Papers

The $\ell_p$-Lewis weights of a matrix are defined by a fixed-point equation. For $p<4$, Cohen and Peng [CP15] showed that iterating an equivalent rearrangement of this equation computes Lewis weights to high precision; for $p\geq4$, prior high-precision methods instead use optimization-based approaches. We show that the direct Lewis fixed-point iteration, appearing in the thesis of Lee [Lee16], computes Lewis weights to high precision for every $p>2$. For a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ partitioned into row blocks $\mathbf{A}_{[1]},\ldots,\mathbf{A}_{[k]}$, we compute, for $p>2$, coordinatewise $\varepsilon$-approximate $\ell_p$ block Lewis weights in $O\left(p\log\frac{p\sqrt{\sum_{i=1}^k\operatorname{rank}(\mathbf{A}_{[i]})}}{\varepsilon}\right)$ rounds of exact leverage-score-vector computations. For ordinary Lewis weights, this becomes $O\left(p\log\frac{p\sqrt{m}}{\varepsilon}\right)$, improving the $O\left(p^2\log(m/\varepsilon)\right)$ bound of Gribling, Sidford, and Zhang [GSZ26] for $p\geq4$. Our guarantees also extend to approximate leverage-score computations. Our main observation is that each direct Lewis update contracts the KL divergence to the true weights by a factor of $1-\frac{2}{p}$. We also give an alternate explanation of this contraction through volume sampling and entropic independence. Synthetic experiments closely match our predicted local contraction rates and iteration counts grow approximately linearly with $p$; real-data experiments illustrate the information-concentration tradeoff of finite-$p$ block Lewis designs.

We study two notions of positivity for smooth currents on singular spaces. We show that they are not equivalent by establishing a necessary condition on the fourth Whitney cones. We also construct an explicit compact normal projective variety $X$ and a real smooth $(1,1)$-form $T$ on $X$ such that $T$ has local smooth ddc-potentials and is a Kähler current, but $T$ is not a positive form (hence not Hermitian).

We explore the use of simulated data for training a model for protein annotation in crowded cryo-electron tomography volumes reconstructed from images collected at limited tilt angles and severely corrupted by the measurement operator. Firstly, we leverage the corruptions imposed by the forward model to generate domain-specific augmented paired views of the exact same scene for an invariance objective integrated into the LeJEPA self-supervised training framework. Secondly, we use additional information from the simulation pipeline such as the positions and identity of proteins in the simulated volumes to inform the architecture of the model and the loss function, so that semantic information is localised at protein positions in the resulting dense feature volume. The resulting model, CARNIVAL, is evaluated without finetuning on classification and detection tasks in real tomograms, using a benchmark dataset containing multiple protein types and two tomogram processing types. We show that CARNIVAL outperforms a state-of-the-art model trained using a contrastive objective on simulated data but without forward model-based paired views or privileged information.

We prove that harmonic morphisms between sub-Riemannian Lie groups are smooth, are symmetries of the sub-Riemannian Laplacian and are conformal submersions that satisfy a particular PDE. Moreover, we give some partial results for harmonic morphisms beyond Lie groups. We show the existence of harmonic coordinates for harmonically homogeneous spaces, and that smooth harmonic morphisms are symmetries of the Laplacian in all sub-Riemannian manifolds. Finally, we compute the first variation of the horizontal energy along a harmonic morphism of sub-Riemannian Lie groups: it is given by a linear form on the Lie algebra of the target, the modular mismatch, which vanishes for Riemannian and Carnot targets, but surprisingly not in general. Therefore, unlike in the Riemannian case, harmonic morphisms of sub-Riemannian Lie groups need not be harmonic maps.

Given a finite family $\mathcal F$ of forbidden $r$-graphs, the Turán problem asks for the maximum asymptotic edge density of $\mathcal F$-free $r$-graphs and the structure of near-extremal examples. We show that both questions can encode arbitrary computation. Fix a universal Turing machine $\mathsf U$. For every sufficiently large fixed $r$, there is a rational $\tau_r\in(0,1)$ such that, from each binary word $\beta$, one can construct a finite family $\mathcal F_{r,\beta}$ with $\pi(\mathcal F_{r,\beta})=\tau_r$ if $\mathsf U$ does not halt on $\beta$, and $\pi(\mathcal F_{r,\beta})>\tau_r$ otherwise. The same dichotomy governs extremal structure. We construct finite families $\mathcal G_{r,\beta}$ such that nonhalting gives a unique extremal limit and Erdős–Simonovits stability, whereas halting gives two nonempty compact extremal phases separated by the sign of a fixed continuous statistic. Hence uniqueness and connectedness of the extremal space, symmetry breaking, two-phase behavior, and stability are all undecidable. The reductions are effective and verifiable in ZFC by finite certificates. Consequently, for every consistent computably axiomatized extension of ZFC and every sufficiently large fixed $r$, there is a finite family $\mathcal F$ for which the true equality $\pi(\mathcal F)=\tau_r$ is neither provable nor refutable; analogous independence holds for the five structural properties above. We also obtain effective approximation, classify exact comparison complexity, and show that the smallest improvement witnesses have Busy-Beaver growth, with no uniform computable positive lower bound on the density gain.

Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is to do similarity matching or clustering. The most common metrics used for measuring similarity of curves are Dynamic Time Warping (DTW) and Fréchet distance. These metrics are sensitive to sampling rate and outliers respectively, and do not yield robust outcomes. Continuous Dynamic Time Warping (CDTW) is a more robust distance metric that improves upon DTW and Fréchet distances. Existing algorithms for CDTW are either exact algorithms that focus on non-Euclidean norms and piecewise linear curves, or approximation algorithms limited to piecewise linear curves. We present an approximation algorithm for computing the CDTW distance under Euclidean norm between piecewise (higher degree) algebraic curves. That is, we present a fully polynomial-time approximation scheme (FPTAS) of multiplicative error $\varepsilon$, with $O \left( (m+n)^{\frac{19}{6}} (\frac{1}{\varepsilon})^{\frac{10}{3}} \log \left( \frac{ (m+n) }{\varepsilon^2} \right) \right)$ complexity, where $m$ and $n$ are the number of pieces of the two input curves.

Human mobility predictability concerns the best prediction performance attainable from a given target and input information, but its ground truth is not directly observable on real mobility data. We present BER-PEF, a Bayes-error-rate-based framework that converts BER estimation into mobility predictability estimation and provides a unified protocol for comparing estimators without observable ground truth. The framework maps symbolic sequences, numeric trajectories, contextual features, and learned representations into a common feature–label space, then evaluates estimator outputs along controlled perturbation curves against a shared predictability reference interval by measuring deviations below the interval, above the interval, and across the full interval. Experiments on Foursquare NYC and TKY, GeoLife, and T-Drive show that several BER-based estimators achieve lower reference discrepancy than existing predictability methods on symbolic sequences and numeric trajectories, while their estimates track changes in empirical prediction performance under perturbation. Additional analyses show that contextual inputs and multiple structured representations can be evaluated under the same protocol, and that aggregating evidence across multiple perturbation levels provides a more reliable basis for estimator selection than relying on a single unperturbed observation. BER-PEF therefore offers a unified and verifiable path for evaluating predictability estimators on heterogeneous mobility data when ground-truth predictability is unavailable.

In this paper, we study the rational realization problem for the classifying space $B\mathrm{aut}_1(X)$. We prove that if $S^{2n}$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected space $X$, then $X$ is $\pi$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $\pi$-finite space $X$. We also prove that, for all $n\geq 2$ and $s,t\geq 1$, the product of Eilenberg–MacLane spaces $K(\mathbb{Q}^s,n)\times K(\mathbb{Q}^t,n+1)$ cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $\pi$-finite space $X$. Moreover, we prove that if $r\geq 2$, $n\geq 3$ and $K(\mathbb{Q}^r,n)$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected $\pi$-finite space $X$, then $X$ has the rational homotopy type of $K(\mathbb{Q}^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.

Using the universal-cover ball comparison theorem of Chodosh–Vikman, we have an easy consequence that every closed oriented Riemannian surface of positive genus is Loewner, that is, it satisfies a systolic-area inequality.

Let $\Sigma$ denote the class of functions $f(z) = z + b_0 + b_1 z^{-1} + \cdots$ that are analytic and univalent in the exterior unit disk $\Delta^* = \{z \in \mathbb{C} : |z| > 1\}$. In 1962, Ch. Pommerenke proved that if $F$ and $G$ are convex functions in $\Sigma$, then every convex linear combination $H = \lambda F + (1-\lambda)G$ ($0 < \lambda < 1$) remains univalent and belongs to $\Sigma$. In Hayman's problem collection (Research Problems in Function Theory, Problem 6.10), Pommerenke raised the question of whether $H$ is necessarily also a convex function. We resolve this question in the negative by constructing an explicit counterexample in $\Sigma$ with parameters in $\mathbb{Q}(i)$. The argument is self-contained and has been formally certified in the Lean 4 proof assistant.

This paper studies the computation of strong solutions of monotone variational inequalities (VIs) with Lipschitz continuous operators. Building on the idea of accumulative regularization (AR), we develop a general framework for VIs, with particular emphasis on stochastic settings. Under unbiased stochastic oracles with uniformly bounded variance $ \sigma^2$, AR computes an approximate solution with expected operator residual bounded by $\varepsilon$ using at most $ \widetilde{\mathcal{O}}\left(\tfrac{LD_0}{\varepsilon}+\tfrac{ \sigma^2}{\varepsilon^2}(\log\tfrac{LD_0}{\varepsilon})^3\right) $ stochastic oracle calls, where $L$ is the Lipschitz constant and $D_0$ bounds the initial distance to the solution. This substantially improves the existing $\mathcal{O}( \sigma^2/\varepsilon^4)$ complexity for residual reduction and matches the lower bound up to logarithmic factors. For strongly monotone VIs, measured by the distance to the solution, AR achieves the optimal oracle complexity when the strong monotonicity modulus is known. By treating the problem as merely monotone, AR still achieves nearly optimal complexity without knowledge of this modulus. We further introduce a state-dependent noise model applicable to general monotone VIs with potentially nonunique solutions, extending state-dependent noise analysis beyond the strongly monotone setting. We apply these results to policy evaluation in reinforcement learning, where we show that estimators of the projected Bellman operator satisfy our state-dependent noise condition. To the best of our knowledge, the AR framework, the improved complexity under uniformly bounded noise, the state-dependent noise model and its guarantees, the adaptivity to an unknown strong monotonicity modulus, and the improved guarantees for FTD learning are all new in the VI literature.

This dissertation studies Cartan representations: representations of certain Coxeter groups as the Weyl groups of Kac-Moody algebras. It sets out and proves conditions for these representations to be isomorphic to each other, the reflection representation of the Coxeter group, and their dual representations, and gives partial results for the same over fields of finite characteristic, as well as clarifying what it means for a Coxeter group to be crystallographic and giving conditions for the Weyl group of a Kac-Moody algebra to be such. Some illustrative examples are also provided.

Whether the constant $$G=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}=\frac1{1^2}-\frac1{3^2}+\frac1{5^2}-\frac1{7^2}+\cdots$$ introduced by Catalan in the nineteen century is irrational, is a long-standing open problem. In this paper we prove the irrationality of $G$ via using suitable weights.

We derive nonasymptotic upper bounds on the probability that a generalized current of a geometrically ergodic diffusion observed for any amount of time, or its sample mean over any arbitrary sample size, deviates from the stationary mean by more than any given amount. The concentration-of-measure behavior of generalized currents is universally governed by the relaxation time of the underlying dynamics, the locally observed dissipation rate, and the intrinsic local fluctuations of the observable. We uncover stark qualitative and quantitative differences in fluctuations in and out of thermodynamic equilibrium. We further obtain refined inverse thermodynamic uncertainty relations, bounding the variance of generalized currents from above. We construct nonasymptotic confidence intervals for controlling uncertainty in thermodynamic inference from small data, i.e., from short trajectories and small samples, and provide the first quantitative answer to when a trajectory is sufficiently long and a sample is sufficiently large. As an illustration, we apply our results to currents observed on a two-dimensional Ornstein-Uhlenbeck process in and out of equilibrium and show how they can be used to rigorously detect broken detailed balance.

We study the job shop scheduling problem with a conflict graph, in which adjacent jobs cannot be processed simultaneously on different machines, with the objective of minimizing the makespan. We establish a polynomial equivalence with a resource-constrained job shop problem having unit-capacity resources and binary resource requirements The problem is strongly NP-hard even on two machines with unit-time operations. For this setting, we derive an optimal-makespan formula and a linear-time algorithm when the conflict graph is a cluster graph, given the clique partition. We also develop a polynomial-time $4/3$-approximation algorithm for arbitrary conflict graphs on two machines when each job has exactly two unit-time operations. For the general problem, we propose four mixed-integer linear formulations, lower bounds, and a genetic algorithm with active, non-delay, and hybrid decoding procedures. Computational experiments on instances derived from the Lawrence and Taillard benchmarks and on randomly generated generalized job shop instances evaluate the proposed methods and show that the genetic algorithm generally improves on constructive heuristics.

In his seminal paper, Bell [Physics Physique Fizika 1, 195 (1964)] considers the correlations that result from space-like separated measurements on a pair of entangled particles. He uses relativity theory to motivate the Bell causal structure, then shows the existence of quantum correlations that cannot be explained classically within this causal structure. Classical explanations of such quantum correlations are possible in alternative causal structures, for instance, those that allow superluminal causal influences, but, as shown in [New Journal of Physics 17 033002 (2015)], all such alternative explanations require fine tuning (causation without correlation). Here we discuss the existence of spurious quantum correlations — correlations that look quantum in one causal structure, but have a natural classical explanation in another. More precisely, there are causal structures that admit non-classical quantum correlations, but for which the same correlations have a classical explanation in another causal structure without fine tuning. The realisation in the other causal structure can also be achieved without breaking any natural constraints on the causal structure that follow from relativity theory. However, similarly to non-classical quantum correlations in the Bell causal structure, we find other causal structures with non-classical quantum correlations that do not have a classical causal explanation in any alternative causal structure without fine tuning.

We study the properties of the $dd^{\Phi}$-operator on $8$-dimensional Spin(7)-manifolds with torsion-free Spin(7)-structures $\Phi$. These operators were first introduced by Harvey and Lawson (An introduction to potential theory in calibrated geometry. Am.J. Math. 131.4 (2009), arXiv:0710.3920). We prove a Hodge decomposition theorem for the $dd^{\Phi}$-operator and obtain an analogue of the $\partial \bar{\partial}$-lemma in Kähler geometry. Using this, we define Bott–Chern-type cohomologies for Spin(7)-manifolds. We relate the Bott-Chern-type cohomology spaces to the moduli space of torsion-free Spin(7)-structures and calibrated geometry of Spin(7)-manifolds. These relations naturally give rise to the notion of Cayley-positive cones. In the course of proving the results, we state and prove various identities for the exterior derivative and its decompositions into irreducible Spin(7)-representations as well as identities for second order derivatives and Laplacians. The identities we prove are for any Spin(7)-structures and the specialized torsion-free ones are Spin(7)-analogoues of Kähler identities and Bryant–Harvey's identities in the $\mathrm{G}_2$-case (R. Bryant, Some remarks on $\mathrm{G}_2$-structures, Proceedings of the 11th and 12th Gökova geometry-topology conference, arXiv:math/0305124) and are results of independent interest.

In this paper, we address the subconvexity problem in level aspect for symmetric square $L$-functions for cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{Q})$ with a prescribed local ramification at prime $p$. To be more precise, let $\pi$ be a tempered cuspidal automorphic representation of conductor $q(\pi)=p^2$ with a non-quadratic central character of conductor $p$. We prove that if the corresponding local representation $\pi_p$ belongs to a suitable class of representations $\mathcal S$, then \[ L\left(\frac{1}{2},\,\mathrm{Sym}^2\pi\right)\ll_{\varepsilon, \pi_\infty} q(\mathrm{Sym}^2\pi)^{\frac{1}{4}-\frac{1}{168}+o(1)}, \] where implied constant depends polynomially on the spectral parameters of $\pi_\infty$. This is the first instance of level-aspect subconvex bound for $L$-functions of a $\mathrm{GL}_3(\mathbb Q)$ automorphic representation. Our approach is based on the delta-symbol method. Apart from some standard analytic number theoretic tools, Katz's theory of hypergeometric sums, and Deligne's proof of Weil-conjectures play an important role in the proof.

We construct a nonempty $C^k$-open family of smooth, autonomous, divergence-free velocity fields on $\mathbb{T}^3$ that generate fast dynamos. The proof proceeds by first establishing fast dynamo action for a large class of smooth, time-periodic velocity fields exhibiting the classical stretch–fold–shear mechanism. We then realize the associated period map within a smooth autonomous flow using carefully tuned return dynamics to a global Poincaré section. In both settings, the proof first isolates unstable distributional eigenmodes for the ideal dynamo operator on an anisotropic Banach space of distributions, and then shows that this spectral instability persists under the singular perturbation $\varepsilon \Delta$. The resulting estimates are uniform as $\varepsilon\to0$ and stable under $C^k$ perturbations of the velocity field, yielding an open set of smooth fast dynamo vector fields. This result resolves the Fast Dynamo Conjecture of Zeldovich and Sakharov, as recorded in Arnold's book of problems.

We determine the growth scale of the optimal constants in the real anisotropic Bohnenblust–Hille inequality. For an exponent vector $\mathbf q^{(m)}$, write $C_{\mathbf q^{(m)}}^{(m)}$ for its optimal constant and $d_m$ for its diameter. These constants are superpolynomial precisely when $m d_m/\log m\to\infty$; throughout this regime, their logarithm has the sharp scale $m d_m$. If also $d_m\to0$, then $\log C_{\mathbf q^{(m)}}^{(m)}/(m d_m)$ lies asymptotically in the interval $\left[\frac{\log 2}{4},\frac{2-\log 2-\gamma}{4}\right]$, where $\gamma$ is the Euler–Mascheroni constant; this interval has width less than $10^{-2}$. We also solve the extremal problems at fixed diameter and fixed total deficit. The normalized reciprocal-deficit profiles order the canonically arranged optimal constants by majorization, yield exact formulas on a full-dimensional region, and recover the lower coefficient $(\log 2)/4$ throughout a broad class of anisotropic regimes.

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $\kappa=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_\kappa$ as $\kappa \downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_\kappa$ graph metric converges as $\kappa \downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.

We continue our study of the conformal loop ensemble (CLE) with parameter $\kappa=4$, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE$_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE$_4$ metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE$_4$ non-intersection bounds for metric balls.

We consider the conformal loop ensemble (CLE) with the parameter $\kappa=4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE$_\kappa$ loops as $\kappa \downarrow 4$. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE$_4$ which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.

We study probabilistic numerical methods for solving nonlinear PDEs from a Bayesian nonparametric perspective. Given noisy evaluations at random collocation points, we place a truncated Gaussian series prior on the unknown solution and establish contraction at the minimax nonparametric rate, up to a logarithmic factor. Our main results give Gaussian approximations of the posterior in positive-order Sobolev spaces and, under suitable conditions, in the uniform topology. This contrasts with classical ill-posed inverse problems, where Bernstein–von Mises theorems typically require substantially weaker topologies. Here, the observation operator is differential rather than smoothing, and inversion of its linearisation gains regularity, making these strong-topology results possible. The posterior may be centred at either the posterior mean or the posterior mode. We further prove that the Gaussian Laplace approximation is asymptotically equivalent to the true posterior at a $\sqrt{N}$-scale.

Synchronization strings provide deterministic position labels for recovering coordinates after insertions and deletions. Haeupler and Shahrasbi introduced these objects, and subsequent work proved that four symbols suffice for some fixed parameter epsilon < 1, whereas two symbols cannot support arbitrarily long synchronization strings. We resolve the remaining ternary case: every length admits a ternary 2001/2002-synchronization string. Thus three is the exact minimum constant alphabet size. A computer-assisted refinement based on a larger 54-uniform family yields ternary epsilon-synchronization strings for every epsilon > 215/216. The previous four-symbol construction uses a ternary square-free backbone to exclude short repetitions and a fourth symbol to carry long-range synchronization marks. Our main technical contribution is a local-entropy transfer theorem: every square-free block-local source with a positive interval conditional min-entropy rate supports synchronization strings with a fixed gap. We instantiate this theorem using occurrence-wise branching in a Brinkhuis family. Every outcome remains ternary and square-free, while every long interval retains linear conditional min-entropy after all choices outside it are exposed. A deletion-ball estimate converts this entropy into an exponentially small probability of a near-complete common subsequence between adjacent intervals, and an asymmetric Lovasz Local Lemma enforces all interval constraints simultaneously. The same framework also yields exponentially many valid words, synchronization circles, and synchronization within a class of extremal square-free words. Adding constraints on distant intervals gives a Las Vegas construction in expected O(n^2 log^3(n+2)) time.

Quantum thermalization explains how an isolated many-body system naturally evolves towards a thermal state, rendering information about the initial conditions inaccessible to local measurements. This is precisely the mechanism utilized in quantum error correction, where information is protected by design through a nonlocal encoding. In this work, we leverage this connection to port the rigorous framework of (approximate) quantum error correction to the study of quantum thermalization. Treating typical late-time states as codewords, we characterize the error-correcting properties of generic thermalizing dynamics. We numerically uncover a universal relationship between the encoding rate, distance, and thermal entropy density of the emergent code. At infinite temperature, this universal curve saturates the quantum Singleton bound, achieving the same optimal limit as Haar-random codes. At finite temperature, we introduce a code family based on the Scrooge ensemble, the natural thermal analogue of the Haar ensemble, and prove it saturates the entropic quantum Singleton bound, establishing this family as optimal within entropic constraints. Our extracted universal curve independently saturates this same bound, revealing that finite-temperature thermalization is itself optimal. Finally, we show how conserved quantities limit the error-correcting behavior of thermalization: codewords with differing energies, or other conserved charges, leak only classical information, and correctability persists until the difference reaches the scale of thermal fluctuations. Our results reveal a universal optimal coding structure in thermalizing dynamics, while introducing new optimal codes that achieve fundamental limits of approximate quantum error correction.

In this paper, we consider a class of Riemannian nonsmooth composite expectation optimization problems, which arises in various machine learning, signal processing, and statistics applications. Noting that these problems admit structured minimax reformulations, we propose an efficient algorithm, named stochastic Riemannian alternating descent ascent method with recursive momentum (StoRADA-RM), to tackle them. StoRADA-RM performs one or multiple Riemannian stochastic gradient descent steps and then a proximal gradient ascent step at each iteration. To compute the Riemannian stochastic gradient, we propose a vector transport-free recursive momentum estimator that requires only $O(1)$ stochastic gradient evaluations per iteration. We prove that StoRADA-RM returns an $\epsilon$-Riemannian-stochastic-stationary point of a given problem in the said class in $O(\epsilon^{-3})$ iterations while making $O(\epsilon^{-3})$ calls to a stochastic first-order oracle (SFO). Both the iteration complexity and SFO complexity bounds are the best known in the literature for the said class of problems. The latter even matches the optimal lower bound for smooth nonconvex optimization with stochastic first-order algorithms. We then present numerical results on sparse principal component analysis and coordinate-independent sparse estimation to demonstrate the superior performance of our proposed method.

Probability plays two distinct roles in modern data-analytic practice: (i) as an internally coherent, filtration-relative language for sequential forecasting and, together with a stated loss or utility, decision making; and (ii) as a foundation for empirical claims such as stabilization, calibration, and repeated-sampling validity. The first role is relative to an assessment law $\mathbb Q$, whereas the second is evaluated under a governing law $\mathbb P$ and a declared repetition regime. In classical settings the roles are often aligned by conditional correctness under $\mathbb P$ together with i.i.d.\ or other stable-law assumptions under a fixed design, or by exchangeability. In adaptive and AI-mediated settings they can diverge because forecasts operate within feedback loops while reliability is evaluated under a potentially different law and regime. We synthesize relevant results from prequential forecasting, calibration, martingale and game-theoretic validity, adaptive data analysis, conformal prediction, and long-context AI reliability into a formal account of role separation. The resulting Role Separation Principle distinguishes internal coherence from conditional-mean correctness under the evaluation law and from target-specific reference and stability conditions. Two separation results show that a proper coarsening of the limiting-frequency distribution need not identify the governing law, and that even conditionally correct forecasts under that law need not accompany stabilization. We then develop a unified stabilization framework spanning filtration-based, shift-based, policy–environment, and design-based regimes and operationalize it through a regime-conditional audit checklist. Applied illustrations clarify calibration claims, synthetic-data validity, and structural mismatches relevant to some high-confidence unsupported AI outputs.

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}.$ We prove that if $\Phi$ is a Young function satisfying $ \Phi(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}_{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ as well, does not satisfy a corresponding $\Phi$-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type $(1,1)$. In the positive direction, we establish an $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)^2 \log_4L$ estimate for $\mathcal{C}_{2,\mathsf{lac}}.$

This paper investigates a two player zero-sum stochastic differential game characterized by three distinct regimes, reflecting the regime switching dynamics within the system. We aim to derive an explicit solution for a number of configurations of the switching system by means of the viscosity solutions approach. In particular, we analyze the associated Hamilton-Jacobi-Bellman-Isaacs equations and examine how the interactions between the different regimes affect the structure of the value function. The proposed framework provides a systematic approach to characterize the optimal strategies of the two players and to obtain explicit representations of the value function un der suitable assumptions. We also illustrate how to derive the value functions in case we know the qualitative structure of switching regions.