Papers

I study the largest eigenvalue \(\Lambda_j\) of \(X^{(j)}(X^{(j)})^*\), where \(X^{(j)}\) is the \(M_j\times j\) northwest rectangle of one infinite complex Gaussian array. For two upper-tail events and, separately, for two lower-tail events at the largest-eigenvalue soft edge, I prove finite-\(N\) one-sided product-scale upper bounds for their joint probabilities. The logarithmic walls have polynomially small marginals, so an additive \(o(1)\) covariance estimate may be much larger than the vanishing product that must be controlled. Uniformly for levels in a short macroscopic window, bounded deterministic shifts, and separations at least \(N^{2/3+\epsilon}\), each joint probability is at most \((1+o(1))\) times the product of its marginals. The classical \(N^{2/3}\) correlation window corresponds to order-one extended-Airy time; the theorem works at supercritical separation. Exact Laguerre operators along admissible row–column paths provide the finite-dimensional input. The upper-tail proof uses occupancy counts and cross-block trace estimates, whereas the lower-tail proof uses gap determinants, diagonal resolvents, and a Schur complement. I transfer the result between the half-integer Laguerre and physical rectangular normalizations and deduce intrinsic first- and second-moment estimates for separated sparse grids under the rarity, count-growth, spacing, and macroscopic-window budget.

In this paper, we investigate the structure of locally generalized radical linear groups over certain non-commutative rings and algebras. The base rings and algebras we consider include division rings, left artinian rings, locally finite algebras over fields, and PI-algebras over fields. Among various results, we get the positive answers to the General Burnside Problem and to Baer's Conjecture for some particular cases of linear groups.

The consensus of continuous-time multi-agent systems with unbounded and heterogeneous constant delays is investigated by combining frequency-domain analysis and algebraic graph theory. Several types of signed Laplacians are constructed to characterize consensusability under delays. The core results are established based on the defined delay-embedded signed Laplacian, where a small-delay link creates a cooperative interaction and a possibly unbounded large-delay link creates an antagonistic interaction between the agents. The dividing line between small and large delays is given by $\tau_{ij}=\pi/2\lambda_{\max}(L_0)$, where $\lambda_{\max}(L_0)$ refers to the maximum eigenvalue of the conventional graph Laplacian. It is proved that the consensusability is preserved if the delay-embedded signed Laplacian is positive semi-definite with a simple zero eigenvalue. Moreover, we derived some consensus conditions in terms of the extended effective resistance which measures the overall coupling between two sets of agents. The obtained results provide new insights into the mechanism of delayed consensus from the interplay between the small-delay-induced cooperativeness and large-delay-induced antagonism in the underlying network topology.

We establish quantitative uniqueness and a fractal uncertainty principle for the Fourier Bessel transform. In arbitrary dimension, we prove a quantitative uniqueness estimate on relatively dense sets for functions whose Fourier Bessel transforms decay according to a quasi-analytic weight. In dimension one, if $X\subset[0,1]$ and $Y\subset[a,a+h^{-1}]$ are$\delta$-regular on the relevant scales, then, for $a\geq a_0h^{-1}$, \[ \operatorname{supp}\mathcal H_\nu f\subset Y \quad\Longrightarrow\quad \|\mathbf 1_Xf\|_{L^2_\nu} \leq Ch^\beta\|f\|_{L^2_\nu}. \] The lack of translation invariance prevents a direct application of the classical Fourier argument. We overcome this by constructing damping functions adapted to translated regular sets and combining Beurling Malliavin multipliers with large-argument Bessel asymptotics and a Bourgain Dyatlov multiscale iteration.

Let \[ A_{\min}(x)=\bigl(x_{\min(i,j)}\bigr)_{i,j=1}^n, \qquad A_{\max}(x)=\bigl(x_{\max(i,j)}\bigr)_{i,j=1}^n \] be the Min and Max matrices generated by a real sequence \(x=(x_1,\ldots,x_n)\). Using their classical cone parametrizations, we give exact characterizations of entrywise maps preserving positive semidefiniteness, total nonnegativity, Loewner order, and Loewner convexity. Our main results concern the Loewner structure.Without assuming continuity or differentiability, entrywise Loewner-order preservation is equivalent to \(f\) being nondecreasing and convex. On the Min and Max cones, requiring the Loewner-convexity inequality on arbitrary pairs is rigid and forces \(f\) to be affine. Under the standard convention of restricting the inequality to Loewner-comparable pairs, the condition automatically forces \(f\in C^1([0,\infty))\) and is equivalent to convexity of both \(f\) and \(f'\). The analogous statement holds for Loewner concavity. In each case, the condition is already detected in dimension two. We also show that \(f:[0,\infty)\to\mathbb R\) preserves positive semidefiniteness entrywise on all positive semidefinite Min or Max matrices if and only if \(f\) is nonnegative and nondecreasing; the same condition characterizes total-nonnegativity preservation. Finally, we determine the power-function ranges and Loewner-order automorphisms, characterize the entrywise preservers of the strict Min and Max classes, and relate these strict cones to inverse \(M\)-matrices and oscillatory matrices.

In contextual optimization, the decision-maker seeks optimal decisions to minimize a cost function, that varies based on observed features. This context is common in many business applications ranging from on-demand delivery and retail operations to portfolio optimization and inventory management. In this paper, we study the learning and optimization approach, which first learns how outcomes result from the features, and then selects optimal decisions based on these outcomes. We focus on the integrated learning and optimization literature, and identify that a lack of control for prediction accuracy can lead to overfitting and a loss of decision effectiveness against simple separate learning and optimization models. Instead, we propose a bi-objective formulation that balances prediction accuracy and cost minimization, termed decision-driven regularization. It also addresses ambiguity in the definition of the cost function via a surrogate that depends on a new hyperparameter. We additionally show that alternative perspectives for formulating the problem, namely robust optimization and regret minimization, lead to models that are closely related to our proposed model. As a consequence, our framework generalizes models such as SPO+. Our model is shown to be numerically superior to other benchmarks, such as OLS, Random Forest, XGBoost, SPO+, Perturbation Gradient, and Learning and Rank, in our synthetic studies.

Given a field $K$, let $K((T^{-1}))$ be the field of formal power series. Continued fractions in $K((T^{-1}))$ can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of $K((T^{-1}))$ arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that $K$ has characteristic zero, we improve the known analogues of the Maillet–Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport–Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy.

The theory of quantum channels is traditionally studied either on finite-dimensional state spaces or within the Heisenberg picture as completely positive maps on C^*-algebras. In this paper, we consider quantum channels as completely positive maps on the duals of general von Neumann algebras in the Schrodinger picture. We investigate the construction of such channels through Pettis integrals using representations of topological groups.

We study the class of congruences on $F$-inverse monoids that respect the unary operation $a\mapsto\mx{a}$, assigning to each element the maximum element of its $\sigma$-class, which we refer to as $F$-congruences. We characterize kernel normal systems and congruence pairs on $F$-inverse monoids which give rise to such congruences and establish a number of further related results including a classification of $F$-congruence-free $F$-inverse monoids.

Solving the time-dependent Schrödinger equation (TDSE) via traditional numerical methods is computationally intensive. Transformer models offer a compelling alternative, but standard implementations rely on soft constraints that cannot rigorously guarantee probability conservation. Here, we introduce a Transformer architecture that enforces TDSE probability conservation as a hard constraint. The design intrinsically ensures unitarity across temporal evolution without requiring repeated retraining. Our empirical findings show that this hard-constraint approach is not only physically exact but also computationally superior to conventional soft-constraint methods.

Diffusion models learn to reverse a predefined corruption process, but sampling still requires a costly time discretization and depends on the chosen noise schedule. We study these two issues for variance-preserving diffusions with matrix-valued schedules. Our analysis transfers reverse-time discretization errors to the forward corruption law and treats two numerical schemes within a common framework. The first freezes the score and yields, through a matrix-sensitive local comparison and forward information dissipation, an ambient-dimensional step complexity with leading factor $d/\varepsilon^2$ for KL accuracy $\varepsilon^2$. The second keeps the known Gaussian drift exact and freezes the posterior mean. For data of metric-entropy dimension $k$, a forward Markov identity, an anisotropic covering estimate, and Stieltjes integration by parts give the corresponding factor $k\log k/\varepsilon^2$. In both cases, the proof identifies a local error, accumulates it through the forward evolution, and inserts the result into a common KL decomposition. The local errors further provide directional criteria for matrix schedules and an asymptotically optimal square-root adaptive grid. A high-dimensional Gaussian-mixture experiment illustrates the resulting schedule and grid improvements.

In the study of the generating series of Demazure characters, Lascoux used isobaric divided differences to define a family of polynomials $\mathcal{E}_{\sigma}(t)$ indexed by permutations $\sigma$, and asked for a satisfactory expression of these polynomials. In this paper we obtain a combinatorial interpretation of $\mathcal{E}_{\sigma}(t)$ for the permutation $\sigma=[2,3,\ldots,n,1]$ or its inverse in terms of the descent statistic of parking functions of length $n-1$. Based on this progress on Lascoux's open problem, we find that the polynomial $\mathcal{E}_{\sigma}(t)$ for this special case coincides with the $h$-polynomial $h(\Delta(\mathrm{NC}_W),t)$ of the order complex of the noncrossing partition lattice associated to the irreducible Coxeter group $W$ of type $A_{n-1}$. We are inspired by this coincidence to give an operator approach to $h(\Delta(\mathrm{NC}_W),t)$ for any finite Coxeter group $W$. As an application, we completely solve an open problem on $h(\Delta(\mathrm{NC}_W),t)$ which was proposed by Athanasiadis, Douvropoulos and Kalampogia-Evangelinou. For any $k$-divisible noncrossing partition poset $\mathrm{NC}^{(k)}_W$, we also obtain the interlacing symmetric decomposition property of the $h$-polynomial $h(\Delta(\mathrm{NC}^{(k)}_W),t)$.

The weighted Max-Cut problem is an NP-hard problem with application implications. It is investigated on a cubic lattice with 113 nodes and mixed-signed random edge weights. For a fixed upper bound on edge weights, it has been demonstrated that the computational difficulty increases as the lower bound on edge weights becomes more negative. The solution time for the problem using a novel error-mitigated quantum annealing approach is compared with standard D-Wave quantum annealing (QA) and BQM hybrid solvers, as well as various classical solvers. For the QPU-embeddable weighted Max-Cut instances with mixed-signed edge weights, it has been quantitatively demonstrated that the SEMO (spin-error mitigation for optimisation) error-mitigated quantum annealing achieved substantially shorter time-to solution than standard D-Wave QA, D-Wave BQM, simulated annealing and Tabu search baselines. The error-mitigated quantum annealing approach presented in this article potentially elevates the efficiency and application scope of quantum annealing and would be applicable in solving other discrete optimisation problems that can be formulated as QUBO or Ising instances. The promising solution time advantage would be particularly impactful for time-critical optimisation applications.

The Awareness Logic of Ambiguity (ALA) is introduced as a formal framework for preserving the structure of judgment before scalar projection. An evaluative act is represented by a triadic interpretive state with three functionally distinct roles: interpretive valuation, contextual alignment, and epistemic restraint. ALA uses a cognitive order with reversed polarity for restraint and shows that the resulting state space forms a bounded distributive lattice. It develops weakly and strictly admissible scalar projections as declared interfaces, together with a role-preserving arithmetic core and canonical permissive and conservative aggregation operators. The Non-Collapse Theorem establishes that two states that are equal after scalar projection may become distinguishable under an ALA lattice operation; hence scalar equality does not imply operational equivalence. The paper also establishes the conditional semantic minimality of the triad and treats the unit cube as the normalized real-valued realization of a more general role architecture. A high-stakes clinical scenario illustrates why evidential support, contextual alignment, and restraint toward action should remain separate until a scalar reporting or decision interface is explicitly chosen. ALA therefore provides a structure-preserving calculus for reasoning under ambiguity before projection.

We prove a sharp $2$-systolic inequality for four-dimensional products $\mathbb S^2\times P$, where $P\subset\mathbb R^2$ is an arbitrary convex polygon. Let $h=g_{\mathbb S^2}+g_{\mathrm{eu}}$ be the standard product metric. If a Riemannian metric $g$ on $\mathbb S^2\times P$ has scalar curvature $\geq \sigma>0$, nonnegative mean curvature on every codimension one face, and dihedral angles no larger than the corresponding dihedral angles of $h$, then both its homotopy and homology $2$-systoles are at most $ \frac{8\pi}{\sigma}.$ This confirms a conjecture of Gromov.

We study the full three-dimensional dynamics of the $L^2$-critical Zakharov-Kuznetsov equation with the fractional nonlinearity $|u|^{4/3}u$, equivalently $u^{7/3}$ for real-valued functions. This equation is a higher-dimensional extension of the generalized Korteweg-de Vries equation. In the critical setting solutions to this 3D ZK equation may blow up in finite time or exhibit global time dynamics. The novelties of this work is to treat a non-integer power and to study the dynamics of solutions in a higher dimension. We first review the finite time blow-up in 2D critical ZK, then do a formal analysis of the slightly mass-supercritical blow-up dynamics, deriving the corrections to the blow-up rate and profile for the critical ZK equation in any dimension. We then perform a computational study of solutions, utilizing full 3D numerical simulations. In particular, we use a Fourier pseudospectral discretization and an integrating factor fourth-order Runge-Kutta method on a full three-dimensional grid. A multi-GPU implementation makes it possible to follow blow-up solutions in a full 3D setting. We examine perturbations of the ground state, Gaussian data, and nonsymmetric two-bump configurations. The computations show dispersive and concentrating regimes, in both cases with radiation emitted in a conic-type region opposite to the direction of propagation and convergence of the concentrating core toward a rescaled ground-state profile. The two-bump experiments also demonstrate that total mass alone does not determine the blow-up dynamics. We discuss the numerical evidence for the predicted blow-up rate and identify the pre-asymptotic and resolution limitations that remain near the blow-up time.

We revisit the problem of computing mean-field equilibria (MFEs) in discrete-time, monotone, finite-horizon mean-field games (MFGs). We show that, when the transition kernel is independent of the state-measure term and the reward function satisfies the usual weak monotonicity condition and is Lipschitz continuous, anchored proximal gradient descent methods can be used to compute a monotone MFE. We also establish last-iterate convergence results for these methods. Our approach relies on formulating the computation problem as an optimization problem over the space of occupation measures. Using this formulation, we show that the problem is equivalent to a class of constrained Lipschitz monotone inclusion problems. We then apply iterative methods for this monotone inclusion formulation to derive a tractable algorithm. The resulting algorithm achieves a convergence rate of \(O(1/\sqrt{T})\) after \(T\) iterations, without requiring any regularization. This rate holds even in the absence of a uniqueness assumption for the corresponding MFE.

We study low-lying zeros in families of even holomorphic cuspidal newforms of fixed weight and prime level, with particular emphasis on the number of forms having a zero in a prescribed normalized window about the central point and on the distribution of the number of such zeros among the forms. We quantify the number of forms having at least one zero in the window and study the distribution of the number of zeros in that window among the forms. Assuming the Generalized Riemann Hypothesis, we obtain new lower bounds for the number of forms having a low-lying zero. We first prove that, along an infinite sequence of prime levels $N$, the number of such forms is $\gg N^{7/8}\log N$. We then use higher centered moments and an appropriate test function to show that a positive proportion of the family has a zero in a prescribed window. Finally, we obtain polynomial upper-tail bounds for the number of zeros occurring there and show that a positive proportion of forms have a bounded, nonzero number of low-lying zeros.

By employing the Moore$-$Penrose inverse of a bounded linear operator, we derive several bounds for the numerical radius and operator norms of the sum of operators in semi$-$Hilbertian space that generalize and improve the classical bounds. We establish novel inequalities pertaining to the $\mathbb{A}$$-$Davis$-$Wielandt radius for $n \times n$ operator matrices and further explore their ramifications, particularly concerning $\mathbb{A}$$-$Davis$-$Wielandt radius bounds for $2 \times 2$ operator matrices, where diagonal operator matrix $\mathbb{A}$ contains positive bounded operator $A.$ Ultimately, we get an improved upper bound for the $A$$-$numerical radius inequalities relating to the commutators of operators.

We investigate the étale fundamental group of the complement of a horizontal divisor on $\mathbb{P}^{1}_{\mathbb{Z}}$. We prove that this group has no nontrivial finite solvable quotient if and only if the divisor is normal crossings at the prime~$2$. Moreover, if the divisor is normal crossings at the prime~$2$ and either has three irreducible components or is normal crossings at the prime~$3$, we show that no quotient isomorphic to $\mathrm{PSL}_{2}(q)$ can occur for certain prime powers~$q$.

In this paper, we study the effects of employing multiplier bootstrap to analyze the asymptotic distributions of the largest eigenvalues of high-dimensional sample covariance matrices in both spiked and non-spiked models. Our findings demonstrate that the multiplier bootstrap establishes several phase transitions in the limiting edge distributions of both unconditional and conditional bootstrapped covariance matrices, provided the different classes of multipliers. In the nonspiked setting, unbounded multipliers lead to Frechet or Gumbel limits for the largest eigenvalue of the bootstrapped covariance matrix, both conditionally on the observed data and unconditionally. For bounded multipliers, the unconditional model exhibits transitions among Tracy-Widom, Gaussian, or Weibull limits, determined jointly by the aspect ratio p/n, the upper-endpoint behavior of the multipliers, and the population covariance matrix. The conditional model displays analogous Gaussian and Weibull regimes; in contrast, the conditional counterpart of the unconditional Tracy-Widom regime collapses to a point mass. In the spiked setting, under suitable signal-strength conditions, the leading eigenvalues of both the unconditional and conditional bootstrapped sample covariance matrices are asymptotically Gaussian for bounded as well as unbounded multipliers, under some mild assumptions. Our theoretical results also clarify the feasibility and adaptability of the multiplier bootstrap for spectral inference in high-dimensional sample covariance models. Numerical simulations confirm the accuracy of our results and the effectiveness of the proposed spectral inference procedures, which may be of independent interest.

Andy is an autonomous mathematical research agent that turns a mathematical problem into a traceable proof. It solves or verifies a submitted problem, formulates a literature-grounded new problem through a research-value gate, and carries it through proof construction and final verification. It organizes proof steps in an executable DAG, verifies each step independently and binds the result to a certificate, retains verified work whose interfaces remain unchanged during local repair, and records the full path from problem formulation to final proof. The system separates proof generation from correctness evaluation and can acquire, retain, retrieve, and reuse knowledge from existing results. Starting from a self-triggered impulsive consensus result, Andy formulates a global exponential leader-follower synchronization problem for delayed heterogeneous networks with switching communication topologies. The proposed hybrid control combines self-triggered impulses with execution delay and continuous feedback over a recovery window. After each delayed impulse, the feedback cancels the delayed error channel until the pre-impulse history leaves the active delay interval. Sufficient conditions for global exponential synchronization are established, Zeno behavior is excluded for both timing sequences, and a numerical example illustrates the result.

Let $\varphi$ be a minimal counterexample to Vaught's conjecture in the sense of Montalban; such a $\varphi$ exists if Vaught's conjecture fails, by Steel and Harnik-Makkai. We bring into contact, at the level of statements, two bodies of work on the models of such a $\varphi$: the analysis of Gonzalez-Rossegger-Turetsky, in which at every countable level $\beta$ exactly one back-and-forth class $C_\beta$ is uncountable and, at fixed points of an associated function, has a distinguished member of least Scott rank (its label); and the supply of models with prescribed $\omega_1^A$ from higher recursion theory, namely Montalban's Gandy-basis lemma and Sacks' $\Sigma_1$-hull club. All results are theorems of ZFC under standing hypotheses (H0)-(H3). We prove: (i) a per-witness trichotomy – for every limit $\lambda$ in the fixed-point club above $qr(\varphi)$, every model $A$ of $\varphi$ with $\omega_1^A=\lambda$ and Scott rank at least $\lambda$ either is the label $K_\lambda$, or lies outside $C_{\lambda+1}$ and so forces two non-isomorphic models of Scott rank $\lambda+1$, or is the label $K_{\lambda+1}$ and attains the Nadel bound; (ii) coordinate identities: the first and third branches are equivalent, level by level, to computations of $\omega_1$ of the labels; (iii) a seeding theorem: on a club, Sacks' construction supplies at every level a model of top rank with prescribed $\omega_1$, and his atomic chain consists of the labels; (iv) an unconditional stationary dichotomy: on that club, either stationarily many successor levels carry two non-isomorphic models of Scott rank $\lambda+1$, or stationarily many labels attain the Nadel bound. On the fiber of models with $\omega_1=\lambda$ we further prove uniqueness of the node-saturated model and give a complete isomorphism invariant with countable spectrum. We prove nothing bearing on Vaught's conjecture itself.

We answer a question raised by Vogt by proving that the growth bound of an individually eventually positive $C_0$-semigroup on an $L^p$-space ($1 < p < \infty$) coincides with the spectral bound of its generator. Our proof follows the general strategy of Vogt's argument for the uniformly eventually positive case. The key new ingredient is a variant of an operator-range uniformisation principle of Arora and Glück, adapted here to time-dependent operator ranges. When applied to principal ideals, this principle turns individual eventual positivity into the uniform domination estimate required for Vogt's stability argument.

Structured nonconstant large stepsizes can improve the convergence of gradient descent in the deterministic setting. However, in stochastic optimization, aggressive stepsizes can amplify oracle noise and hinder the convergence of stochastic gradient methods. We characterize the globally optimal two-step stepsize schedule for stochastic gradient methods applied to strongly convex and smooth functions, assuming access only to unbiased stochastic gradient estimates with finite support and bounded variance. The optimal schedule depends on the ratio of the initial optimality gap to the noise level and exhibits several distinct regimes. As the influence of stochastic noise diminishes, the optimal two-step stepsizes become larger, reflecting a balance between the benefits of faster iterate convergence and the perturbations induced by stochastic noise.

We prove a quantitative central limit theorem for linear functionals of regularized empirical-risk minimizers in the proportional-dimensional regime \(p=O(n)\). The data columns are independent, not necessarily identically distributed, and satisfy a uniform columnwise Poincaré inequality. Under uniform curvature and smoothness assumptions, and for a quadratic regularizer, we show that every nondegenerate statistic \(\sqrt n\,u^\top\hat\theta\), centered by its expectation and normalized by its standard deviation, converges to a standard normal random variable in Wasserstein distance, with rate \(O((\log n)^7n^{-1/4})\). The proof is based on moment and stability bounds for the minimizer, a second-order leave-one-out expansion, and a perturbative normal-approximation argument for functions of independent variables. We also prove the variance upper bound \(\Var(u^\top\hat\theta)\le C\norm{u}_2^2/n\), identifying the \(\sqrt n\) fluctuation scale.

In 1991 Rickard asked whether every derived equivalence between finite-dimensional algebras over a common field is standard. Hu, Xi and Zhang constructed non-standard derived equivalences over the field with two elements and conjectured that Rickard's question has a positive answer in characteristic different from two. We disprove this conjecture by constructing non-standard derived autoequivalences over every field. In fact, each field admits infinitely many finite-dimensional algebras with such autoequivalences. The proof is characteristic-free and is based on a supertrace identity for matrices over truncated polynomial algebras. We also give a second family over finite fields. Consequently, Rickard's question has a negative answer over every field.

We treat the question of upgrading Guth-Manolescu's real Heegaard Floer theory to a $\mathbb Z/2$-graded invariant defined over $\mathbb Z$, specifically for the hat version with one basepoint on each component of the branching link. We provide if-and-only-if obstructions to the existence of relative $\mathbb Z/2$-gradings and $\mathbb Z$-lifts, in terms of purely homological information associated to the 3-manifold with involution. When the corresponding obstruction vanishes, we study the set of such $\mathbb Z$-lifts, which a priori is only an invariant up to a torsor action. The obstruction theory could be of independent interest in various Floer theories more broadly, particularly Lagrangian Floer theory.

Extreme weather leaves power systems exposed to residual outage risk even after physical resilience investments. Parametric insurance can provide pre-agreed contingent liquidity, but its physical value depends on how trigger thresholds and payout levels are designed. This paper proposes a resilience oriented parametric insurance framework that couples a three tier wind-index contract with post-event network restoration. Insurance payout expands the budget available to activate emergency resources, so the contract changes the physical restoration feasible set rather than merely offsetting accounting losses. Trigger thresholds and payout levels are jointly designed to balance actuarial premium, expected post-event system cost, and the conditional value-at-risk (CVaR) of scenario energy not supplied (ENS). A response-library method precomputes the restoration mixed-integer linear program for each scenario-payout pair and then evaluates admissible contracts efficiently. On the IEEE RTS-24 with 80 extreme-wind scenarios, the optimized contract reduces expected EENS and CVaR0.90 of ENS by 21.1% and 21.4%, respectively, relative to no insurance, while requiring 48.8% less premium than a fixed parametric contract with comparable resilience. The results show that insurance design should target the nonlinear liquidity-to-resilience response rather than loss compensation alone.

We construct an optimal transport map between uniform distributions on two antipodal annuli of a sphere.