Papers

The classical sphere packing problem, which remains unsolved, consists of determining how densely a large number of identical spheres can be packed together. In some sphere packings, the centers of the spheres in a sphere packing form a Euclidean lattice, which is a discrete additive subgroup of $\mathbb{R}^n$. Free $\mathbb{Z}$-modules in the ring of integers of an algebraic number field yield algebraic lattices via the canonical embedding. In this work, we present new constructions of algebraic lattices from certain families of $\mathbb{Z}$-modules in the ring of algebraic integers of subfields of the $p$-th cyclotomic field, where $p$ is a prime number. Within this framework, we compute lower bounds for the center density of these algebraic lattices and construct algebraic lattices having the best known packing density in dimensions 2, 3, and 5.

A bounded agent may obtain information for a decision from its own past, from peers, or from both sources. Retaining task-relevant history can reduce later communication, while a peer message can supply what memory lacks. Under limits on both resources, how should an agent allocate its information budget? Given a fixed task and decision rule, the memory and message rate pairs attaining a performance threshold form an achievable region under specified rules for using history and peer observations. We call its efficient boundary the remembering–signaling frontier. Across conditions where history permits the same maximum reduction in task loss, we hypothesize that a bounded agent will need less peer communication when it obtains a larger loss reduction from history. In preliminary referential games, target repetition coincided with shorter successful messages, while predictability from a hidden cyclic rule did not shorten them. Experiments varying memory and message rates can estimate the frontier and test this prediction across cooperative tasks.

Let $d(n)$ be the divisor function and let $H<H'$ be consecutive highly composite numbers. We study directed unit moves between their prime-exponent vectors under the hard ceiling $z\le H'$. The normalised capacity of such a geodesic is its smallest divisor count divided by $d(H)$, giving a finite fixed-endpoint maximin problem. The exact record enumeration first finds a failure of the static surrogate $d(\gcd(H,H'))\ge d(H)/2$ at $48,886,437,600<64,250,746,560$, where the ratio is $4/9$. For every state $z$ in the exponent box, however, we prove $$d(z)d(HH'/z)\ge d(H)d(H')$$ and deduce the record-box gap: no box state lies numerically strictly between the two records. We also give an exact dynamic-programming recursion, solve the strata $L_-\le 1$, and reduce the complete $L_-=2$ problem to an explicit divisor-selection functional. A computer-assisted enumeration through $10^{70}$ produces $889$ records and $888$ transitions. Although the static half-gcd bound fails $119$ times, every computed geodesic capacity is at least $1/2$; equality occurs in exactly the $124$ transitions that lose an exponent-one support prime. Independently checked certificates cover all $301$ transitions with $L_-=2$. The corresponding universal half-capacity bound and equality classification remain open beyond the proved strata and the verified range. COMMENTS

Given a finite rank Coxeter system, we construct a hyperbolic space by coning off the wide parabolic subcomplexes of its Davis complex. The resulting space is “stability recognizing”, in the sense that stable subspaces of the Davis complex are exactly the quasigeodesically connected subspaces whose images in the coned-off space are quasiisometrically embedded. Equivalently, the coned-off space is “Morse recognizing”: a quasigeodesic in the Davis complex is Morse if and only if it is quasigeodesic in the coned-off space. Similar results are true for the mapping class group of a hyperbolic surface acting on its curve graph. We strengthen this analogy with additional results. First, the action of the Coxeter group on the coned-off space is acylindrical and gives the largest acylindrically hyperbolic structure for the Coxeter group. Second, the stable subgroups of a Coxeter group are characterized as the finitely generated subgroups that are undistorted and purely loxodromic with respect to the action on the coned-off space. The coned-off space, being constructed directly from the Coxeter structure, facilitates concrete characterizations of Morseness and stability for two important Coxeter-native families of objects: combinatorial geodesics and special subgroups.

In this work, we propose a new Arrow-Hurwicz iterative scheme designed to solve the steady inductionless magnetohydrodynamics system. The main feature of the proposed scheme is the introduction of new penalty terms in the current density equation. These terms play a central role in effectively controlling the unfavorable mixed terms that commonly arise in such formulations. The proposed method is shown to achieve geometric convergence. Numerical tests affirm the efficiency of the new scheme without compromising accuracy.

We prove a shape theorem for a multidimensional periodic branching diffusion, which is a branching particle system where particle trajectories evolve according to a diffusion, drift, branching rate and offspring distribution which all depend periodically on space. More precisely, we show that almost surely as $t\to\infty$, the particle cloud at time $t$, normalized by $t$, converges in Hausdorff distance to a deterministic convex set $\mathcal{W}$. We further establish that the boundary of the asymptotic shape $\mathcal{W}$ is of class $C^2$, strictly convex, and has positive Gaussian curvature bounded away from zero. This work generalizes the approach of arXiv:2507.10515 for $g$-BBM and furthers our understanding of the shape $\mathcal{W}$ in that setting.

We characterize the simplicity of reduced crossed product C*-algebras in terms of stabilizer subgroups. Specifically, we prove that if $G$ is a countable group and $X$ is a minimal $G$-flow, then the reduced crossed product C*-algebra $\mathrm{C}(X) \times_\lambda G$ is simple if and only if there is a point in $X$ with a C*-simple stabilizer subgroup. Further, these conditions are equivalent to a generic point in $X$ having a C*-simple stabilizer subgroup. We also provide an example demonstrating that this result does not extend to uncountable groups. This completely resolves a question of Ozawa.

We analyze lattice-based algorithms for recovering integer-valued signals from partial discrete Fourier transform (DFT) measurements. These algorithms formulate signal recovery as the problem of finding short vectors in an appropriately constructed lattice. We derive parameter estimates that predict the regimes required for successful recovery and quantify how these estimates depend on the signal length, the error of an initial guess, and the number of sampled DFT coefficients. The analysis also reveals a close connection between the lattice parameters and the precision of the measured DFT coefficients, providing insight into the numerical stability of the inversion. Numerical experiments demonstrate close agreement between the theoretical predictions and observed recovery thresholds over a broad range of problem parameters.

We study the underdamped (kinetic) Langevin dynamics confined to a bounded domain $\Omega\subset\mathbb{R}^d$ by specular reflection of the velocity at the boundary. This process is the natural momentum-based analogue of the normally reflected overdamped Langevin diffusion, and it is used in practice for constrained sampling. Assuming only that the position marginal $\mu_x\propto e^{-U}$ satisfies a Poincaré inequality on $\Omega$ with constant $m>0$, $\nabla^2U\succeq-K\,\mathrm{Id}$ and $\Omega$ is a convex domain, we prove that the law converges to the Gibbs measure exponentially fast in $L^2$, with an explicit rate that scales like $\sqrt m$, which is optimal when $U$ is convex. Since the normally reflected overdamped dynamics converges exactly at rate $m$, this establishes a square-root acceleration for constrained sampling in the small-gap regime when $m$ is small, matching the acceleration known in the unconstrained case. The explicit rate is the same in the unconstrained setting of Fan–Li–Lu. The proof adapts the modified $L^2$ hypocoercivity method of Dolbeault–Mouhot–Schmeiser with the gap-shifted corrector of Fan–Li–Lu. The specular symmetry makes the transport operator antisymmetric, and that the corrector automatically selects the Neumann realization of the overdamped generator, which is precisely the boundary condition that keeps every auxiliary function inside the specular class. The Bochner identity used in the whole-space argument is replaced by a weighted Reilly formula, whose boundary contribution involves the second fundamental form of $\partial\Omega$ and is nonnegative for convex $\Omega$. Finally, we extend our results to the setting where the domain $\Omega$ is non-convex. We obtain an explicit contraction rate that depends on the domain.

This work studies the relations between shadowing, topological stability, Lipschitz structural stability, and pseudo-hyperbolicity for linear dynamical systems on Banach spaces. The main result establishes that pseudo-hyperbolicity implies both topological stability and strong Lipschitz structural stability, whereas strong Lipschitz structural stability implies the shadowing property. In addition, a spectral characterization of pseudo-hyperbolicity is obtained, yielding an equivalence between pseudo-hyperbolicity, topological stability, strong Lipschitz structural stability, and the shadowing property for invertible operators whose eigenspaces associated with unimodular eigenvalues admit closed complements. In particular, these four notions are equivalent on Hilbert spaces. Combined with a recent result of Dragičević and Pituk, these results yield, for a large class of Banach spaces, an invertible operator that has the shadowing property but is not generalized hyperbolic, thereby answering an open problem in linear dynamics.

A multiplicity-free subvariety of a product of projective lines is a subvariety with the property that the expansion of its Chow class in the usual basis has all coefficients equal to 0 or 1. We show that, over a field of positive characteristic, a multiplicity-free subvariety has F-rational singularities. This strengthens Brion's result that multiplicity-free subvarieties are normal and Cohen–Macaulay, and that they have rational singularities over a field of characteristic 0.

We generalize the unified framework in [Phys. Rev. A 113, 022438 (2026)] to accommodate arbitrary stabilizer codes (referred as non-CSS for emphasis). The generalization has immediate consequences in areas including logical measurement, quantum weight reduction and Euclidean embedding. For example, currently CSS (pure $X,Z$-type) logical measurements are well-understood based on the (height-1) cone, while non-CSS (e.g., $Y$-type) logical measurements are ad hoc (e.g., rely on local Clifford transform). Similarly, quantum weight reduction and optimal Euclidean embeddings only exist for CSS input codes. Here, we show that our generalized framework addresses this issue, and thus many constructions, including qLDPC surgery, Layer Codes and quantum weight reduction generalize to non-CSS codes in a relatively straightforward fashion. The previously mentioned examples are derived in detail for clarity.

In this work, we compute the operator product expansion density for the genus-two handlebody by exploiting its duality with extremal conformal field theories arXiv:0710.2129. Using a Poincaré construction, we analytically determine the first- and second-order corrections in the pinching parameter to the genus-two OPE density through a direct inversion of the partition function. As a consistency check, we also compute the genus-one density within the same Poincaré framework and find perfect agreement with the result obtained from the lightcone bootstrap analysis arXiv: 1906.04184. It is known that, at genus one, the OPE density of extremal CFTs exhibits a negativity pathology for states lying just above the black-hole threshold. We show that this pathology continues to persist at genus two, despite the emergence of a nontrivial inversion kernel. In particular, the OPE density associated with states of both odd and even spin $j_i$ remains negative within a very narrow band above the black-hole threshold. We further analyze the sharp transition from large positive to large negative values of the OPE density across a curve along which the density vanishes.

We study the spread complexity of the infinite-temperature thermofield double state in large-$N$ random matrix theory and investigate the physical meaning of its Lanczos coefficients $b_n$. The leading density of states defines a family of orthogonal polynomials whose recursion coefficients coincide with the Lanczos coefficients of the associated Krylov dynamics. Starting from the Coulomb-gas equation, we derive a pair of nonlinear relations for these coefficients, which we call the Lanczos equations. For normalizable one-cut ensembles, the Lanczos coefficients approach constants at large Krylov index, rendering the asymptotic Krylov chain translationally invariant and the late-time growth of spread complexity linear, with the growth rate given by the spectral average of the group velocity obtained from a Bloch-wave analysis. For an even potential, we further show that the difference between consecutive squared Lanczos coefficients, $b_{n+1}^2-b_n^2$, is the generating function for two-legged maps whose marked legs are separated by the exact geodesic distance $n$. Consequently, the spread complexity acceleration operator equals twice the geodesic generating operator. We illustrate these results in the quartic matrix model, where the expansion in the quartic coupling counts tetravalent planar maps, and in the double-scaled Sachdev–Ye–Kitaev model, where the auxiliary Hilbert space can be identified with the physical chord Hilbert space.

AI systems are increasingly capable of contributing to mathematical research. In research practice, frontier-model reasoning is a limited resource, and expert mathematical review is even more sharply constrained. Allocating these scarce resources well is therefore central to making AI-assisted mathematical discovery efficient. In most current AI-for-math workflows, human effort is concentrated at the beginning and end, in selecting suitable research problems and later reviewing the resulting artifacts. These two stages are becoming bottlenecks for research-level mathematics. We address them by proposing a new human-AI discovery paradigm. The human input is no longer a single problem selected in advance, but a research direction in which the experts have interest and expertise. The system then searches a broad literature corpus for candidate problems in that direction. Inspired by search and recommender systems, we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade that automates the search for suitable problems and focuses human attention on artifacts that have passed several stages of filtering. In a combinatorics pilot, the pipeline starts from 5,245 combinatorics papers, recovers 6,453 candidate conjectures or open problems, and filters them to 4,717 apparently well-posed and still-open conjectures. Subsequent reasoning and automated triage stages surface 598 potential resolutions and select 77 items for author-team review. Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies–Jenssen–Perkins–Roberts, Erdős–Straus, Ikenmeyer–Pak–Panova, and Lund–Saraf–Wolf. These results demonstrate the effectiveness of this new mode of human-AI collaboration for mathematical discovery.

In this paper, we consider methods for the diagonal multi-omics integration of heterogeneous datasets. Several approaches to the nature of biological heterogeneity are analyzed and developed to comprehend more clearly the generated differences. Specifically, the extremal trace problems for the coupled Laplacian on sets homeomorphic to the Stiefel manifold embedded in the complex Euclidean space are investigated. The gradient ascent method for the maximization problem is elaborated in the classical terms of functional analysis, which is of significant interest in itself. On this basis, we introduce a novel characteristic of dataset heterogeneity by employing the norm of the difference between the maximum and minimum points.

Snow is a complex geomaterial whose macroscopic response is governed by density, temperature, and the topology of its evolving microstructure. Its mechanical behavior spans elastic, plastic, viscous, and failure dominated regimes, imposing significant challenges for numerical methods, which intends to simulate large deformations, evolving free surfaces, and complex boundary interactions. This work presents the first integration of a Modified Cam-Clay constitutive formulation for snow into a purely meshfree strong-form collocation framework based on the Generalized Finite Difference Method. The main methodological contribution is a numerical coupling that combines a global implicit mixed formulation for pressure and velocity with a constitutive return-mapping algorithm. The hydrostatic pressure contribution is obtained from a Poisson equation and subsequently corrected through the Modified Cam-Clay return-mapping procedure, while the deviatoric response is treated semi-implicitly using a numerical viscosity formulation. This partitioned treatment of the volumetric and deviatoric stress contributions enables stable simulations with comparatively large time steps while producing smooth spatial pressure fields. As a result, forces on complex boundary geometries can be evaluated accurately. Numerical results of this coupling illustrate the algorithmic stability of the framework, the effective imposition of boundary conditions, and the suitability of local spatial refinement. The feasibility of applying the framework to vehicle-snow interaction through rigid-body coupling is also illustrated. The presented formulation provides a robust basis for future simulations of dynamic snow loading on vehicle structures.

For $q> p$, let $T(n,q,p)$ be the minimum number of translates of the cube \(\{0,1,\dots,p-1\}^n\) required to cover the $n$-dimensional torus $(\mathbb{Z}/q\mathbb{Z})^n$. We show that for each $q$ there exists a constant $1\le \Lambda_q \le 2$ such that $T(n,q,2)=(\Lambda_q + o(1))(q/2)^n$.

Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sawhney confirmed this conjecture for every constant $p$, leaving the regime where $p=o(1)$ open. In this regime, the only case addressed in the literature is when $H=K_3$, where, in a recent paper, Araújo and Mattos confirmed the conjecture for $p \in (4n^{-1/2}, 1/2)$. This, together with a general result of Röllin and Ross, essentially settles the conjecture for the triangle. We generalise these results by showing that an LCLT holds for $H = K_r$ (for any fixed $r \ge 3$) in the regime $n^{-1/m(H)}\ll p\leq 1/2$, essentially settling the conjecture for cliques.

This paper captures and extends some of the core results from the tech memo "A New Semantics for Belief Revision in Simplicial Complexes". As such, we set out to explore the implementation of action models in the setting of simplicial semantics for modal logic. Such an idea is not entirely new to the literature, showing up in both "A simplicial complex model for dynamic epistemic logic to study distributed task computability" and "Knowledge and Simplicial Complexes". However, we will explore action models in a more general setting. In particular, we will allow for action models for simplicial models for belief, as in "A Semantics for Belief in Simplicial Complexes". This will let us incorporate the notion of belief revision, as developed in "Simplicial Semantics for Belief Revision", into these action models. Moreover, we explicitly connect action models in the simplicial setting to distributed protocols as defined in the textbook "Distributed Computing Through Combinatorial Topology". We conclude with some speculation on how we might interpret distributed protocols with revision.

The only Einstein metrics currently known on complex projective spaces are homogeneous: the Fubini-Study metric, arising via the Hopf fibration; and, in odd complex dimensions, Ziller's metric, obtained as a canonical variation along the twistor fibration over the quaternionic projective space. In 1965, Berger proved that the Fubini-Study metric is the unique Kähler-Einstein metric on the complex projective space, and posed the question of whether other Einstein metrics exist. We construct the first inhomogeneous Einstein metrics on complex projective spaces of complex dimension $n$ for $3\leq n \leq 7$, answering Berger's question affirmatively for the complex even dimensional cases $n=4$ and $n=6$.

There are several non-equivalent notions of infinitesimal symmetry in the literature of second order linear differential equations: Lie point symmetries, vertical (gauge) symmetries, operator symmetries, infinitesimal contact symmetries, and Lie–Bäcklund operators. We construct an explicit correspondence among the first three, We then describe the Lie algebra $\mathcal L_\Omega(\mathbb U)$ of infinitesimal contact transformations of the contact system $\Omega=\langle dy-y'\,dx\rangle$, with coefficients in a differential field $\U$ of functions of $x$ and $y$. We obtain a canonical decomposition $\mathcal L_\Omega(\mathbb U)=\prod_{k\ge0}\mathcal L_\Omega^k\mathbb U$ into $\mathbb C$-vector spaces, each parametrized by $\mathbb U$ (by $\mathbb U\oplus\mathbb U$ for $k=0$), and we compute the algebraic differential formulae for the Lie bracket in these coordinates. Applied to the symmetry problem, we prove that a contact vector field with generating function $W$ is a symmetry of if and only if $A^{2}W=aW+b\,AW$, where $A$ is the vector field in the jet space corresponding to the equation; equivalently, if and only if $W=F_1(u_1,u_2)\phi_1+F_2(u_1,u_2)\phi_2$ for arbitrary functions $F_1,F_2$ of the two first integrals of $A$ and a basis $\phi_1,\phi_2$ of solutions. The symmetry algebra is always parametrized by two arbitrary functions of two variables. It also shows that the decomposition of $\Lom(\U)$ never captures the whole symmetry algebra: for $a\neq0$ the graded part reduces to the point symmetries, while for $y''=0$ it is an infinite dimensional but still proper subspace, and in neither case is it a Lie subalgebra. Finally we make precise the transformation law $W\mapsto\mu^{-1}(W\circ\varphi)$ for characteristics under a contact transformation with conformal factor $\mu$, which governs the transport of evolutionary representatives.

For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $\Omega(1/(n^2 C_{\mathsf{PI}}))$, where $C_{\mathsf{PI}}$ is the Poincaré constant of the uniform distribution $\pi$ over $\mathcal{K}$. This implies that Hit-and-Run converges to a distribution within $\chi^2$-divergence $\varepsilon$ of the uniform distribution $\pi$ in $O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))$ steps from any starting distribution $\pi_0$ with $M=\chi^2(\pi_{0}\,\|\,\pi)$, thus refining the known bound of $O(n^2 R^2 \log(M/\varepsilon))$ by Lovász and Vempala (2004) in terms of the outer radius $R$; for nearly isotropic bodies, together with progress on the KLS conjecture, the complexity is $O(n^2\log n\log(M/\varepsilon))$, improving the dimension dependence from cubic to nearly quadratic while maintaining logarithmic dependence on the initial distance. It was an open problem to connect the convergence of Hit-and-Run to Poincaré/KLS constants as was done for the Ball walk by Kannan, Lovász and Simonovits (1997). Unlike Hit-and-Run, the Ball walk has an unavoidable linear dependence on (a stronger notion) of the initial warmness. We directly bound the spectral gap of the Hit-and-Run Markov chain by connecting it to functional isoperimetric constants, inspired by the recent analysis of In-and-Out. Rewriting the spectral gap in terms of dual certificates leads to the Babuška–Aziz constant studied in the analysis of PDEs; it is asymptotically bounded by the improved Poincaré constant, which we show can be bounded in terms of the usual Poincaré constant. The proof is based on duality and calculus, unlike known proofs of convergence for Hit-and-Run which are based on bounding the conductance. The same technique can be applied to Coordinate Hit-and-Run, resulting in a much improved mixing time of $O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$.

We introduce Automatic Symbolic Regression (AutoSR), a fully automated system that instantiates Research-Space Symbolic Regression by searching persistent scientific investigations rather than isolated equations. Finite, noisy data often yield numerically competitive expressions that imply very different behavior outside the observed regime, making numerical fit and syntactic complexity insufficient measures of scientific credibility. Existing approaches largely focus on improving expressions, yet the search typically retains little beyond the resulting formula and score, losing the scientific record, such as motivations and probes, that inform what to try next. AutoSR preserves this record in a Research State, coupling each candidate equation with the reasoning, computational evidence, and independent review developed along its branch. Proposer–reviewer agents develop these states under progressive-widening Monte Carlo tree search (PW-MCTS), which allocates computation across competing investigations, while the accumulated research record is ultimately synthesized into a final report that explains the leading relation and the basis for its selection. Across nine selected challenges from two benchmark suites, AutoSR recovers algebraically equivalent relations in every case, including three cp3-bench problems that no published system recovers and six structurally diverse LSR-Transform problems. Overall, AutoSR extends symbolic regression from equation-level search toward automated scientific investigation, allowing scientific knowledge and accumulated evidence to shape both what is explored and how the resulting equation is justified.

There is a well-known circle of conjectures relating unitary solutions of the Yang–Baxter equation, unitary braided fusion categories, topological quantum computation, and link invariants. Progress is limited by the lack of a classification of unitary Yang–Baxter operators. We propose a conjectural classification with three generating sources: monomial solutions, group-type solutions arising from Yetter–Drinfeld modules, and solutions from twisted group-algebra towers. We conjecture that, up to unit scalars and local unitary basis changes, all solutions are generated by these sources. We provide evidence for our classification conjecture by means of computational searches for unitary Yang–Baxter operators among Clifford groups and a polynomial-in-Pauli ansatz. This search yields only a few solutions that do not appear to be equivalent to monomial or group type solutions. Direct calculations identify their link invariants with specializations of the HOMFLYPT and BMW/Kauffman invariants; the latter is a squared Jones specialization at a primitive sixth root, and the corresponding qutrit operator strictly localizes the semisimple quotient of the BMW algebra specializations $\mathcal{C}_n(q^3,e^{\pi\ii/6})$ associated with the braided fusion category $SO(4)_2$. The quaternionic Family III specialization realizes a two-eigenvalue class and gives tensor-power realization of the semisimple Hecke algebra tower associated with \(SU(3)_3\). Significantly, this answers a recent existence question of Lechner, and verifies an open case of the Rowell-Wang localization conjecture from 2010.

We initiate a polyhedral study of the graph multi-separator problem proposed by Irmai et al. (2024) as an alternative to the lifted multicut problem for application to the task of image segmentation. Starting with an integer linear program (ILP) formulation and the multi-separator polytope spanned by its feasible solutions, we characterize in terms of efficiently-decidable, graph-theoretic conditions all facets induced by inequalities of the ILP. We proceed by strengthening these inequalities and describing additional facets of some multi-separator polytopes induced by the stronger inequalities. Specifically, we obtain a totally dual integral description of the multi-separator polytope for paths in the case where separation is considered for all vertex pairs. Finally, we relate the multi-separator polytope to the boolean quadric polytope, showing that facets induced by odd-cycle inequalities do not transfer generally, and to the lifted multicut polytope, showing that either polytope is a projection of a face of the other.

We show how to compile an arbitrary classical circuit into a fault-tolerant circuit, which performs the desired computation even when an almost-linear number of bits are adversarially chosen and corrupted in each timestep. Using a variant of this fault-tolerance scheme that only detects (rather than corrects) corruptions, we give a new construction of probabilistically checkable proofs (PCPs) for NP with polylogarithmic query complexity. This PCP construction from fault-tolerance presents a promising candidate for quantization by the work of Anshu, Breuckmann, and Nguyen (STOC'24), who provided a roadmap for constructing quantum PCPs via fault-tolerance.

Time-critical interactive systems increasingly require ultra-low-latency device identification for multiple users, yet prevailing approaches such as passwords, QR codes, and RFID/NFC are constrained by human input, frame-based sensing, or near-contact range. This paper presents ECO-ID, an event-camera-based optical system for multi-user, ultra-low-latency identification over visible light communication (VLC). Leveraging microsecond-resolution, asynchronous observations of brightness transitions, ECO-ID employs a spatiotemporal coding design: disjoint LED subsets provide spatial separation among users, while user-specific timing delays encode identities without inter-user synchronization. The optical channel and event-driven sensing reduce full-scene capture relative to frame cameras and limit the RF attack surface, while enabling rapid token verification with freshness and replay protection. We implement a prototype and demonstrate that ECO-ID can practically achieve approximately 99.8% localization and 98.7% identification with 0.64 ms mean latency, while theoretically supporting identification at the scale of tens of concurrent users. Overall, ECO-ID provides a fast, privacy-conscious, and security-aware alternative for scalable multi-user identification in time-critical interactive environments.

We prove a fault-tolerance theorem for quantum computation against adversarial noise. For every quantum circuit on $\bar{N}$ logical qudits of depth $\bar{T}$, we construct a fault-tolerant circuit on $N=\text{poly}(\bar{N})$ physical qudits of depth $\bar{T}\cdot\bar{N}^{o(1)}$, which is robust against an adversary who may arbitrarily choose and corrupt an almost-linear number $N^{1-o(1)}$ of physical qudits at each time step. This robustness significantly improves upon prior fault-tolerance theorems, which assumed corruptions were either local and stochastic, or else only act on a polynomially vanishing fraction of qudits. Our fault-tolerance scheme addresses a key bottleneck towards constructing quantum PCPs via the circuit-to-Hamiltonian mapping of Anshu, Breuckmann, and Nguyen (STOC'24). More fundamentally, our result demonstrates that fault-tolerant quantum computation remains possible under noise models that are global, worst-case, and non-Markovian over the full duration of the computation, directly countering concerns that correlated noise could fundamentally undermine quantum fault tolerance. Our construction is based on a new family of subsystem product codes we develop, which have large dimension and distance along with low-weight parity-checks, and which support transversal non-Clifford gates. We show how to perform single-shot fault-tolerant error correction on these codes using a Floquet-like procedure based on the local testability of classical tensor codes. We then obtain a universal fault-tolerance scheme using repeated code switching in a hypercubic qudit architecture. Finally, we recursively compose our scheme with itself to reduce an initially exponential qudit dimension down to a constant.

Interconnected systems have been widely studied, with a focus on interconnected systems whose subsystems are solely input-to-state stable (ISS) or passive systems. The focus of this work is on the interconnected systems that appear in adaptive gradient methods. In adaptive gradient methods, one subsystem seeks to move parameters of a cost function towards a minimizer while the other subsystem works to estimate some derivative information about the cost function to help determine the direction of the parameter update. This work studies instances of such interconnected systems and gives various Lyapunov function constructions for them using different techniques. Global asymptotic stability (GAS) of adaptive gradient optimizers follows from small-gain theory; the Lyapunov functions constructed here certify it explicitly.