Papers

Low-rank matrix denoising is a central primitive in patch-based image restoration and many other inverse problems. Classical SVD-based image denoising methods often choose a truncation rank by matching residual singular-value energy with an estimated noise energy, but this rule is not a finite-sample risk principle because a fitted low-rank approximation inevitably absorbs part of the noise. This paper develops a mathematically rigorous alternative based on Stein's unbiased risk estimate (SURE). Since singular value hard thresholding is discontinuous and does not satisfy the hypotheses of Stein's lemma, we introduce a logistic smooth hard-threshold spectral estimator. We prove that the smooth shrinker satisfies the regularity conditions required by a spectral-estimator version of Stein's lemma, and therefore admits an exactly unbiased fixed-threshold risk estimate under Gaussian noise. For a fixed observed matrix and a finite set of candidate thresholds separated from the observed singular values, the ordering of the fixed-threshold smooth SURE objective eventually agrees with a simple limiting score. The limiting score has the same algebraic form as the biased hard-threshold SURE formula, but here it is used only as a computational device for ranking finite candidates. Selecting the minimizing threshold is a data-adaptive tuning step; the selected SURE value should not be interpreted as an unbiased risk estimate of the finally selected estimator.

The present study provides a closed-form characterisation of Nash equilibria in N-player binary games with unidirectional dependencies. While general network games are PPAD-complete, prior work has established that trees or paths admit polynomial-time solutions via dynamic programming. We provide a deterministic characterisation for the subclass of directed cycle graphical games, demonstrating that non-zero boundary incentives linearize the topology into a feed-forward propagation. Under this Robust Incentive Structure, resolution is achieved in O(N) time: strict dominance guarantees a unique equilibrium; in its absence, pure strategy equilibria are governed by the Parity Condition, while a unique fully mixed equilibrium is guaranteed via induced payoff indifference. For non-robust regimes, we deliver branching rules. The transition-matrix formulation evaluates the search tree size beforehand. This transparency enables the inverse design of target equilibria in circular networks, making explicit the mechanics that remain opaque in numerical solvers.

In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed $G_2$-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed $C^{1,\alpha}$ convergence. Our principal result shows that this \(C^{1,\alpha}\) convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive \(3\)-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to \(G_2\)-solitons. Since the available \(C^{1,\alpha}\) control does not directly close an elliptic bootstrap for the \(G_2\)-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy \(\varepsilon\)-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed \(C^{1,\alpha}\) convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with \(G_2\)-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted \(L^{\frac{7}{2}}\) -curvature bound then yields pointed $C^\infty$ compactness.

Long-range interactions such as electrostatics and dispersion remain a central bottleneck for machine learning interatomic potentials (MLIPs), especially in ionic, polar and interfacial systems. Ewald-based reciprocal-space mechanisms provide a physically grounded route for capturing these nonlocal effects, but often require dense Fourier grids and can become memory-limited at scale. This problem is particularly pronounced in molecular dynamics, where high efficiency requirements make accurate long-range modelling particularly costly. Here we introduce PSWF-LR, an exponent-aware long-range framework based on prolate spheroidal wave functions (PSWFs) that can be easily incorporated into existing model architectures. Its core components are PSWF-based mollification and atom-grid spreading, which enable compact and efficient representation of arbitrary inverse-power channels $1/r^p$ while treating the decay exponent as a physical prior. Across diverse long-range benchmarks, PSWF-LR reduces Fourier-mode requirements, improves energy and force accuracy, accelerates production-level simulations by about threefold, and extends long-range MLIP simulations beyond the memory limits of conventional MLIPs.

Consider the $(2+1)$D Discrete Gaussian model (ZGFF) on an $L\times L$ box with a hard floor at height zero and zero boundary conditions, at low temperature. The second author, Martinelli and Sly (2016) showed that the surface has a plateau, filling nearly the full square, at height either $H$ or $H+1$ for an explicit function $H(L)$. In a companion paper, we studied the local laws of the top level lines near the four sides of the box, and showed that after rescaling each by $(L^{2/3-o(1)},L^{1/3-o(1)})$, they converge to a product of Ferrari–Spohn diffusions. Two key features of the top level lines remained unaddressed: their global limit shape, and the critical window marking the transition from a top plateau at height $H$ to one at height $H+1$. These features are intrinsically linked: deriving the global limit of the top level line is needed for determining whether it is preferable to be at height $H$ or $H+1$ near criticality. This work completes this picture as follows. First, we obtain the global limit of the top level lines: for every fixed $n$, the $n$-th from-the-top level line converges in Hausdorff distance to a deterministic shape $\mathscr{L}_n$ that features the Wulff shape at scale $N_n=L^{1-o(1)}$ near the four corners of the box. Second, we identify, for every $h$, the point of emergence of a macroscopic $h$ level line: the probability of this event is monotone increasing in $L$ (up to a $o(1)$ error), and undergoes a sharp transition from near $0$ to near $1$ in a critical window of width $\leq L^{1/2+o(1)}$ around a side length $L=L_c^{(h)}$. This transition is discontinuous in that, once a macroscopic level $h$ emerges, it immediately occupies nearly all the box, and the above global and local scaling limits (Wulff, Ferrari–Spohn) hold for it. The new results extend to the $(2+1)$D $|\nabla\phi|^p$-models (ZGFF is the case $p=2$) for every fixed $p> 1$.

A guiding principle in P. Scholze's $p$-adic geometry asserts that the points of $\operatorname{Spec}\mathbf{Z}$ over an algebraically closed perfectoid field of characteristic $p$ are classified, up to equivalence, by its untilts. In this paper, we give a concrete geometric realization and a generalization of this paradigm. We construct the absolute $\mathbf{F}_1$-arithmetic curve $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ by pulling back the $\mathbf{F}_1$-structure sheaf of the arithmetic site to $\operatorname{Spec}\mathbf{Z}$. We demonstrate that $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ provides a common geometric origin for fundamental structures in $p$-adic Hodge theory, complex analytic geometry, and the adelic scaling site. The moduli space of points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over an arbitrary perfectoid field, modulo intrinsic symmetries, canonically parameterizes the space of all perfectoid fields with the same tilt, providing a universal, characteristic-independent geometric realization of Scholze's heuristic. Evaluating the points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over the field $\mathbf{C}$ of complex numbers reveals, at each prime $p$, that the non-trivial points canonically form two principal homogeneous spaces (torsors) over the Weil groups $W_p=\mathbf{Q}_p^\times$ and $W_\infty=\mathbf{C}^\times$. Quotienting the archimedean orbit by the discrete Frobenius symmetries yields the complex Tate curve with modulus $q=p^{-1}$. We show that this elliptic curve canonically decomposes as the product of its real locus, which exactly recovers the adelic periodic orbit $C_p=\mathbf{R}_+^\times/p^\mathbf{Z}$, and a $p$-independent phase space that emerges naturally as a real analogue of the Fargues–Fontaine curve.

We investigate a nonlinear extension of the Madelung transformation based on a hyperbolic phase–amplitude coupling of the form \[ \Psi = R e^{\imath\theta \coth R}, \] where \(R\) is a real amplitude field and \(\theta\) is an auxiliary phase coordinate governed by Schrödinger's equation. In contrast to the conventional polar decomposition, this construction imposes a singular hyperbolic relation between amplitude and phase, thereby endowing the Bohmian or hydrodynamic description with an intrinsically geometric structure. We show that the associated velocity field acquires a density-gradient contribution, producing generalized continuity equations and modified quantum force terms. When interpreted as a complex macroscopic order parameter, this generalized phase structure leads to modified superconducting electrodynamics; in particular, the London equations acquire additional contributions that render the Meissner response sensitive to spatial density gradients. The proposed framework is motivated by broader developments involving complex group velocities, dissipative wave propagation, and amplitude-sensitive transport in quantum systems.

In this paper, we investigate uncertainty principles for random signals associated with the linear canonical transform (LCT). First, the LCT of random signals is formulated on the probability space. Based on this representation, the Heisenberg uncertainty principle is established to characterize the relationship between the expectations in the time and frequency domains. Furthermore, the Donoho-Stark uncertainty principle, developed from a measure theoretic perspective, reveals that a random signal cannot be simultaneously concentrated on arbitrarily small sets in both the time and frequency domains. The bounds obtained in these two uncertainty principles explicitly depend on the LCT parameters, indicating that the LCT offers greater flexibility than the Fourier transform (FT). The corresponding results in the fractional Fourier transform and FT domains are also given as special cases.

We prove that if $j>0$ is not an integer, then there is a choice of signs $\varepsilon_k\in\{\pm1\}$ such that the partial sums $ \sum_{k=1}^{N}\varepsilon_k k^j $ are dense in $\mathbb R$. The proof groups consecutive terms into Thue–Morse blocks, whose Prouhet–Tarry–Escott cancellation produces nonzero block sums tending to zero but with divergent total variation. A standard steering argument then chooses block signs so that the resulting partial sums visit arbitrarily small neighbourhoods of every real number.

For the better part of a century, isolated singularities were treated as pathological obstructions requiring elaborate avoidance strategies. Pierre Alphonse Laurent (1813–1854), a French military engineer at Le Havre, ended this avoidance in 1843 by extending Cauchy's Taylor-type theorem to doubly connected (annular) domains, producing the doubly infinite power series that now bears his name. Negative-power terms in the expansion encode precise geometric information about the singularity rather than signaling a breakdown of the formalism. Laurent's contribution arrived through an unhappy institutional trajectory – submitted after a prize deadline, subjected to a priority claim by Cauchy, and issued in full only posthumously in 1863 – yet it became indispensable to every branch of mathematics and mathematical physics that touches on complex function theory. We reconstruct the mathematical problem Laurent solved, place it within Cauchy's analytic program of the 1830s–1840s, examine the institutional failure that prevented publication, document the independent parallel proof by Weierstrass (1841, published 1894), and trace the series' absorption into the standard toolkit via Briot and Bouquet and the residue calculus. Drawing on Laurent's 1843 Comptes rendus notice, Cauchy's Academy report, Bertrand's memorial notice (1890), and the secondary literature (Neuenschwander 1978, 1981; Manning 1975; Bottazzini 1986; Gray 2015), we analyze the philosophical significance of the series, which we term “exile mathematics”, and survey its reach into perturbation theory, number theory, probability, and quantum field theory. Readers familiar with the theorem but not its institutional history will find here a documented account of why a foundational result was withheld for two decades and how it nevertheless achieved canonical status.

Central Limit Theorems for integrals of third degree Hermite polynomials of Berry's random wave model on increasingly large domains are proved in dimensions 2 and 3. These were the missing cases for a complete description of limit theorems for integral functionals of monochromatic random waves based on the Wiener chaos decomposition.

We prove modified scattering for small data solutions to the Vlasov-Poisson system in a functional framework where the initial data, scattering states, and asymptotic convergence are measured in the same topology. In addition, we show that the corresponding wave operators define homeomorphisms between the spaces of initial and scattering data, while enjoying a local Lipschitz continuity property in weaker norms. As a consequence, in the repulsive case, large spherically symmetric solutions are asymptotically stable. The proof relies in particular on the introduction of a suitable system of dynamic coordinates adapted to the asymptotic nonlinear flow.

Given multiple data matrices, many problems in statistics and data science rely on estimating a common subspace that captures certain structure shared by all the data matrices. In this paper we investigate the statistical and computational limits for the common subspace model in which one observes a collection of symmetric low-rank matrices perturbed by noise, where each low-rank matrix shares the same common subspace. Our main results identify several regimes of the signal-to-noise ratio (SNR) such that estimation and inference are statistically or computationally optimal, and we refer to these regimes as weak SNR, moderate SNR, strong estimation SNR, and strong inference SNR. First, we propose an estimator based on projected gradient descent initialized via spectral sum of squares and show that it achieves the optimal $\sin\Theta$ error rate under strong estimation SNR. These results are complemented by both statistical and computational lower bounds identifying the weak and moderate estimation SNR regimes. Next, we turn to statistical inference for the $\sin\Theta$ distance itself, and we show that our estimator has an asymptotically Gaussian distribution in the strong inference SNR regime. Based on this limiting result we propose confidence intervals and show that they are adaptively minimax optimal in the strong inference SNR regime, where adaptivity is measured in terms of the SNR. Finally, we show that adaptive confidence intervals are information-theoretically impossible below the strong inference SNR regime. Consequently, our results unveil a novel phenomenon: despite the SNR being “above” the computational limit for estimation, adaptive statistical inference may still be information-theoretically impossible.

In many settings one is often interested in determining whether two networks share some joint structural connectivity patterns such as communities. However, while communities may be shared across networks, edge probabilities may differ significantly. Therefore, in this paper we consider testing a general null hypothesis that two networks have the same underlying subspace, which in particular includes the setting that communities are the same for either stochastic blockmodels or mixed-membership stochastic blockmodels (even if edge probabilities are different). We propose a test statistic based on the Frobenius norm of the difference of the leading subspace projection matrices, and we prove that our test statistic, after appropriate centering and scaling, converges in distribution to a Gaussian random variable as long as the average expected degree grows at least logarithmically in the number of vertices. We then provide estimators for the asymptotic mean and variance and show consistency under a stronger signal condition, and we give the local power of our test when the networks are sufficiently dense. Our theoretical results are based on a limit theorem for the projection difference of empirical and true eigenvectors which can also be viewed as the one-sample version of our test statistic, and this result may be of independent interest. We demonstrate our results through numerical simulations and an application to US Flight data.

Let $S$ be the set of unit norm linear classifiers $\theta \in \mathbb{R}^d$ which correctly classify every point of a labeled dataset $(X_i,y_i)_{i=1}^n$, $X_i \in \mathbb{R}^d$, $y_i \in \{-1,+1\}$, with a possibly negative margin $\kappa$ fixed in advance. Under two natural data-generating distributions of the $(X,y)$ pairs – a Gaussian mixture model and a logistic model with Gaussian features – and in the proportional regime $n/d \to \alpha$ with small enough $\alpha$, we establish a large deviation principle on the event that a point $\theta$ chosen uniformly at random from $S$ achieves a given generalization error, with high probability over the choice of the data. The associated large deviation rate function is deterministic and describes the proportion, at the exponential scale in $d$, of interpolating classifiers having a given desired performance. As a consequence, we establish the following concentration phenomenon: all but an exponentially small fraction of interpolating classifiers have approximately the same generalization performance given by the unique maximizer of this rate function. We numerically compare this maximizer to the performance of empirical risk minimization by gradient descent and to the performance of a natural linear program, both finding a point in $S$, and deduce that in the overparametrized regime of small $\alpha$, these efficient procedures outperform the vast majority of interpolators, pointing to their nontrivial benign overfitting in this setting.

We study the defocusing $k$-dispersion generalized Benjamin-Ono equation. For every even integer $k\geq 4$, we prove that solutions with initial data in the energy space $H^{\frac{\alpha}{2}}$ are global in time and scatter. The proof combines the concentration-compactness-rigidity method of Kenig and Merle with techniques based on the Caffarelli-Silvestre extension and Tao's monotonicity formula adapted to the fractional dispersion setting.

High-rate quantum low-density parity-check (qLDPC) codes are a leading candidate for fault-tolerant quantum computing. They feature higher encoding rates than planar alternatives such as the surface code, but their implementation often entails significant hardware hurdles like the need for long-range couplers. We leverage the flexibility of a trapped-ion quantum computer to demonstrate nine quantum error-correcting codes with starkly different qubit connectivity requirements on a single device without any hardware reconfiguration. These experiments span three families of quantum error-correcting codes: qLDPC codes, topological codes, and concatenated codes. With a qLDPC code encoding 4 logical qubits into 18 physical qubits, we achieve a logical error rate up to $9\times$ better than a previous demonstration of a similar code on superconducting solid-state qubits. Moreover, our implementation exhibits breakeven performance, with some instances achieving qubit lifetimes comparable to or slightly exceeding that of our trapped-ion qubits. We use a novel implementation of the optical-metastable-ground (OMG) architecture for addressable mid-circuit measurement and reset, which enables us to perform these experiments without any ion transport or dedicated coolant ions, requirements that typically consume a large fraction of the runtime or ion count of trapped-ion quantum computers.

We explicitly compute the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on $\Sigma\times\mathbb{C}$, for any connected smooth projective curve $\Sigma$. The computation is carried out in the natural basis given by the $\mathbb{C}^*$-upward-flow cycles. The result is described by a simple symmetric-function dictionary: upward-flow classes correspond to products of complete homogeneous symmetric functions, while the perverse-homogeneous basis corresponds to products of power-sum symmetric functions. This gives an explicit triangular change-of-basis between the two bases.

We study maximum matching problems in temporal graphs whose underlying graph is a tree. We consider two temporal models. In a $\Delta$-matching, selected time edges sharing an endpoint must have time ticks differing by at least $\Delta$. In a $\gamma$-matching, the selected objects are blocks of $\gamma$ consecutive appearances of the same underlying edge. We also consider the related ordered static problem of $d$-distance matchings. We show that maximum $\Delta$-matching remains NP-hard on temporal trees for every $\Delta\geq 2$, even in the sparse case where each edge appears at most twice. Using a reduction between the temporal models, we obtain the analogous result for maximum $\gamma$-matching on temporal trees, even when each edge admits at most two $\gamma$-edges. We also show, via a reduction from $d$-distance matching, that maximum $\gamma$-matching is APX-hard even when the underlying graph is bipartite. Complementing these hardness results, we identify several tractable cases. We prove that maximum $\Delta$-matching is polynomial-time solvable on temporal trees in which every edge appears exactly once, and that maximum $\gamma$-matching is polynomial-time solvable when each edge admits at most one $\gamma$-edge. We also give dynamic-programming algorithms under bounded local-use and local-sparsity assumptions, and derive polynomial-time solvability of maximum $d$-distance matching when the input bipartite graph is a tree. Finally, we prove that both maximum $\Delta$-matching and maximum $\gamma$-matching admit polynomial-time approximation schemes on temporal trees.

We study an inverse source problem for fully nonlinear elliptic equations of the form \[ F(D^2u)=f \quad \text{in } \Omega. \] The question is whether the source term can be recovered from the Dirichlet-to-Neumann map. In two dimensions, the first linearization does not immediately give uniqueness: it leaves a natural conformal ambiguity in the linearized coefficients. For homogeneous nonlinearities $F$ with injective differential $DF$, we show that this ambiguity has a precise meaning at the level of the equation itself, namely that the source is determined up to an explicit scalar factor. The main point of the paper is to show how this remaining factor can be removed. We use the second linearization to extract information which is invisible at first order, and combine it with an algebraic nondegeneracy condition on the nonlinearity. Under this condition, the residual ambiguity is forced to be trivial, and the Dirichlet-to-Neumann map uniquely determines the source. The result applies, in particular, to homogeneous admissible Hessian equations of Monge–Ampère type and related examples.

We study an inverse problem for translation-invariant symmetric stable operators of the form \begin{equation*} L_a u(x)=\mathrm{P.V.}\int_{\mathbb R^n}(u(x)-u(y))\frac{a((x-y)/|x-y|)}{|x-y|^{n+2s}}\,dy, \quad 0<s<1, \end{equation*} where the unknown is the even angular density $a$ on $\mathbb Sn$. For a bounded open set $\Omega\subset\mathbb R^n$, with $\Omega_e=\mathbb R^n\setminus\overline\Omega$, we consider restricted exterior Dirichlet-to-Neumann maps $\Lambda_a^{W_1,W_2}$, where exterior data are supported in $W_1\Subset\Omega_e$ and the nonlocal Neumann data are observed on $W_2\Subset\Omega_e$. We prove three recovery results for the leading angular density. In the overlapping regime $W_1\cap W_2\ne\emptyset$, the exterior diagonal singularity determines every smooth elliptic angular density. In the separated regime $\overline W_1\cap\overline W_2=\emptyset$, where this singularity is absent, we prove uniqueness in the finite harmonic angular class by an exact factorization of the stable symbol. We also prove separated-data uniqueness for real-analytic angular densities when the source and observation sets lie in the unbounded exterior component, using analytic continuation of the off-diagonal Dirichlet-to-Neumann kernel and a far-field asymptotic argument.

The goal of this article is to show the categorical links between on the one hand the category of reductive Lie algebra pairs $\mathcal{RLP}$ and on the other hand the category of Lie-Yamaguti algebras $\mathcal{LY}$. The fact that the well-known construction of an enveloping algebra associating to a Lie-Yamaguti algebra a reductive Lie algebra pair is not functorial leads us to the main construction of the article, namely a left adjoint to the natural restriction functor $G:\mathcal{RLP}\to\mathcal{LY}$. As a final result we observe that the construction of the enveloping algebra becomes functorial when one restricts the morphisms of the categories $\mathcal{RLP}$ and $\mathcal{LY}$ to the surjective ones. Then it becomes a right adjoint to the restriction functor.

We present a novel approach to eigenvector universality for generalized Wigner matrices. Our main consequences are asymptotic normality of joint eigenvector projections everywhere in the spectrum as well as, under a uniform subexponential decay assumption, convergence of the extremal process of eigenvector entries and the resulting Gumbel law for the largest entry of an eigenvector. In the case of smooth entries, we are able to obtain joint normality of an explicit growing number of eigenvector projections, and we are also able to obtain explicit rates of convergence in Kolmogorov distance, which are nearly optimal for a fixed number of projections. This is based on a new analysis of the Dyson vector flow which does not rely on the eigenvector moment flow.

Lens spaces are useful test examples in spectral geometry because their spin Dirac eigenspaces admit explicit congruence descriptions. We use these descriptions to study equivariant $\eta$ invariants for three-dimensional lens spaces with the round metric and the standard coordinate-torus action, retaining the spin-Fourier character of each eigenspace rather than only the ordinary scalar $\eta$ value. For the square family $L(\ell^2,\ell-1)$ and $L(\ell^2,2\ell-1)$, with $\ell\geq 5$ odd, we obtain a residual-circle equivariant $\eta$ separation: the ordinary $\eta$ values agree, and the first derivative of the residual $\eta$ germ vanishes by symmetry, but the second derivative is nonzero. For $L(25,4)$ versus $L(25,9)$, the normalized second derivative is $-6080$. Thus, the residual-circle equivariant $\eta$ germ detects a distinction invisible to the ordinary $\eta$ invariant. The calculation uses spin-Fourier residues directly; perturbative Hessian signs serve only as motivation and are not part of the invariant.

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with a faithful action of a finite group $G$. Let $\text{Rep}^G(\mathcal{A})$ be the $G$-crossed balanced $\mathrm{W}^*$-tensor category of $G$-twisted representations of $\mathcal{A}$ as introduced in arXiv:2606.03623. We show that there is an equivalence of balanced $\mathrm{W}^*$-tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ between the $G$-equivariantization of $\text{Rep}^G(\mathcal{A})$ and the category of representations of the fixed-points conformal net $\mathcal{A}^G$. This generalizes to the non-rational case the equivalence of braided tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ for $\mathcal{A}$ rational appearing (in the language of localized endomorphisms) in arXiv:math/0403322, and it also includes the balances.

Classical chance-constrained programs are solved by safe approximations based on the empirical CVaR, which uses a uniform measure over scenarios and systematically underweights tail events under heavy-tailed distributions. We introduce q-CCP, a non-extensive safe approximation grounded in the Riemannian geometry of the Tsallis statistical manifold: the rank-based q-CVaR escort weights are the $g^{(q)}$-geodesic projection onto the tail simplex face, and the q-CCP feasible set is a Tsallis-divergence ball (Proposition~12). This geometric foundation yields three results. First, q-CCP is a provable strict tightening of CVaR-CCP for all $q > 1$ (Theorem~7). Second, the empirical violation ratio satisfies $\rho(q) = [1-(1-\varepsilon)^{q+1}]/\varepsilon$, independent of the tail index $\nu$ (Proposition~10). Third, the feasible-region volume cost is monotone increasing in $q$ and $\nu$ (Proposition~11), providing a data-adaptive safety knob. The formulation inherits convexity and coherence from the q-CVaR functional and admits an iterative LP reformulation converging in 2–3 iterations. Experiments on 15 Ibovespa equities confirm the theory (violation ratio $0.241$, $q^* = 1.50$); an M5 inventory newsvendor experiment generalises the method to supply chain ($q^* = 1.88$, cost premium $1.155\times$, zero OOS stockout violations).

The extension of the analytic fractal curve trees of (2601.17490} to analytic surface patch trees reveals a new geometric structure: branch points are replaced by interface curves that transmit the full analytical state of parent patches to their children. These interfaces prove to be central in determining the topology of the surface patch trees, including for the conditions for self-similarity of the interfaces, the patches and thus the trees. We establish the analytic conditions for the integrability and well-posedness of the surface patch trees and introduce further restrictions for conformality. We demonstrate that patch trees have a natural foliation that slices the trees into one dimensional curve trees, each of which has their own Hausdorff dimension, jointly creating a smooth dimension field. We extend the two dimensional surface model to arbitrary dimensions $n$ where $n-1$ interface manifolds transport the $n$ field state of the parent patches to their child branches. We note that the balance or discrepancy between patch field dimension and the dimensions in which the branches may evolve, determine the analytical regime from essentially geometrical to essentially operational.

We construct a periodic weighted graph whose discrete Laplacian has a spectrum with precisely $n$ gaps. Moreover, we show that by an appropriate choice of the weights, the endpoints of these gaps, as well as the upper edge of the spectrum, attain the prescribed values. The underlying graph has a brush-like geometry: it consists of an infinite chain of vertices, each of which is connected to $n$ additional pendant vertices by extra edges. Semi-explicit formulae for the weight coefficients are provided: some of the coefficients are determined explicitly, while others are given as roots of an explicitly determined polynomial.

We give a complete classification of doubly totally-umbilical submanifolds in the probability simplex. The probability simplex is one of the most standard statistical manifolds, and information geometry initiated by S. Amari and H. Nagaoka studies the statistical submanifold theory of the probability simplex. On the other hand, H. Furuhata defined doubly totally-umbilical submanifolds in the geometry of statistical manifolds, inspired by the surface theory of Euclidean space.

The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable $X$ is defined as $\xi^2_\star = \sup_{\lambda \in \mathbb{R}} L(\lambda)$ where $L(\lambda) = \frac{2}{\lambda^2} \log \mathbb{E} e^{\lambda X}$ is a weighted cumulant generating function. We study the estimation of $\xi^2_\star$ and prove that the minimax risk is governed by a non-increasing function $\delta_P(C) = \sup_{|\lambda| \geq C} L(\lambda) - \sup_{|\lambda| \leq C} L(\lambda)$ which captures the influence of the tail behavior of the distribution $P$. Over the class of distributions with $\delta_P \leq r$ for a non-increasing function $r$, the minimax risk is, up to a multiplicative constant, lower bounded by $r(\sqrt{\log n}) + n^{-1/2}$ and upper bounded by $r((\log n)^{1/2-\varepsilon}) + n^{-1/2 + \varepsilon}$ for any $\varepsilon > 0$. Our estimator for the upper bound is based on constrained maximization of the empirical analogue of $L$. In addition to being almost minimax optimal and adaptive, we further prove that the estimator is asymptotic normal under suitable conditions and that, if the underlying distribution is not sub-Gaussian, the estimator diverges with a rate determined by the heaviness of the distributional tail.