Papers

This paper proposes a system-aware adaptive channel state information (CSI) feedback framework for massive multiple-input multiple-output (mMIMO) systems, aiming to dynamically optimize the trade-off between reconstruction fidelity and signaling overhead. While deep learning-based autoencoders (AEs) have enabled significant CSI compression, conventional fixed-ratio schemes fail to adapt effectively to non-stationary channel conditions. To address this limitation, we develop a reinforcement learning (RL)-driven control framework that operates over a bank of pretrained multi-rate AEs, each corresponding to a distinct compression ratio (CR). At each time step, a centralized RL agent selects the most suitable CR for each user based on observed channel conditions and system performance indicators. Distinct from conventional mean squared error (MSE)-centric designs, we introduce a system-aware reward formulation that jointly accounts for spectral efficiency via signal-to-interference-plus-noise ratio (SINR), feedback overhead constraints, and the computational cost of model adaptation. Simulation results on high-dimensional delay-domain CSI datasets demonstrate that the proposed RL-guided framework effectively balances the overhead-accuracy tradeoff and adapts to dynamic channel environments. The proposed method improves spectral efficiency and feedback efficiency compared with fixed compression schemes and adaptive baselines, while maintaining a modest computational and memory footprint. Averaged over different numbers of users and across all considered baselines, the proposed RL framework reduces the CSI feedback cost by more than 53.4%, improves the average downlink sum rate by 53.64%, and reduces the NMSE by 22.38%. These results demonstrate its ability to achieve a more efficient rate-accuracy-feedback tradeoff under dynamic wireless conditions.

In this paper, we introduce a minimal dynamical model for the diffusion of a new social norm, in which individuals transition among three states: non-supporters, silent supporters, and vocal advocates. Despite its simplicity and its close relation to the classical SI-type and SIS-type spreading dynamics, this model exhibits a nontrivial latent-outbreak dynamic pattern: an initial small adoption wave is followed by a long quiescent period and then an abrupt, endogenous explosion of support. Through a complete analytical characterization of equilibria, stability, convergence, and phase-transition conditions, we identify the nonlinear mechanism that generates this incubation phenomenon and derive estimates of the latent period. Our results reveal a dynamical route to sudden social change driven purely by internal interactions rather than external shocks.

Let $G$ be a finitely generated group acting by isometries on a geodesic metric space with two independent strongly contracting WPD elements. We prove that, for every finite generating set of $G$, strongly contracting WPD elements for this action are exponentially generic in word-metric balls. The proof establishes a growth gap for elements with small displacement relative to word length in an auxiliary projection complex. For finitely generated acylindrically hyperbolic groups, we establish growth tightness and cogrowth tightness for every finite generating set. We also show that the number of conjugacy classes meeting a ball of radius $n$ is comparable to the cardinality of that ball divided by $n$.

Degenerate fixed points of maps appear in many interesting problems in Celestial Mechanics, Economics and Chemistry. Lattice systems, that is, dynamical systems consisting in an infinite array of finite dimensional subsystems interacting locally among them appear in many models of Biology, Physics and Mathematics. In this work, we extend the results concerning the existence of stable and unstable invariant manifolds of degenerate fixed points to lattice systems with decay properties. As an example, we find such manifolds in perturbations of the Toda lattice.

We study the long time dynamics of a selection-mutation integro-differential Lotka-Volterra system of $N$ populations. In our model, fitness depends on a continuous phenotypic trait, but the effect of one population on another is independent of this trait. We establish that, under some usual assumptions on the interactions between populations, the long time behaviour of solutions is exactly determined by the well studied Generalised Lotka-Volterra (ordinary differential) Equation. We also show that, all else being equal, the minimal mutation rate is selected for in a static environment, whereas in an a changing environment, an intermediate mutation rate could be selected instead. The key step in our proofs is establishing that the total population sizes are asymptotically governed by a Generalised Lotka-Volterra Equation, which then allows the use of a general result on asymptotically autonomous dynamical systems.

In the present note, we consider a semilinear damped wave equation on a compact Lie group with a non-autonomous nonlinearity $\varphi(t)|u|^p$. We are interested in describing how the nonnegative time-dependent factor $\varphi$ affects the global in time prolongability of a local solution. In particular, the summability of the function $\varphi$ provides a criterion to distinguish between the blow-up in finite time and global existence of small data. Finally, we derive sharp lifespan estimates for local in time solutions when $\varphi\not\in L^1([0,+\infty)))$ and satisfies a certain scaling condition, that we named uniform upper scaling condition.

Recent advances in 3D Gaussian Splatting (3DGS)-based wireless radiance field (WRF) reconstruction provide an efficient solution for wireless channel modeling. However, existing WRF reconstruction methods rely on pre-collected observations and offline optimization, and thus struggle to provide real-time channel knowledge. To bridge this gap, we propose CORF-GS, a real-time WRF reconstruction framework that processes sequential optical and radio frequency (RF) keyframes. Specifically, CORF-GS constructs a unified Gaussian representation for optical and RF with shared geometry and modality-specific appearance, allowing high-resolution optical images to provide structural priors for WRF reconstruction. When a new keyframe arrives, CORF-GS first employs optical-guided Gaussian sampling to densify the WRF in under-represented regions. Since light and radio waves may respond differently to the same object surfaces due to wavelength mismatch, relying solely on optical guidance may neglect RF-informative areas. Therefore, CORF-GS performs coupled optical-RF optimization to jointly refine the shared Gaussians. Compared with the existing two-stage training pipelines, this prevents WRF from passively adapting to a frozen optical geometry and encourages the shared Gaussians to adapt to both optical structures and RF power distributions. Simulations show that CORF-GS achieves state-of-the-art RF spectrum synthesis quality and reduces the reconstruction time by $6.4\times$ compared with existing WRF methods.

We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree $T(\xi,\omega)$ consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\partial G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.

We study projection phenomena in word metrics of finitely generated acylindrically hyperbolic groups. For a loxodromic WPD element acting on a hyperbolic space, we prove that shortest projection in the word metric to the corresponding cyclic subgroup sublinearly tracks the pullback of shortest projection to its axis in the hyperbolic space. As applications, we obtain effective upper bounds for growth functions and construct proper quotients whose growth rates converge to that of the original group. We further prove a growth–cogrowth inequality for confined subgroups in both acylindrically hyperbolic groups and Morse local-to-global groups with Morse elements.

We focus on refined Humbert invariants of principally polarized superspecial abelian surfaces, introduced by Kani in 1994. The main contributions are to enumerate principal polarizations on a superspecial surface, and for each polarization, to compute the refined Humbert invariant of a principally polarized superspecial abelian surface. Then, we present several applications of computing this invariant for isogeny-based cryptography. First, we provide a decision algorithm to check if two given polarizations are isomorphic. Second, we present an efficient algorithm to determine the geometric type of a principally polarized superspecial surface. Third, we prove an upper bound on the largest minimal isogeny degree among pairs of supersingular elliptic curves, independent of their endomorphism-ring structures, and our experimental evidence verifies this claim up to $p=659$, $p\equiv 11\pmod{12}$. Fourth, we present experimental evidence for a minimum isogeny frequency within the proven upper bounds. Lastly, we provide a different perspective on the fixed isogeny degree problem using refined Humbert invariants and analyze it without explicit endomorphism rings.

This is the first in a series of papers studying symplectic dynamics on Liouville domains $(W, \lambda)$ from a geometric approach. We employ Usher's tube construction to build a higher-dimensional Liouville domain from $(W,\lambda; H)$, where $H$ is a Hamiltonian function on $(W, \lambda)$, possibly non-autonomous. We establish dynamical stabilities with respect to a new Thompson-type pseudo-metric on Hamiltonians, bounded below by Banach–Mazur type distances. We also explore relations between the topological entropies of the Hamiltonian dynamics and its induced Reeb dynamics, examine symplectic capacities, especially the Gromov width, under the tube construction, and give a categorification of the Thompson-type metric via a generalized tube construction (called the tunnel construction). Finally, we investigate the relation between the filtered symplectic homology of the resulting tube and the Hamiltonian Floer homologies of the input Hamiltonians, as well as their iterates, on the given base $(W, \lambda)$.

Informal statements of Gödel's incompleteness theorems often run: "any consistent formal system with arithmetic has true but unprovable statements." That reading assumes a classical notion of truth, under which a universal statement is true as soon as each of its numeral instances is. This paper presents reflective grounded arithmetic (RGA), in which truth is grounded in computation and universal quantification is grounded reflectively: a universal statement is true when the system's own proof search certifies its schematic instance, and false when the system refutes one numeral instance. RGA allows recursive definitions without restriction and proves addition, multiplication and Ackermann's function total as theorems quantified inside the system. RGA is consistent, and it is complete for its own, grounded notion of truth: every well-formed statement that evaluates to true is provable. Gödel's theorems still hold for RGA, as a companion paper shows. The statements that classical reasoning calls true but unprovable have every numeral instance provable but no certified schematic instance, so they are ungrounded in RGA; read instance by instance, in an $\omega$-semantics for which RGA is also sound, they are true. All results are machine-checked in Isabelle/HOL: soundness and consistency; completeness for grounded truth, open statements included; $\omega$-soundness and $\omega$-consistency; that grounded truth is recursively enumerable and RGA represents exactly the recursively enumerable sets; and $\omega$-incompleteness. The logic is neither classical nor intuitionistic: double negation is eliminable as an entailment, excluded middle fails wherever a statement has no boolean value, RGA has the disjunction and witness properties, and the deduction theorem holds exactly at provably decided hypotheses.

Let $Y=\Gamma\backslash G/H$ be a compact standard quotient of a non-Riemannian semisimple symmetric space $X=G/H$. We investigate the spectral decomposition of the algebra ${\bf D}(X)$ of $G$-invariant differential operators on $X$ acting on $L^2(Y)$. The absence of elliptic invariant differential operators makes the spectral theory fundamentally different from the Riemannian case. We first show that standard quotients arise from transitive actions on $X$ of real reductive subgroups $L$ of $G$ containing the discrete subgroup $\Gamma$. Our approach is based on the geometry of properly transitive triples $(G,H,L)$. We derive explicit formulas expressing the Casimir operator of $G$ in terms of Casimir operators of $L$. Triples fall into two classes: Type I and Type II. For triples of Type I, we prove essential self-adjointness of invariant differential operators and discreteness of the corresponding spectral decomposition. This decomposition is illustrated by a detailed analysis of compact standard quotients of anti-de Sitter spaces. In contrast, Type II triples exhibit genuinely continuous spectral phenomena. A central theme of the paper is the interaction between the representation theories of $G$ and $L$. For Type I triples, we prove $L$-admissibility of $H$-spherical $G$-representations of finite length and establish multiplicity formulas. We show that the resulting correspondence defines a map between irreducible spherical $L$-representations and spherical $G$-representations. For triples of both types, we obtain a representation-theoretic description of eigendistributions via distributional matrix coefficients. As an application, we show that every integrable discrete series representation of $G/H$ contributes an infinite-dimensional family of $L^2$-eigenfunctions on every compact standard quotient of Type I.

The ferromagnetic magnetohydrodynamic equations are a non-linear PDE system that models the flow of an electrically conducting fluid that also has intrinsic magnetisation. In this paper, we propose a fully-discrete linear finite element scheme based on Euler's method to approximate the solutions to these equations and perform an error analysis for the scheme. Numerical experiments have been included to corroborate our theoretical results.

We study the type A nilpotent Hessenberg varieties associated with Hessenberg functions that satisfy $h(i)\le i+1$. We call these the generalized parabolic Peterson varieties. We show that such varieties can be decomposed into the union of specific generalized parabolic Peterson varieties $\operatorname{Pet}_{\lambda,\alpha}$, such that $\lambda$ is an integer partition, $\alpha$ is an integer composition, and $\alpha$ is dominated by $\lambda$. We prove that when $\alpha$ is dominated by $\lambda$, the cardinality of the maximal dimensional components of $\operatorname{Pet}_{\lambda,\alpha}$ equals the Kostka number $\mathcal{K}_{\lambda\alpha}$, and its dimension is determined only by $\lambda$ and the length of $\alpha$. We provide a recursive formula for the Poincaré polynomial of $\operatorname{Pet}_{\lambda,\alpha}$. These results partially answer open questions about the geometry of Hessenberg varieties but raise further questions about the representation-theoretic reasons for these facts.

In this work, we extend the protomodularity of the category of cocommutative Hopf algebras to the quasitriangular setting. Every quasitriangular Hopf algebra admits a minimal quasitriangular Hopf subalgebra and, as we show, can be regarded as a cocommutative bimonoid in the tensor-braided duoidal category of bimodules over it. This leads us to investigate protomodularity in the broader context of Hopf monoids in duoidal categories. To this end, we adopt a slight modification of Böhm's notion of antipode associated with a reversion, further refining an earlier one due to Böhm-Lack. This framework allows us to study Hopf monoids in this setting, Galois and co-Galois maps, and the factorization of Hopf monoids. Using these tools, we prove a factorization of points, the Split Short Five Lemma, and the existence of pullbacks of split epimorphisms along arbitrary morphisms in the category of cocommutative Hopf monoids with monic unit in any tensor-braided duoidal category with a reversion; hence this category is protomodular. As applications, we recover the protomodularity of cocommutative Hopf algebras in symmetric monoidal categories under mild assumptions, and we obtain that of the coslice category of quasitriangular (resp. triangular) Hopf algebras under a fixed subobject; in the triangular case, this can be traced back to a category of generalized internal groups, introduced in the present work. When the fixed subobject is minimal, we infer the protomodularity of the category of quasitriangular Hopf algebras whose minimal quasitriangular Hopf subalgebra is isomorphic to the fixed subobject, which we interpret as the protomodularity of an essential fibre of a functor. As a byproduct, our results extend the double cross product of cocommutative Hopf algebras to the quasitriangular setting.

The hypernatural numbers ${}^*\mathbb{N}$ of nonstandard analysis have recently proven to be an effective tool in arithmetic Ramsey theory. After introducing the fundamental “nonstandard" notions, we present several examples to illustrate the use of this technique in practice. In particular, we provide brief nonstandard proofs of some recent results concerning the monochromaticity of certain families of finite and infinite configurations.

We study extension algebras arising from Lusztig's construction of perverse sheaves on quiver representation spaces equipped with an admissible automorphism $a$, leading to folding Khovanov–Lauda–Rouquier (KLR) algebras. Extending the method of M. Varagnolo and E. Vasserot to Lusztig's symmetrizable setting, we construct the skew group algebra $R(\nu)\langle\mathbf{a}\rangle$, where $\mathbf{a}$ is induced by the admissible automorphism $a$, and show that $\Ext^{\bullet}(\ ,L)$ preserves the signed basis and the crystal structure. In this framework, for finite types, we define standard and dual standard modules, analyze their behavior under dualities, and study the resulting Grothendieck groups. We prove that the transition matrix from the classes of indecomposable projective modules, which give the canonical basis, to the dual standard modules, which give the PBW basis, is upper triangular with diagonal entries equal to one. This provides a geometric categorification of these bases and of their relation in the symmetrizable setting.

Finite-precision arithmetic unavoidably introduces numerical approximation errors. Numerical computations may use insufficient precision or an improper formulation, which leads to numerical instability. In this paper, we introduce a unified software tool for stochastic numerical validation of deep-learning operators. The tool follows CESTAC on supported exposed operations and uses an operator-level data-perturbation approximation for GEMM-like kernels. Our developed software not only enables numerical validation with a single computation pass but also detects the sources of numerical instability and provides numerical stability monitoring during deep learning training and inference. We verified its effectiveness on the detection of polluted operators with injected numerical instabilities across various tasks. We believe that our developed method and tools provide valuable insights into developing numerically stable computing kernels, which are particularly critical for numerically stable and efficient deep learning training and inference.

We prove that the path $P_{11}$ does not admit uniform average mixing under any probability distribution on $\mathbb R$, answering a question of Baptista, Coutinho, and Marques in the negative. The proof is exact: we construct an explicit rational symmetric matrix $Y$ such that $\langle Y,M(t)\rangle_F=1$ for every $t\in\mathbb R$, whereas $\langle Y,J/11\rangle_F=12/11$.

We study the asymptotic spectral properties of high-dimensional Spearman correlation matrices for scale-mixture data. We consider observations of the form $x_t=\sigma_t \xi_t \in \mathbb{R}^N,$ where the coordinates of $\xi_t$ are i.i.d.\ and the scalar mixture variable $\sigma_t$ is shared by all coordinates. Under natural symmetry assumptions, the coordinates of $x_t$ are pairwise uncorrelated in both the Pearson and Spearman sense. Nevertheless, they are not independent when the mixture variable is non-degenerate. We show that this higher-order dependence survives the rank transformation and leaves a nontrivial spectral signature. In the proportional regime $N/T\to q\in(0,\infty),$ the empirical spectral distribution of the Spearman correlation matrix converges almost surely to a generalized Marčenko–Pastur law governed by the limiting distribution of an effective rank variance. We also formulate a broader latent-variable extension, which covers, in particular, some scale-mixture models with correlated directional components. We discuss solvable examples and numerical approximations, motivated in part by heavy-tailed data in robust multivariate statistics, econometrics, and finance.

There is no general method for designing proper parameters to achieve faster convergence in distributed optimization algorithms. In this paper, we consider the distributed inexact gradient tracking (DIGing) algorithm with the objective function being the unweighted sum of squares. By representing the iteration algorithm as a dynamical linear system, we decompose it into different graph frequencies and obtain a set of decoupled subsystems, on which we can easily analyze the convergence rate. By using Routh stability criterion from control theory, we derive the explicit formula of the optimal worst-case convergence rate and the corresponding parameters. We can see that the convergence rate of DIGing is slow even for the simplest objective functions, thus acceleration is necessary for general application. The proposed method can be viewed as the first step toward optimal parameter design of DIGing algorithm in solving general objective functions.

We show how a standard constructive notion of ordinal supports a useful constructive theory of transfinite recursion. We do this by giving constructive proofs of various initial algebra theorems, like Adamek's theorem, and a version of Quillen's small object argument for constructing cofibrantly generated algebraic weak factorisation systems.

We define a mesh refinement algorithm based on the rule of dividing the largest angles of triangular elements of planar partitions in focus into $n$ equal parts, and analyse the (geometric) properties of triangulations generated by this technique. This largest-angle $n$-section rule is compared with the classical longest-edge $n$-section rule, where it is the longest edges which are split into $n$ equal parts. The longest-edge bisection and trisection are known to produce nondegenerate triangulations (possibly with hanging nodes), but the longest-edge $n$-sections with $n\geq 4$ always produce (infinite) sequences of triangles with minimum angles tending to zero (moreover, their relevant maximum angles tend to $\pi$), thus breaking the minimum and maximum angle conditions. We show that this degeneration effect is not a consequence of $n$-section itself. For every $n\geq 2$, the largest-angle $n$-sections produce partitions satisfying the minimum angle condition (and, therefore, the maximum angle condition). More precisely, if the initial triangle has its smallest angle $\gamma_0>0$, then all descendant triangles have angles bounded below by $m_n=\min\left\{\gamma_0,\frac{\pi}{3n}\right\},$ and, correspondingly, bounded above by $\pi-2m_n<\pi$. We also show that the recursive largest-angle $n$-section algorithm always produces a family of triangular partitions, i.e. the maximum diameter of level-$k$ descendants tends to zero as $k \to \infty$.

Let $P (Y_1, ..., Y_d)$ be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along $\Omega (|P (n_1, ..., n_d)|)$. Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergelson and Richter: if $P$ is an irreducible binary cubic form and $ (X, T)$ is a uniquely ergodic system with unique invariant measure $\mu$, then for any $x \in X$ and $f \in C(X)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac 1 {N^2} {\mathop{\sum\sum}_{n_1, n_2 \leqslant N}} f \big( T^{ \Omega (|P (n_1, n_2)| ) } x \big) = \int_{X} f \ \mathrm{d} \mu . \end{equation*} Moreover, we prove in the appendix a related conjecture of Céspedes and Donoso over number fields.

Seasonal population models must combine diffusion, inactive periods and abrupt biological transfers while remaining differentiable with respect to localized interventions. We formulate a finite-dimensional timing-and-placement control problem for a Stieltjes-time reaction–diffusion model and derive the exact adjoint of the fully discrete residual. Continuous finite elements and an implicit Stieltjes–Euler scheme represent propagation, compartment replacement, resets and historical phase averages. For the resulting box-constrained penalized problem, we prove well-posedness and differentiability of the discrete control-to-state map, existence of a minimizer, first-order stationarity, and equivalence between direct-sensitivity and adjoint gradients. Benchmark tests show machine-precision agreement (below $3\times10^{-15}$) and adjoint speed-ups from about $7.5$ to $30.9$ as the number of traps increases from one to four. The method is applied to Vespa velutina trap campaigns on a realistic Galicia mesh. Smoothed activation windows, domain-normalized moving kernels and the complete adjoint are independently verified. The final two-cluster control remains admissible and has a scaled gradient infinity norm of $2.97\times10^{-5}$ with 120 temporal subdivisions per month. Fixed-control evaluations at 60, 120 and 240 subdivisions yield a refinement-increment ratio of $0.4984$. At the finest level, the joint control outperforms time-only and reference controls, reducing the diagnostic objective by $0.49209%$. This reduction depends on uncalibrated trap-to-mortality intensities and should not be interpreted as field capture efficacy. The framework is fully reproducible, without claiming global optimality or a calibrated field-management prescription.

The problem of finding lower and upper bounds on multivariate homogeneous polynomials is both difficult and important given its applications to questions ranging from dynamical stability in complex potential landscapes to data analysis. From the standpoint of tensor eigenvalue theory, the question is equivalent to finding the smallest and the largest eigenvalues of the coefficient tensor corresponding to the given polynomial. Standard approaches outlined in the literature amount to running nonlinear iterations in search for the optimal rays along which the growth of the polynomial is fastest or slowest. Unlike the case of matrices (or their corresponding multivariate quadratic forms) convergence of such algorithms for higher-rank tensors is capricious due to the complex topography of polynomial objective functions. In this essay, a very different strategy, inspired by quantum-mechanical variational methods, is introduced for finding bounds on polynomials. The original polynomial is replaced by an operator acting in a suitably chosen (large) space of states, such that in an appropriate "classical" limit this operator approaches the original polynomial expression made of commutative variables. As a result, approximating the smallest and largest eigenvalues of the coefficient tensor, and thus finding bounds on polynomials, amounts to diagonalizing the resulting quantum operator, represented as a large matrix, and then inspecting the smallest and largest eigenvalues of this matrix. This approach is then successfully applied to standard test examples from tensor eigenvalue literature and other problems of interest in mathematical physics including Strichartz-type inequalities.

We will show that the twelve families of minimally representation-infinite incidence algebras are brick-continuous and admit a generic brick, which will be explicitly constructed. After that, reversing the reduction process between posets, we will show that any representation-infinite incidence algebra is brick-continuous and admits a generic brick, which can be explicitly computed using what will be called ”extension functors”. Lastly, a method to apply these results to a broader class of algebras is given.

Let $d \geq 1$ be an integer, $G$ be an abelian group and $\nu, w_1, \dots, w_{2d}: G \to [0, \infty)$ be functions with finite, non-empty supports. Define the generalised additive energy \[ E_{2d, \nu}(w_1, \dots, w_{2d}) = \sum_{y,y' \in G}\sum_{a_1, \dots, a_{2d} \in G } w_1(a_1) \dots w_{2d}(a_{2d}) \nu(y) \nu(y') 1_{\sum_{i=1}^d (a_i - a_{i+d}) = y-y'} .\] Moreover, for every $1 \leq i \leq 2d$, let $E_{2d, \nu}(w_i) = E_{2d, \nu}(w_i, \dots, w_i)$. A standard Fourier analytic argument delivers the estimate \[ E_{2d,\nu}(w_1, \dots, w_{2d}) \leq \prod_{1 \leq i \leq 2d} E_{2d, \nu}(w_i)^{1/2d}.\] In this note, we present a purely combinatorial proof of the above inequality. In particular, our proof does not use any Fourier or spectral analysis and relies on repeated applications of Cauchy–Schwarz inequality combined with a discrete convexity extension type argument. We also record a variation of this upper bound in the non-abelian setting via spectral inequalities following work of Hatami on graph norms, as well as a relevant sumset analogue obtained via iterative applications of the Plünnecke–Ruzsa inequality.

We record the Calabi-Yau version of Kontsevich's formality morphism from deformation quantization. As a special case we find that any unimodular holomorphic Poisson Calabi-Yau has a canonical closed deformation quantization of its resolved algebra of functions. A broader motivation is to find an explanation of Kontsevich's deformation quantization in the setting of Calabi-Yau categories via Sen-Zwiebach's string vertices.