Papers

When does a noisy quantum sampler yield an end-to-end polynomial-time optimization algorithm with performance guarantees? Building on finite-depth and finite-shot guarantees for Constraint-Enhanced QAOA, we show that inverse-polynomial ideal probability on the optimal set, together with independent sampling, polynomial-time feasibility repair, and scoring, produces an exact-hit fully polynomial randomized approximation scheme, which we call an FPRASq. This guarantee survives device noise within an instance-dependent window. For effective circuit depth linear in the product of layer count and problem size, preserving an inverse-depth fraction of the ideal optimal mass increases the required shot complexity by one power of the problem size. Beyond this window, deterministic repair guarantees feasibility and provides an instance-dependent approximation guarantee whenever the induced objective inflation is controlled. The resulting NP-HQ algorithm fits the Chen-Cotler-Huang-Li oracle model. On any NP-hard kernel-admissible promise family, reproducing its inverse-polynomial optimal overlap with a polynomial-time classical sampler would imply that NP is contained in BPP, even with identical repair and perfect access to the constraint structure. Thus, the separation lies in generating the sampling distribution. We further introduce Heavy-Hitter QAOA, which preserves these conditional guarantees while reducing the retained candidate set and classical post-processing cost by one power of the problem size. Hardware experiments on IBM Eagle r3 processors cover instances with up to one hundred logical variables and match or improve every tested QOptlib reference tour.

Biochemical networks with unit stoichiometry arise naturally in receptor-ligand binding and multisite phosphorylation systems. A central question is to identify which networks admit multistability. Because this property can be inherited from smaller subnetworks to larger ones, it is natural to seek the smallest networks with it. In this paper, we completely determine all minimal quadratic zero-one networks that exhibit multistability. Building on recent progress that has narrowed the search to the zero-one networks with 3 species, 6 reactions, and dimension 3, say (3, 6, 3) family. We develop a computational pipeline to provide a complete characterization of multistability within this class. The primary theoretical contribution is a structural simplification showing that for (3, m, 3) quadratic zero-one networks, the Jacobian determinants at consecutive positive steady states have opposite signs, and exactly half of these states are stable. This reduces multistationarity detection and stability verification to a local sign-checking problem, eliminating the need for boundary or asymptotic analysis. Applying this result, we identify 373 quadratic zero-one networks that exhibit bistability (two stable and one unstable positive steady states). Strikingly, no (3, 6, 3) quadratic zero-one network admits more than 3 positive steady states–a sharp bound that falls well below the theoretical BKK bound of 5 and the Bezout bound of 8. Among the 375 networks that admit exactly 3 positive steady states, 373 are multistable and only 2 are not. These minimal networks provide essential test cases for understanding how bistability emerges in cell signaling without requiring higher-order molecular collisions.

In this paper, we study the Connes spectral distance between states on the fuzzy torus. We construct a Dirac operator by commutators and anticommutators. Based on this Dirac operator, we construct a spectral triple of the fuzzy torus. We study some properties of the spectral distance on the fuzzy torus. We find that there is a reciprocal Pythagorean theorem between the spectral distances. We construct a conditional expectation function of the optimal element which can lead to a contraction of the corresponding Lipschitz seminorm. We find that for any diagonal states, the corresponding optimal elements of spectral distances are also diagonal. We explicitly calculate the spectral distances of some simple states, including basic states and some simple mixed states. We find that there are some kinds of cyclic symmetry in both the optimal elements and the spectral distances between the diagonal states. Furthermore, we also construct a fuzzy torus with some type of conformal twist, and study the relation between conformal parameters and spectral distances.

Problem B22 in Unsolved Problems in Geometry asks which triangles occur as the common face of a convex polyhedron, how many copies are needed, and how they may be arranged. We settle the existence and minimum-face-count questions for triangles in the version that allows reflected copies; we do not classify all attainable face counts, nor the possible arrangements. Every nondegenerate Euclidean triangle occurs: we exhibit an explicit convex polyhedron, combinatorially an octahedron, all eight of whose faces are congruent to a prescribed triangle. We then determine the minimum number of faces for every triangle. It is four for an acute triangle; six for a right or obtuse isosceles triangle with side lengths $(\lambda,\lambda,\beta)$ satisfying $\lambda\sqrt2\leq\beta<\lambda\sqrt3$; and eight in all remaining cases.

We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$ of convex points $x \in \partial\Omega$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(\Gamma_0) > \frac{2}{3}\mathcal{H}^{n-1}(\partial\Omega)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\Omega$ and attains the prescribed boundary data at least on $\Gamma_0$ in the classical sense.

We derive transformation and summation formulas for terminating and nonterminating series involving the digamma function. Our principal results are obtained by a limiting process starting with duality relations for the generalized hypergeometric functions and their consequences. Selected formulas are further extended by parameter differentiation of Euler's transformation and by using contiguous relations. Most of our identities express products of hypergeometric and digamma series in terms of hypergeometric functions, and some evaluations of terminating digamma sums involve Bernoulli polynomials. In several cases the digamma contributions cancel, producing identities involving only products of hypergeometric functions.

In anabelian geometry, Kummer-faithful fields are expected to be suitable as base fields. In recent years, to characterize Kummer-faithfulness in terms of Galois representations, Ozeki and Taguchi defined a notion of (quasi-)highly Kummer-faithful fields as a variant, and posed the following natural question: Are sub-$p$-adic fields quasi-highly Kummer-faithful? In this paper, by studying the weights of $\ell$-adic étale cohomology over mixed-characteristic local fields and discussing the $p$-adic analogue, we give an affirmative answer to this question. Furthermore, we extend the class of base fields from mixed-characteristic local fields to complete discrete valuation fields whose residue fields are algebraic extensions of some finite field, and give equivalent conditions for the vanishing of the coinvariants of $\ell$-adic and of $p$-adic étale cohomology. As a result, we show that, for mixed-characteristic complete discrete valuation fields whose residue fields are algebraic extensions of some finite field, Kummer-faithfulness and quasi-high Kummer-faithfulness are equivalent.

We establish new Bohr-type inequalities for holomorphic mappings from the unit ball of a complex Banach space into the closed unit disk. Using Fréchet derivatives, Schwarz mappings of prescribed orders and area functionals, we obtain sharp Bohr radii characterized by explicit equations. We further prove a refined Bohr inequality involving derivative, coefficient and area terms and show that the corresponding radius is the unique positive solution of $r^m=(\sqrt{17}-3)/4$. The sharpness of the obtained radii and constants is also established. Our results extend several recent Bohr-type inequalities for bounded analytic functions on the unit disk to the setting of holomorphic mappings on complex Banach spaces.

Let $X$ be a centered isotropic log-concave random vector in $\mathbb{R}^n$. For $\theta\in S^{n-1}$, let $\mu_\theta$ be the law of $\langle X,\theta\rangle$, and let $\Theta$ be uniformly distributed on $S^{n-1}$, independently of $X$. We prove the sharp estimate \[ \textsf{E} W_1(\mu_\Theta,\gamma_1) \le \frac{C}{n}. \] Here the Wasserstein distance is computed after the direction is fixed and is then averaged over the sphere. No symmetry assumption is imposed. A product measure with centered exponential coordinates gives a matching lower bound of order $n^{-1}$. The proof separates the averaged-direction law from the fluctuation among fixed directions. For the first part, a Taylor expansion in the random radius retains a mean-zero cancellation and yields an $O(n^{-1})$ error. For the second, a weighted $L^2$ distance between distribution functions is converted into an exact spherical kernel depending only on $|x|^2$, $|y|^2$, and $\langle x,y\rangle$. Expanding this kernel in $\langle x,y\rangle$, we control its linear, quadratic, and cubic terms using the quadratic variance inequality $\operatorname{Var}(X^\top M X) \le 8\operatorname{Tr}(M^2)$, while fixed-order moment estimates control the remainder.

We develop a theory of proper spaces and proper maps in the context of finite posets.

In this paper, the logarithmic $p$-Laplacian operator $\log (-\Delta_{\mathbb H ^n})_p$ on the hyperbolic space $\mathbb H^n$, with $n\geq 2$, is introduced. We prove that if $f$ is a locally Lipschitz function of exponent $\alpha \in (0,1)$ with compact support in $\mathbb H^n$, then, for a suitable constant $A_{n,p}>0$, $$ \lim_{s\rightarrow 0^+}(-\Delta_{\mathbb H ^n})_p^sf(x)=A_{n,p}|f(x)|^{p-2}f(x),\quad x\in \mathbb H^n, $$ where $(-\Delta_{\mathbb H ^n})_p^s$ denotes the $s$-fractional $p$-Laplacian on $\mathbb H^n$. We establish a pointwise integral representation for the operator $\log (-\Delta_{\mathbb H ^n})_p=\frac{d}{ds}(-\Delta_{\mathbb H^n})_p^s\,_{|s=0}$. Furthermore, we show that $\log (-\Delta_{\mathbb H ^n})_p$ can be realized as the solution of a suitable extension problem and provide an extension theorem that yields the operator $\log (-\Delta)_p$ in $\mathbb R^n$. To the best of our knowledge, this property has not been established for the Euclidean logarithmic $p$-Laplacian $\log (-\Delta)_p$.

We propose a unified framework for bijective parameterizations of $n$-dimensional manifolds that jointly control angular and volumetric distortion through a weighted combination of conformal and volume-preserving energies. A logarithmic barrier based on signed simplex Jacobians prevents element inversion and degeneration during energy minimization. On oriented simplicial manifolds, the gradients of all three discrete energies admit a unified cotangent Laplacian-type representation that preserves the sparsity of the mesh connectivity. Based on these formulations, we develop a three-stage algorithm that constructs an initial parameterization, repairs foldings to restore strict feasibility, and subsequently minimizes the weighted distortion energy while maintaining local injectivity. The framework is formulated in a general setting, with particular attention to parameterizations onto the unit sphere $\mathbb S^n$ and the unit ball $\mathbb B^n$. We derive sufficient conditions for discrete spherical maps to have and retain degree one, and apply classical topological criteria to establish global bijectivity. Numerical experiments demonstrate that the proposed framework achieves simultaneous control of angular and volumetric distortion, while maintaining bijectivity and eliminating surface and volumetric folds. In a brain-tumor label-transfer study, the method improves tumor-label reconstruction compared with a non-bijective ablation.

Bandit algorithms generate data for downstream inference, but adaptive sampling biases post-bandit sample means. We analyze this bias for stable index algorithms, including UCB1 and its generalizations, and derive sharp leading-order expressions for the sample-mean bias and expected $Z$-statistic, in bandit experiments of fixed horizon $T$. Our characterization reveals the algorithmic origin of bias through a key index-function-dependent quantity, which we term effective exploration rate. For example, under UCB1, the effective exploration rate is of order $\sqrt{\log T}$, and the standardized bias of any arm (that is not uniquely optimal) decays at the extremely slow rate $1/\sqrt{\log T}$. We also show how the choice of the index function affects both regret and bias, which reveals a regret-bias trade-off: more exploratory algorithm reduces bias but increases regret. We further show how bias most severely distorts confidence intervals and hypothesis tests when the tested arm is one of the tied-optimal arms. Our sharp characterization for bias uses a novel empirical fluid approximation of the algorithm's sampling dynamics, which may be of independent interest.

We establish boundary second derivative estimates for convex graphical solutions of the special Lagrangian curvature potential equation. Since the curvature matrix depends on both $Du$ and $D^2u$, a phase subsolution alone does not provide the full linearized separation needed for the mixed derivative estimate. We introduce a mixed recession compatibility condition imposed only on doubly degenerate level jets. It yields a uniform mixed derivative bound and is sharp within the class of fixed smooth zero-order barriers considered here. For the double-normal derivative, an exact complex Schur-complement identity gives \[ u_{\nu\nu}=\beta+\alpha\cot\delta, \qquad 1\leq\alpha\leq C, \qquad |\beta|\leq C, \] where $\alpha$ and $\beta$ are explicit Schur-complement coefficients, $\delta$ is the actual boundary limiting-phase gap, and $C$ depends only on uniform bounds for the gradient and the mixed boundary derivatives. Thus curvature blows up if and only if this gap collapses, with optimal rate $\delta^{-1}$. Smooth radial solutions attain the rate, while a rank-loss model shows that a strict lower subsolution need not force strict convexity.

We establish new quantitative bounds for asymptotic approximate groups arising from finite subsets of lattices and, more generally, semi-linear subsets of abelian groups. Our approach combines Khovanskii's theorem on sumsets with Rogers and Zong bounds for the covering numbers.

For a matroid $M$, a join is a set $J\subseteq E(M)$ that meets every circuit $C$ in at most $|C|/2$ elements. Let $\mu(M)$ denote the maximum size of a join. Motivated by Frank's min–max theorem for graphic matroids, we compare $\mu(M)$ with an ear-decomposition parameter $\eta(M)=(r(M)+\varphi(M))/2$, where $\varphi(M)$ is the minimum number of even lobes in an ear decomposition of $M$. Frank's theorem implies $\mu(M)=\eta(M)$ for connected graphic matroids. Here we study how far this equality extends beyond graphic matroids. We show that the exact equality does not hold in general: it already fails for cographic matroids, hence within the binary class. Furthermore, the class of matroids satisfying $\mu(M)=\eta(M)$ is not minor-closed, thus there is little hope for a forbidden minor characterization. We also prove that computing a maximum join is NP-hard for cographic matroids, hard to approximate within a factor of $519/520$, and NP-hard for sparse paving matroids given by their list of bases. Despite these negative results, we show that the two parameters remain quantitatively comparable in several natural classes. We prove comparison bounds for binary, paving, cographic, and arbitrary connected matroids. In particular, using Seymour's decomposition theorem, we combine the equality for graphic matroids, the bound for cographic matroids, and a direct analysis of $R_{10}$ to obtain $\eta(M)\leq 6\mu(M)-2$ for every regular matroid $M$.

We study elliptic curves induced by regular Diophantine triples, with emphasis on their torsion subgroups. We show that an elliptic curve $E$ induced by a regular Diophantine triple in integers necessarily has torsion subgroup $E(\mathbb{Q})_{\mathrm{tors}} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Moreover, we develop a criterion for when such an elliptic curve acquires a point of order $3$ over a quadratic field. For a particular family $\{ k-1, k+1, 4k\}$, we use it to show that this does not happen. Finally, we study both the torsion and the generic rank of a family of elliptic curves induced by the $D(-k^2)$-triple $\{1, 2k^2, 2k^2+2k+1\}$.

In this paper, we provide a new method for constructing a compact and non SI-compact space and thus give a new answer for the question posed by Zhao and Ho. Using this method, we find a countable compact space which fails to be SI-compact. As a corollary, some new non sober frames are obtained and thus we give a new answer for a question raised by A.Jung. Inspired by the definition of SI-compactness, we introduce a new notion of compactness of topological spaces, namely, v-compactness, which is stronger than SI-compactness.Some properties of v-compactness, SI-compactness and weak v-compactness are discussed. Specifically, we prove that (1) all closed subspaces of a v-compact(SI-compact,weakly v-compact) space are v-compact (SI-compact, weakly v-compact);(2) a characterization of v-compactness for the upper topology is given; (3)some characterizations of weakly v-compact subsets of real line and Sorgenfrey line are given.

We prove that any two quadrangulations of the same surface, i.e., decompositions of a surface into squares, can be transformed into one another by a finite sequence of two-dimensional local cubical Pachner moves and non-local subdivision transformations.

We prove that every full-dimensional compact convex body in $\mathbb R^n$ whose barycenter is its unique interior lattice point and whose volume is $(n+1)^n/n!$ is a unimodular image of the simplex $(n+1)\Delta_n-(1,\dots,1)$. This resolves the equality case of Ehrhart's volume conjecture, as a counterpart of the inequality part recently proved by OpenAI. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system.

Blohmann's tangent functor makes the category of elastic diffeological spaces a Cartesian tangent category in the sense of Cockett and Cruttwell. We use this structure to develop Riemannian geometry on elastic diffeological spaces. Following Cockett and Cruttwell, with additional axioms for scalar multiplication, we introduce connections, covariant derivatives and curvature; for a Riemannian metric we prove a Koszul formula and the uniqueness of the Levi-Civita covariant derivative. Our main result is a variational characterization of geodesics: if a full connection whose vertical part is Levi-Civita admits a geodesic flow fixing the zero section, then the critical paths of the energy functional are exactly the paths satisfying the geodesic equation, and regular length-minimizing paths are geodesics after constant-speed reparametrization. On elastic spaces for which the real line is a curve object, completeness of the geodesic spray yields such a flow and uniqueness of geodesics. The results apply to quotients of complete Riemannian manifolds by free isometric actions of countable groups, such as irrational tori, and, under a tangent-commuting hypothesis, to mapping spaces from a closed manifold into elastic spaces for which the real line is a curve object, including free loop spaces of complete Riemannian manifolds.

Aihara developed an algorithm for Brauer tree algebras, which we call a mutation reduction, for getting from a Brauer tree algebra to the simpler Brauer star algebra using a sequence of mutations centered on edges. Schaps and Zvi, using the Schaps-Zakay theory of pointing the tree, showed that different algorithms for the sequence of mutations give permutations of the edges. Kozakai gave a new algorithm for a mutation reduction that depends on a given pointing and describes the evolution of the pointing under the mutation reduction. In this paper, we define a pointed generalized Aihara algorithm and show that its permutation is the identity. We give a general form for the permutations resulting from Kozakai's algorithm, which we illustrate with examples from uni-branch binary trees.

Training error is what we can observe on a training set; test error is the quantity we care about. We study squared-error linear regression in a deterministic $(d+1)$-dimensional single-spike model. Each training vector has spike coordinate $\sqrt{\gamma}$, $\gamma>1$, and equal-norm, mutually orthogonal nuisance components. Training labels are $1$. Fresh test points are centered Gaussian with covariance $\operatorname{diag}(\gamma,1,\ldots,1)$ and noise-free label $x[1]/\sqrt{\gamma}$. We compute the exact train–test tradeoff for linear predictors in the training span. When the spike direction is unknown, allowing all linear predictors does not improve this tradeoff, on average or in the worst case over directions consistent with the data. There is a range of training-set sizes where every span predictor that generalizes well must fit the training data worse than the zero predictor. We call this benign misfitting, or the fourth quadrant. At its sharpest, test error tends to zero while training error diverges, with both zero-predictor errors normalized to one. The best span predictor begins to generalize when $n\gg d/\gamma^2$, while interpolation needs $n\gg d/\gamma$. In the intervening window $d/\gamma^2\ll n\ll d/\gamma$, useful span prediction lies beyond interpolation: training predictions overshoot the labels. One-pass stochastic gradient descent with a large constant learning rate attains small test error throughout this window, matching the best span predictor up to a logarithmic factor, while leaving large empirical training error. Finally, the nuisance component responsible for the training misfit also controls adversarial sensitivity.

In this paper, we introduce the double super Yangian $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$ and $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ associated with any fixed $0^{m}1^{n}$-sequence $\mathfrak{s}$. First, we establish an explicit isomorphism between the Drinfeld and R-matrix presentations of $\mathrm{DY}_{h}(\mathfrak{gl}^{\mathfrak{s}}_{m|n})$. We then generalize the notion of the quantum Berezinian to $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$, and employ it to construct the R-matrix presentation of $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ and prove that it is isomorphic to the Drinfeld presentation. As an application, we present level-1 bosonic representations for $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$ and $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ in terms of their Drinfeld current generators.

This study investigates a new approach for scheduling freight rail engineers based on call windows under the Hours of Service (HOS) regulations. Unlike planned trip assignments, call windows specify a time interval during which an engineer may be required to start work, thus providing greater flexibility to handle uncertain trip demand while offering drivers more predictable off-duty periods. Currently in the U.S., all major freight operators require drivers to be available 24/7 outside of mandatory rest periods, raising concerns over workforce fatigue and safety. We formalize the call-window scheduling problem and propose two solution approaches: a Set-covering-type optimization model and a Direct Algorithm. Both aim to maximize demand coverage while minimizing the number of engineers subject to HOS feasibility. Computational experiments for 2-city and 3-city instances with varying call window lengths, maximum delay allowances, and whether to allow deadhead trips show that the Set-covering-type model yields higher demand undercoverage (94.89% and 91.09% for 2-city and 3-city instances, respectively) than the Direct Algorithm (93.96% and 71.83%). It also offers greater rest opportunities and reduced delays. Sensitivity analysis reveals that the upper limit on engineer availability significantly affects all key performance metrics.

We consider the two-dimensional incompressible Euler equation on $\mathbb T^2$ with Gaussian random initial data having independent Fourier coefficients. For every $\sigma>0$, we prove that such a Gaussian measure on $H^\sigma(\mathbb T^2)$ is invariant under the Euler flow if and only if it is supported either on shear flows or on cellular flows. This settles the invariant-measure classification conjecture posed by Bedrossian and Latocca (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2026). The proof relies on closure of the Fourier support under non-degenerate interactions and an affine relation among the inverse variances along Euler triples.

We introduce and characterize age dispersion as a measure of temporal consistency in status update systems. Age dispersion is defined as the difference between the ages of the two most recently received updates, and its higher-order extension, the {$k$-th} order age dispersion, captures the difference between the ages of the most recent update and the $(k+1)$-th most recent one. We analyze age dispersion in an M/G/1/1 queueing system. Furthermore, we establish connections between the {$k$-th} order age dispersion and the {$k$-th} order age of information (AoI), where the latter quantifies the age of the {$k$-th} most recently received update.

Solomonoff Induction, or SolInd, provides an ideal unbounded model of a priori sequence prediction but cannot naturally describe extrapolation from a given training dataset, as performed by Large Language Models. We apply de Finetti's theorem on exchangeable distributions to SolInd to produce what we call Hierarchical Solomonoff Induction, or HSI, which maintains a hyperprior over all Solomonoff priors that can be conditioned on previously observed sequences. We extend Wood et al.'s proof that universal mixtures of semimeasures are equivalent to SolInd to show that universal mixtures of these mixtures are also equivalent, proving that HSI=SolInd. We also prove that HSI's excess error on any distribution, compared to its true generator, is bounded by that generator's complexity in the hyperprior. This result is directly comparable to SolInd's prediction error being bounded by the Kolmogorov complexity of the sequence being predicted, and forces HSI's average excess error to converge to 0 as a dataset grows, leading to optimal prediction in the limit. We claim that HSI is an ideal unbounded model of sequence prediction given a dataset in the same way that SolInd is ideal over individual sequences.

The computation of the action of a matrix function on a vector, $f(A)b$, is a major computational bottleneck for large, sparse matrices, particularly when unfavorable spectral distributions cause standard Krylov subspace methods to stagnate. In this work, we propose a unified framework for preconditioning $f(A)b$ based on the Cauchy integral representation of the matrix function. By exploiting shift-invariance properties, we decouple the preconditioner evaluation from the Krylov subspace generation. We develop this framework in two distinct directions. First, for rational shift-and-invert preconditioning, we resolve a fundamental trade-off between optimal spectral compression and finite-precision instability. We achieve this by formulating a closed-form extraction stabilized via Double Modified Gram-Schmidt reorthogonalization, which eliminates the formation of spurious phantom poles. Second, we present a matrix-free polynomial approach. To ensure numerical stability, we isolate the continuous numerical quadrature step using a Schur decomposition of the projected Hessenberg matrix. To further stabilize the integration near contour singularities and accelerate overall convergence, we incorporate an exact LR-deflation scheme targeting the critical low modes of the preconditioned operator. We analyze the asymptotic stability and proximity to singularity of these methods, and present numerical experiments demonstrating their efficiency on the 2D Laplacian with $f=\textrm{exp}$, and a highly ill-conditioned Wilson-Dirac operator from lattice quantum chromodynamics with $f=\textrm{sign}$, although the framework can be in principle used with any $f$ and it is particularly beneficial when applying $f(A)b_{i}$ with many different vectors $b_{i}$.

We prove analogues of Tannakian reconstruction theorems of Lurie and Bhatt–Halpern-Leistner in derived algebraic geometry, where the basic geometric objects are spectra of animated rings rather than $\mathbb{E}_\infty$-rings. In this setting, symmetric monoidal $\infty$-categories are replaced by the $\Theta$-categories of Nuiten–Toën. These are enhancements of symmetric monoidal $\infty$-categories which capture the strict commutativity structure on animated rings.