. The local equations also yield separable descent, including canonical triangulations for non-modular equivariantizations and for the Markman–Mehrotra K3 deformation categories.

In this work we establish a connection between the min-max optimal matching in the critical dimension $d=2$ and a simpler, geometrically linear, action of random curves in a Brownian potential. This is done by following Leighton and Shor and working with the dual formulation, which we approximate by a problem of isoperimetric-type with a random volume term of white-noise character. This allows us to extract the leading-order term in the asymptotics of the expected cost, sharpening previous results.

Let $A$ be a split finite-dimensional algebra over a field whose radical $J$ satisfies $J^3=0$, and let $s$ be the number of isomorphism classes of simple $A$-modules. We prove that a non-projective module $M$ with ${\rm Ext}_A^i(M,A)=0$ for all $i>0$ has a non-zero self-extension in some degree between $1$ and $3s+1$. In particular, $A$ satisfies the Auslander–Reiten conjecture, which asserts that every self-orthogonal generator is projective. As a consequence, every finite-dimensional algebra over an algebraically closed field with radical cube zero satisfies the Auslander-Reiten conjecture.

We develop a variational framework for extremal Lin–Lu–Yau curvature under positive edge reweighting with the combinatorial metric fixed. For a connected locally finite graph $G$, define \[ \Kmax(G) := \sup_{w>0}\inf_{e\in E(G)}\kappa_{\LLY}^w(e). \] This invariant relates edgewise discrete curvature to global combinatorial and topological data. In the high-girth regime, we prove for every connected locally finite graph the exact identity \[ \Kmax(G)=\frac4{\mad(G)}-2, \] with the convention $1/\infty=0$. The formula is also equivalent to a normalized Euler-characteristic density: \[ \Kmax(G) = \inf_{\substack{H\subseteq G\text{ finite}\\E(H)\ne\varnothing}} \frac{2(c(H)-\beta(H))}{|E(H)|}. \] For arbitrary locally finite graphs, a nonnegative short-cycle surplus yields girth-dependent density bounds, with rigidity results in the finite case. We also classify the locally finite graphs of girth at least $6$ admitting a positive edge weight with nonnegative curvature on every edge: they are precisely the trees and the finite cycles.

In this article we provide a systematic reduction to static spherical symmetry for (classical) gauge theoretic models with structure group SU(2) x U(1), coupled to gravity. In particular, we derive the most general static spherically symmetric ansatz for the Einstein-Yang- Mills-Higgs-Dirac-Yukawa equation system. This model is reminiscent of the lepton sector of the Standard Model coupled to gravity. The resulting total system of ordinary differential equations is rather large, so that finding global solutions is non-trivial. For this reason, an open-source interactive solver GUI written in C++ is provided together with the article.

We study the localization of $s$ distinct planar point scatterers from sparsely sampled single-incidence, single-frequency acoustic far-field data over a limited receiver aperture. Under the first Born approximation, the coherent data form a two-dimensional off-grid exponential sum and admit a low-rank Hankel representation. A limited aperture generally admits multiple aperture-admissible Hankel pencils with different lifted dimensions, sampling multiplicities, and Vandermonde conditioning. Hence, the choice of the Hankel pencil affects both data completion and subsequent localization. In this work, we introduce a unified framework for structured completion and off-grid localization based on general aperture admissible Hankel pencils. We establish exact recovery in the noiseless case and stable recovery under bounded noise from a number of random aperture samples that is linear in $s$, up to factors determined by the pencil geometry, source conditioning, and logarithmic terms, together with an explicit localization bound in terms of the completion error. The analysis separates the contribution of the pencil geometry from the conditioning induced by the source configuration and provides verifiable sufficient conditions for stable completion and MUSIC resolution. These results yield quantitative criteria for selecting sample efficient Hankel pencils under a prescribed physical aperture. Numerical experiments on connected and disconnected apertures demonstrate the effects of pencil geometry, sampling, and noise.

This article is concerned with a definitive analysis of its long-term dynamics of the CAO-system. We establish the convergence of global strong solutions to constant equilibria in the full $H^1$-topology in the non-perturbative regime. Remarkably, we demonstrate that the CAO-system exhibits a polynomial rate of decay and we determine the optimal rate. This phenomenon stands in stark contrast to the exponential convergence typically expected for uniformly parabolic systems on bounded domains. This algebraic slowing is shown to be a purely nonlinear effect, driven by the emergence of a slow manifold (center manifold) induced by nonlinear coupling conditions. To the best of our knowledge, this provides the first instance of a global polynomial decay result for a non-gradient, non-local dissipative system on a bounded domain.

We derive linear programming (LP) bounds on the minimum distance of binary and ternary linear complementary dual (LCD) codes by imposing arithmetic constraints on their weight enumerators. Special values of the weight enumerator give finitely many Gauss phases, each of which yields linear equations in the ordinary weight-distribution variables. The resulting bounds strengthen the real-valued LCD counting LP without introducing additional variables; both the number of branches and the number of added equations per branch are bounded independently of the code length. Exact certificates establish strict improvements for 62 binary parameter pairs of length at most 20 and 39 ternary pairs of length at most 14. For four binary pairs, the bounds also improve the joint-weight-enumerator LP while using fewer variables per branch. These comparisons show that Gauss-phase information provides a compact strengthening of existing LP relaxations for LCD codes.

For a tournament $T$, let $\omega(T)$ be the minimum clique number among the backedge graphs of $T$, and let $\chi(T)$ be its dichromatic number. We give a template-lifting construction. It turns a $k$-template into a regular, vertex-transitive Cayley tournament that is simultaneously $(k+1)$-$\omega$-critical and $(k+1)$-$\chi$-critical. The output is also a $(k+1)$-template. Iterating the construction, we prove that for every $k\geq3$, there is a positive even integer $m_k$ with the following property. Every $N>1$ with $N\equiv1\pmod{m_k}$ is the order of a regular, vertex-transitive Cayley tournament that is simultaneously $k$-$\omega$-critical and $k$-$\chi$-critical. This proves a conjecture of Aboulker, Aubian, Charbit, and Lopes and gives a negative answer to their bounded-certificate question when the hypothesis is $\omega(T)\geq k$. We also find the clique number of a cyclic substitution when each block satisfies $\omega=\chi$. We then describe exactly when this substitution is $\omega$-critical if the blocks are $\chi$-critical and satisfy $\omega=\chi$.

Can the classical Heavy-Ball method, with arbitrary horizon-dependent parameters chosen in advance, achieve Nesterov's $O(T^{-2})$ last-iterate rate on every smooth convex objective? We provide a negative answer. For every horizon $T\ge2$ and every predetermined schedule with nonnegative step sizes and momenta in $[0,1)$, there exists a convex $1$-smooth objective, with initialization distance at most one and zero initial velocity, for which the last iterate of the Heavy-Ball method satisfies \[ f(x_T)-f^\star=\Omega\!\left(\frac{1}{T^\alpha\log T}\right), \qquad \alpha=\frac{1+\sqrt5}{2}. \] Thus even fully nonstationary, horizon-dependent tuning cannot give the classical Heavy-Ball method a Nesterov-rate guarantee on the smooth convex class.

We study the continuous homology and cohomology of semisimple Lie groups with coefficients in arbitrary unitary representations. The case of irreducible representations was determined by Vogan and Zuckerman; we focus on reducible representations. The (co)homology splits into its Hausdorff and torsion parts. The Hausdorff part is governed by containment of irreducible cohomological representations. We show that the torsion part is governed by weak containment of irreducible cohomological representations. Precisely, we show that a unitary representation admits non-zero torsion if and only if there is a cohomological point which is not isolated in its support. We discuss in length examples of rank-$1$ groups, completely determining unitary cohomology for the group $\mathrm{SO}^\circ(n,1)$. For a simple Lie group, the first and fourth named authors showed that the first degree in which it obtains non-trivial cohomology for some unitary representation with no invariant vectors is related to the rank of the group. We discuss the analogous question regarding torsion cohomology, and show that the corresponding first degree could be much higher: in the presence of property (T), it is bounded below by the square root of the dimension of the symmetric space. A technical device that we use is the restriction to well chosen dense subgroups which satisfy finiteness properties, which we call cohomological witnesses. We combine it with results on the unitary cohomology of groups which satisfy finiteness properties. These results are of an independent interest.

An influential 1989 result of Naddef shows that the diameters of $0/1$-polytopes are at most their dimension. This was extended shortly after by Kleinschmidt and Onn to any lattice polytope in $[0,k]^{d}$, where they showed a bound of at most $dk$. Naddef's argument easily extends to the monotone setting motivated by the simplex method, where one requires paths to increase with respect to a linear objective function. However, the Kleinschmidt-Onn argument does not. In fact, no argument in the 30 years since has managed to fill that gap. Prior to this work, it remained open whether the monotone diameter is bounded by a polynomial in $d$ and $k$ with no lower bounds suggesting any separation between the worst-case diameter and worst-case monotone diameter. Linear upper bounds hold for $k=1$ and $k=2$. However, we exhibit a sharp threshold for this question at $k = 3$ by constructing for each $d \geq 1$ a lattice polytope in $[0,3]^{6d}$ with monotone diameter at least $2^{d}-1$. In particular, the polynomial bound does not hold. Furthermore, we show that Naddef's result does not extend to the unbounded setting by exhibiting a family of unbounded polyhedra with $0/1$-vertices and diameter exponential in their dimension.

We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schrödinger equation on the three-dimensional torus under saturating smooth finite rank Brownian forcing. The mixing rate is measured in the $p$-Wasserstein metric induced by the $H^1$ distance for every $1\le p<\infty$. We also obtain sharp geometric characterizations of saturation. The proof is based on a polynomial mixing criterion built on a stable–compact decomposition of the exact solution differences with polynomial moment control of the logarithmic path amplification. Dense Malliavin range allows the compact defect to be compensated by finite-dimensional Cameron–Martin shifts, producing a block multiplier with negative mean logarithm. A logarithmic transportation gauge, combined with a renewal–reset coupling scheme, then yields mixing at every prescribed polynomial order in the weaker $L^2$ distance. A stationary regularity gain to $H^{2-}$ then enables us to upgrade the convergence to $H^1$.

In this paper, we consider an inverse random source problem for the stochastic Moore-Gibson-Thompson equation driven by fractional Brownian motion with Hurst index $H \in (0, 1)$ of the form $f_1(x)g_1(t)\dot{B}^H(t)+f_2(x)g_2(t)$. Given the random source, existence and uniqueness of mild solutions are verified. For the inverse problem, the uniqueness of recovering the strength $f_i(x)$ if the time functions $g_i$ are known and $g_i(t)$ if the spatial functions $f_i$ are known when $H \in(0,1)$ from the boundary flux on a special nonempty open subset is proved.

We introduce a modulated Gibbs measure for the usual tensorized initial data for stochastic Newton's systems with singular repulsive interactions. Using the uniform-in-$N$ partition-function estimates of Wang–Zhao \cite{wang2026uniform}, we show that the modulated Gibbs measure and its tensorized reference are $O(N^{-1})$-close in normalized relative entropy. A stability argument then transfers propagation of chaos from modulated Gibbs measures to tensorized data and other initial laws at the same entropy scale. Combined with the weighted BBGKY estimates of Bresch–Jabin–Soler \cite{bresch2025new}, this yields the first cutoff-free mean-field limit and propagation of chaos for the three-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) equation on an $N$-independent short-time interval.