Papers

We study the exact discretization of the $L_2$-norm in the space spanned by the first $N$ Rademacher functions. It is shown that the sufficient number of nodes for discretization with the minimal number of nodes is equal to $N$ or $N+1$ and depends on the dimension $N$. The connection with Hadamard matrices and the Hadamard conjecture is demonstrated.

A point lies in the kernel of a polygon if it can see the entire polygon. Thus the kernel measures how much of the polygon is available to a single guard, while the convex hull measures how far the polygon is from being convex. We prove that these two losses are linked by a sharp factor of two: the area lost between a polygon and its kernel dominates twice the kernel-weighted area missing from the polygon's convex hull. In terms of Sibley's guard-point ratio $G$ and exterior ratio $E$, which we denote by $A$, the result is $G \leq A/(2-A)$, which improves Nakano's inequality $G \leq A$ whenever $A < 1$. The proof passes through a convex-body cap union. From a compact convex body $K$ and finitely many points whose convex hull contains it, we join every point to $K$ and take the union $U$ of the resulting caps. Cyclically sorting the directed boundary edges of a polygonal $U$ produces a convex companion $H$. A boundary-reversal argument gives $|H| + |U| \geq 2|\mathrm{conv}\, U|$, while a support-function identity and Minkowski's mixed-area inequality give $|U|^2 \geq |K||H|$. Inner polygonal approximation handles every positive-area compact convex $K$, while a separate null-area branch covers points, segments, and all other lower-dimensional cases. Both main theorems have machine-checked Lean 4 proofs whose final statements were audited against the informal statements after kernel checking.

We develop a rigorous mean-field theory for transformer networks that captures two large-scale limits inherent in the architecture: the number of tokens $N\to\infty$ in the input sequence and the number of attention heads $H\to\infty$ in each layer. It further considers an infinite number of layers leading to a time-continuous formulation. The resulting framework couples two interacting mean-field objects: a token distribution $\mu_t\in\mathcal P(\mathbb R^d)$, which evolves through network depth $t\in[0,T]$ according to a McKean–Vlasov transport equation, and the attention-parameter distribution $\rho_s\in\mathcal P(\Theta)$, that evolves through training time $s\ge0$ according to a Wasserstein gradient flow of the empirical risk, with optional entropic or Tikhonov regularization. We establish a comprehensive analytical foundation for the system coupling the transformer and the training dynamics; in particular, we establish global well-posedness of the resulting nonlinear Fokker–Planck system describing the coupled mean-field and training dynamics. Beyond well-posedness, we connect the mean-field formulation to optimization. For shallow, single-layer attention models, we prove exponential convergence to the entropy-regularized global optimum under a log-Sobolev condition. For genuinely deep, compositional transformers, we establish local linear convergence under a Neural Tangent Kernel non-degeneracy condition.

Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every $\alpha\geq0$ and $\gamma>0$ and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive $\alpha$ and its benefit.

It was recently discovered by Boiti–Manfrin that a variant of the Kirchhoff wave equation introduced by Pohozaev admits infinitely many conserved quantities. In this paper, we obtain explicit formulae for such conserved quantities, as well as introduce a generating function for them that is coercive. These tools are then employed to establish a priori bounds, the propagation of equicontinuity, global well-posedness in $\mathcal H^s$ for $s\geq \frac32$, and the existence of global $C_t \mathcal H^1$ solutions.

A novel dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method (DDM) is introduced for solving Maxwell's equations. The proposed method is built upon a two-channel transmission condition (TC) enforcing both tangential-field and normal-flux continuity across subdomain interfaces. Two interface channels are separately constructed from the Faraday and Ampere-Maxwell equations. Simultaneously enforcing tangential-field and normal-flux continuity reduces interface jumps by several orders of magnitude relative to the widely used Robin TC. The resulting global interface equation exhibits markedly improved iterative convergence. Numerical experiments across different partitioning strategies, subdomain counts and volumes, mesh resolutions, and geometries demonstrate drastic reductions in iteration counts, while accuracy, massive parallelism, and scalability are preserved.

Standard neural architectures often fail to generalize to longer inputs for dynamic programming (DP) targets. We investigate what makes this hard geometrically. Every finite min-plus DP is a shortest path on a DAG, which is equivalently a tropical polynomial whose extended Newton polyhedron encodes the decision boundary of which path wins. We prove these three descriptions (graph, polynomial, polyhedron) form isomorphic semirings at two levels — formal polynomials and their computed functions — connected by operations that characterize all structural redundancies. We then address the length-generalization question geometrically: does the decision boundary at length $T$ decide the boundary at $T+1$? We present two structural negatives. The semiring's two native ways to reduce dimension (setting a variable to each identity) are neither injective nor always closed within the DP. Series and parallel composition fail to construct all DAG topologies from smaller sub-DAGs, and even all terminal-only operations do not capture all DP compositions.

Let $X$ be a smooth projective surface and let $C\subset X$ be a unicuspidal rational curve with one Puiseux pair $(p,q)$. We use the log minimal model program to prove a sharp upper bound on the self-intersection of $C$ conjectured by Weimin Chen in [C, Conjecture 1.15] in the course of his study of contact structures and their symplectic fillings. We then extend this argument to rational curves with arbitrary cuspidal singularities and compare the resulting algebraic bounds with the symplectic bounds of Golla and Starkston [GS]. Finally, we prove that there is no plane curve of degree $102$ and genus $10$ with a $(36,289)$-cusp, answering a question of Evans [E, Remark 7.3.5] about algebraic curves associated with the final post-Fibonacci step of the McDuff-Schlenk staircase.

The entropy production rate (EPR) quantifies irreversibility of a nonequilibrium steady state, yet standard formulas obscure how a complex interaction network generates it. For multivariate Ornstein-Uhlenbeck dynamics on such networks, we express the EPR as a quadratic form in antisymmetric matrices measuring the nonreciprocity of aggregate directed walks at every length, and, equivalently, as two weighted-walk quantities: pairs of directed walks sharing both endpoints, and directed closed walks. For diagonalizable interactions, an exact correspondence translates these walk quantities into eigenvalues and biorthogonal eigenvector overlaps. Across dense, sparse, and deep acyclic random interactions satisfying matched-walk conditions, the mean EPR per node universally follows the square-root law $\phi_*(g)=1-\sqrt{1-g^2}$, where $g \in [0,1)$ parametrizes the interaction strength. Deep acyclic interaction matrices are nilpotent, with all eigenvalues fixed at zero for every $g$, yet, as their depth increases, their mean EPR per node approaches $\phi_*(g)$. Thus, the square-root law arises from directed walk properties, rather than from a shared spectral density or specific network topology.

In this paper, we construct a class of random measures $\mu^{\mathbf{n}}$ by infinite convolutions. Given admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{m}$ and a sequence $\bn=\{n_{k}\}_{k=1}^{\infty}$ of positive integers, for every $\bw\in \Omega$, we write $\mu^{\mathbf{n}}(\bw) = \delta_{N_{\omega_{1}}^{-n_{1}}B_{\omega_{1}}} * \delta_{N_{\omega_{1}}^{-n_{1}}N_{\omega_{2}}^{-n_{2}}B_{\omega_{2}}} * \cdots$. First, we show that the mapping $\mu^{\mathbf{n}}: (\bw, B) \mapsto \mu^{\mathbf{n}}(\bw)(B)$ is a random measure. Next, we introduce the notion of a $t$-equi-positive family, and use it to obtain a general sufficient condition under which an infinite convolution is a spectral measure and possesses a specified set of spectral eigenvalues. We then extend the concepts of spectrality and spectral eigenvalues to random measures, and show that, under the assumption that the corresponding standard infinite convolution is non-degenerate for $\mathbb{P}$-a.e.\ $\boldsymbol{\omega}\in\Omega$, the measures $\mu^{\mathbf{n}}$ are spectral random measures for $\mathbb{P}$-a.e.\ $\boldsymbol{\omega}$, admitting the spectral eigen set \[ \mathcal{E}_m=\{t\in\mathbb{N}_+:\gcd(t,N_k)=1,\,1\le k\le m\}. \] Moreover, for each such $t$, there exist uncountably many spectra $\Lambda_{\boldsymbol{\omega}}\subset\mathbb{Z}$ with $t\Lambda_{\boldsymbol{\omega}}$ also a spectrum of $\mu^{\mathbf{n}}(\boldsymbol{\omega})$. Finally, for the important case where each digit set $B_k$ is a consecutive set $\{0,1,\dots,b_k-1\}$, we completely characterise the positive integer spectral eigenvalues of $\mu^{\mathbf{n}}(\boldsymbol{\omega})$, proving that they are exactly the integers coprime to every $b_k$.

We develop a geometric framework for constructing and organizing Kochen–Specker configurations in real four-dimensional space. The construction uses finite grids on pairs of smooth quadrics related by Euclidean polarity. Their incidence geometry directly produces orthogonal measurement contexts and parity proofs of quantum contextuality, yielding infinite families of configurations and a geometric interpretation and extension of a previously known cyclic construction. For a distinguished subfamily, the same geometry admits a canonical completion determined by secant lines. The first two instances of this completion recover the exceptional root configurations of types $F_4$ and $H_4$, while the smallest case also recovers the Cabello configuration and its embedding in the Peres configuration. Thus several prominent four-dimensional contextual configurations, previously obtained from different constructions, arise from a single projective-geometric mechanism.

We prove three results about refined $SU(N)$ VW invariants on Hirzebruch surfaces. First, we crucially simplify the modular anomaly equation satisfied by the generating series of these invariants by expressing it in terms of complementary generalized error functions, which requires extending these error functions to their singular loci. Second, we evaluate the variation of the modular completions of the generating series upon change of polarization. Finally, a representation of the generating functions in terms of indefinite theta series allows us to compute VW invariants at the walls of marginal stability, and to express them in terms of the invariants on the two sides of the wall and those of lower ranks. All the results exhibit a universal structure suggesting their wider applicability.

Geometry of quantum states is a fundamental research area with applications ranging from topological materials in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are often studied, while mixed states are generally needed, e.g., to have nonzero temperature. Mixed-state geometry, however, is much more challenging and has so far lacked any structure relating distance and curvature, like the Kähler structure does for pure states. This work introduces that structure. We show that it has the physical meaning of fluctuation-dissipation—by identifying distance with fluctuations and curvature with linear response—and that it becomes the known structure for pure states, but is non-Kähler otherwise. This paints a simple picture that unifies fluctuation-dissipation and geometry of quantum states. It also applies on parameter spaces, as encountered in condensed matter or estimation theory. Our picture implies that the geometry contains all linear responses and fluctuations, proves the fluctuation-dissipation relation, explains that quantum geometry appears in any sum rule, and allows one to geometrically characterize generic transport behaviors. Closed and open systems are treated with equal ease.

Equivariant localization in the $\Omega$-background is a powerful technique for the evaluation of the partition function of $\mathcal{N}=2$ supersymmetric field theory on a toric four-manifold. For the complex projective plane $\mathbb{CP}^2$, we apply equivariant localization to supersymmetric Yang-Mills theories with gauge groups $SU(2)$ and $SO(3)$ with $N_f$ massive hypermultiplets in the fundamental representation, and the $\mathcal{N}=2^*$ theory with the hypermultiplet in the adjoint representation. We evaluate equivariant correlation functions, which provide an equivariant extension of intersection numbers of moduli spaces of instantons on $\mathbb{CP}^2$, such as Euler numbers and Segre invariants. In the non-equivariant limit, we compare our results with the evaluation using low energy field theory and integration over the $u$-plane. Using this comparison, we identify a specific overall factor, which captures the difference between a partition function for gauge group $U(2)$ and $SU(2)$ or $SO(3)$ on a compact four-manifold. We also discuss our results in the context of $S$-duality of the $N_f=4$ and $\mathcal{N}=2^*$ theory, which includes the triality automorphism group of the flavor group for $N_f=4$.

The supermoduli stack ${\mathfrak M}_g$, parametrizing smooth unpunctured super Riemann surfaces of genus $g$, is known to be non-projected for $g \ge 5$. We extend this result to the even-spin component of ${\mathfrak M}_4$. This is done by exhibiting an appropriate compact curve in the spin moduli space ${\mathcal S\mathcal M}_4^{+}$, and computing the restriction to it of the class obstructing the projectedness of ${\mathfrak M}_4^{+}$. This restriction turns out to be non-zero, proving that ${\mathfrak M}_4^{+}$ is non-projected.

Multiway data such as image collections and video is ubiquitous, but the usual approach of flattening them into matrices discards the cross-mode structure that often carries the signal. The t-product and its generalization, the $\star_{\mathbf{M}}$-product, give a matrix-mimetic tensor algebra with a tensor SVD whose truncation is optimal in the Frobenius norm, just as in the matrix case. Much real data also has internal reflective symmetry: frontal faces, manufactured parts, and leaves are all bilaterally symmetric. We define a symmetry-preserving $\star_{\mathbf{M}}$-SVD that extends the matrix symmetry-preserving SVD of Shah and Sorensen to the $\star_{\mathbf{M}}$-algebra. When a tensor's transform-domain frontal slices are reflectively symmetric, the left basis of its $\star_{\mathbf{M}}$-SVD is symmetric too, so only its top half must be stored. Bilaterally symmetric images, turned on their side and stored as lateral slices, give such a tensor. This construction keeps the symmetric part of each image and stores only half the basis. We then recognize new images by projecting them onto this basis and matching to the nearest training image in coefficient space. Across ten datasets of faces, leaves, butterfly wings, and other objects, the symmetric basis matches the recognition rate of ordinary tensor SVD at 2-11x less basis storage, with the exception of MUCT. On faces under varying illumination it exceeds the best rate the ordinary tensor SVD attains at any storage level.

We establish a uniqueness theorem in four-dimensional spacetime, demonstrating that a nonlinear spinor system naturally constrains the background geometry to the Kastor-Traschen spacetime, which describes dynamical, cosmic multi-black-holes. By analyzing a nonlinear field equation where the super-covariant derivative of a Dirac spinor is sourced by intrinsic fermion currents, we evaluate its integrability under two physical requirements: a phase-locking condition enforcing a real hermitian product $\overline{\psi}_R \psi_L$, and a pure-electric source condition. We show that the field equation uniquely compels the spinor to be an eigenvector of the super-covariant Dirac operator, with the eigenvalue dynamically identified as the Hubble constant of the de Sitter background. These results provide a novel geometric mechanism wherein both the cosmological constant and dynamical black hole configurations spontaneously emerge from fermion condensates.

Let $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}$ be a Borel subalgebra of a basic classical Lie superalgebra over $\mathbb{C}$ with $\mathfrak{h}$ a Cartan subalgebra. We give an upper bound for the index $\mathrm{ind}(\mathfrak{b})$ of $\mathfrak{b}$; in some instances this bound is $0$, in which case $\mathrm{ind}(\mathfrak{b})=0$. Additionally we prove that $\mathrm{ind}(\mathfrak{b},\mathfrak{n})=0$, which implies $\mathrm{ind}(\mathfrak{b},\mathfrak{i})=0$ for all ideals $\mathfrak{i} \subseteq \mathfrak{n}$ of $\mathfrak{b}$. We also show that $\mathrm{ind}(\mathfrak{b},\mathfrak{a}^*)=0$ for all abelian ideals $\mathfrak{a} \subseteq \mathfrak{n}$ of $\mathfrak{b}$. These results are achieved by extending the theory of strongly orthogonal roots and the Kostant cascade to the theory of generalized root systems developed by Dimitrov and Fioresi.

In this essay I reflect on the status of three of the Millennium Prize problems: the Riemann Hypothesis, P vs. NP, and the solvability of the Navier-Stokes equations.

Published experience with neural surrogates in derivative-free optimisation is contradictory: the same family of models that cuts the evaluation count of one solver leaves another unchanged, or makes it worse. We show that the contradiction dissolves once three factors are stated, and that these, rather than the fit accuracy a training curve reports, are what delimit when a learned local model pays. Role: a surrogate that proposes candidates the true objective must still approve helps, while one that replaces a gradient the solver depends on hurts. Radius: a model fitted to an optimisation path is reliable only inside a bounded neighbourhood, and its error neither vanishes as that neighbourhood shrinks nor survives its growth. Room: a surrogate can only accelerate progress the base method is still able to make. We formalise radius-aware local generalisation, relate it to the classical fully linear condition, and test each factor with the surrogate class, training pipeline and base method held fixed. Over 117 benchmark instances safeguarded assistance raises the instances solved to high accuracy from 67 to 84 while gradient replacement lowers them to 65; removing the gradient term from the training loss cuts surrogate acceptance from 0.703 to 0.148; and 1000 paired comparisons over ten noise levels show no noise threshold, only a base method that stops early. The same factors bound the gain: a model-based trust-region solver, which leaves little room, drops from 88 to 86 when the identical surrogate is attached, and released interpolation software stays ahead at 103, and on a Monte-Carlo inventory model repairing the acceptance interface is worth 10.40 cost units against 0.00 for the surrogate.

Recent advances in artificial intelligence (AI) have sparked growing interest in its use for mathematical research. While some view this as a major opportunity for discovery, others have raised concerns about its impact on traditional research practices. Despite extensive debate, empirical evidence on how AI is actually being used in mathematics remains limited. To address this gap, we collected all 32,944 arXiv submissions posted between March 1 and August 20, 2026, whose primary or secondary categories included Mathematics. We identified 3,575 submissions that explicitly disclosed author use of AI, of which 1,712 involved at least one substantive mathematical contribution. Our analysis reveals several broad patterns. First, disclosed AI use increased sharply over the study period, with substantive use growing from 1.39% of Mathematics submissions in March to 14.09% through August 20. Second, substantive AI use is highly uneven across fields: Combinatorics has the largest number of such papers, while Metric Geometry has the highest substantive-use rate. Third, substantive AI use is geographically concentrated: under weighted author counts, the United States and China together account for about two-thirds of the recognized country weight. Fourth, AI is already being applied to open research problems: among 717 named open-problem records associated with substantive use, 71% are labeled as fully resolved based on the authors' descriptions, with proofs of the conjectured statement more common than counterexamples or disproofs. Finally, AI-system use is also highly concentrated, with OpenAI systems appearing most frequently, followed by Anthropic. Together, these findings suggest that AI-assisted mathematics is expanding rapidly but remains at an early and uneven stage of adoption.

We show Lagrange and Hermite interpolation are possible using Stieltjes polynomials, linear combinations of iterated Stieltjes integrals of a constant function. We introduce divided differences via Newton interpolation and provide an explicit error formula of Peano kernel type via a new Taylor formula. Finally, we use the properties the space of Stieltjes polynomials has to recover two fundamental approximating properties: the uniqueness of the best uniform polynomial approximant and the Chebyshev equioscillation theorem.

For $N$ lattices in $\R^d$ with volume $1$ and pairwise trivial intersections, every nonzero common tiling function has support diameter $\Omega(N^{1/d})$, while for lattice families whose fundamental domains have uniformly bounded diameters, the standard convolution construction gives an $O(N)$ upper bound, leaving a gap that has remained open since the work of Kolountzakis and Wolff \cite{kolwolff-1999Mathematika}. We close this gap by constructing, for every $d\geq 2$ and all sufficiently large $N$, lattice families satisfying the same volume and intersection conditions that admit a nonnegative common tiling function with support diameter $O(N^{1/d})$, thereby also answering Question 1 of Kolountzakis and Papageorgiou \cite{kolPapageorgiou-functions-2022jfaa}. We also obtain the optimal $O(\sqrt N)$ upper bound by constructing, for any prescribed family of plane lattices whose volumes lie in a fixed bounded set independent of $N$, a pairwise trivially intersecting family with the same respective volumes and with bases arbitrarily close to suitable bases of the prescribed lattices.

The oscillation of M(x), the partial sum of the Möbius function has been in the focus of researchers in the theory of primes since the famous conjecture of Mertens in 1905 (formulated in a weaker form by Stieltjes in 1885 in a letter to Hermite). The average order of the modulus of M(x) in an interval of type [0,Y] is clearly in connection with the distribution of primes. The author proved at the beginning of 1980's that this average tends unconditionally to infinity with Y. The present work shows that (similarly to the average order of the error term of the Prime Number Theorem) this average agrees with great accuracy for large values of Y with the largest error term of the Riemann-von Mangoldt prime number formula for the value Y.

Let $\mathcal{A}$ be a finite central real hyperplane arrangement and let $\mathcal{G}(\mathcal{A})$ be its tope graph. Koizumi recently proved that the crossing-graded magnitude homology of $\mathcal{G}(\mathcal{A})$ is torsion-free and is freely indexed by face flags. We refine his result by proving that a fixed crossing vector and terminal chamber determine a summand of rank at most one. Over $\mathbb{F}_2$, we use the canonical cohomology basis to determine the crossing-graded magnitude cohomology ring of $\mathcal{G}(\mathcal{A})$.

We study the Ising model on $\mathbb{Z}^d$ with $d\geq 4$ and derive near-critical bounds on the truncated two-point function $\langle\sigma_0;\sigma_x\rangle_{\beta,h} := \langle\sigma_0\sigma_x\rangle_{\beta,h} - \langle\sigma_0\rangle_{\beta,h}\langle\sigma_x\rangle_{\beta,h}$ at parameters $\beta\leq\beta_c$ and $h\geq 0$. As a corollary, we obtain that the associated mass (or exponential decay rate) is equal to \begin{equation*} \max\bigl((\beta_c-\beta)^{1/2},h^{1/3}\bigr)^{1+o(1)}, \end{equation*} where $o(1)$ tends to $0$ as $(\beta,h)$ tends to $(\beta_c,0)$. The proof combines the corresponding result at $h=0$, recently established by Duminil-Copin and Panis, with an interpolation argument inspired by Aizenman and Fernández and carried out via the random current representation of the model.

For $N\ge2$ and $\beta>0$, consider the canonical 2D Coulomb gas ensemble \begin{equation} \mathrm{d}\mathbb{P}_{N,\beta}(Z) =\mathsf{Z}_{N,\beta}^{-1}e^{-\beta|Z|^2/2} \prod_{i<j}|z_i-z_j|^{\beta/N}\mathrm{d} Z \qquad\text{on }\mathbb{C}^N. \end{equation} The factor $N^{-1}$ in the pair exponent places the ensemble at the diffusive, or high-temperature, scale. We prove that, for every fixed inverse temperature $\beta>0$, this Gibbs measure satisfies a logarithmic Sobolev inequality (LSI) on the closed Dirichlet-form domain generated by collision-free compactly supported smooth functions, with a positive constant independent of $N$. The LSI constant may be chosen uniformly when $\beta$ ranges over a compact subset of $(0,\infty)$. A second main result establishes Poincaré and logarithmic Sobolev inequalities for planar Gaussian measures weighted by finite products of positive powers of distances to points, with constants depending only on the total variance of the underlying Gaussian measure and the total exponent, not on the number or locations of the points. To prove the full labeled Poincaré inequality, we decompose variance into permutation-invariant and label-dependent parts, controlled respectively by a weighted $\bar\partial$ estimate and by relative-coordinate conditioning with random transpositions; separately, one-site logarithmic Sobolev inequalities, a conditional entropy inequality, and a repulsive partition estimate give a defective logarithmic Sobolev inequality. Rothaus tightening combines the resulting full labeled Poincaré inequality with the defective logarithmic Sobolev inequality. The two main theorems on their stated closed form domains and all results used in their formal proofs, including those from the literature, have been formalized in Lean 4.

Let $G$ be a graph of order $n$ with the adjacency eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G) $. Let $c(v)$ denote the maximum order of a clique containing vertex $v$. We prove the vertex-localized positive square-energy inequality \[ \sqrt{s_+(G)} \leq \sum_{v\in V}\left(1-\frac1{c(v)}\right), \] where \[ s_+(G)=\sum_{\lambda_i(G)>0}\lambda_i(G)^2. \] We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.

For a hyper-Kähler fourfold $X$ of generalized Kummer type, we construct an infinitesimal deformation of the tangent bundle $T_X$ with non-zero Yoneda square. This is the first example of a stable hyperholomorphic bundle with obstructed deformations.

Let $G_q$ be an alphabet of size $q\geq2$. We determine the optimal asymptotic rate of generalized covering codes $C\subseteq G_q^n$, whose covering centers in $G_q^{t\times n}$ are constrained to the product form $C^t$. For every fixed integer $t\geq1$ and every $\rho\in[0,1]$, we prove that \[ \kappa_t(\rho,q)= \begin{cases} 1-H_{q^t}(\rho),&0\leq\rho<1-q^{-t},\\ 0,&1-q^{-t}\leq\rho\leq1, \end{cases} \] where $\kappa_t(\rho,q)$ denotes the minimum asymptotic rate $n^{-1}\log_q|C|$ among codes whose $t$-th covering radius is at most $\rho n$, and $H_{q^t}$ is the $q^t$-ary entropy function. When $q$ is a prime power, we prove that the same formula holds under the additional requirement that $C\leq\mathbb F_q^n$. Thus, both the product-form constraint and linearity are asymptotically cost-free: the resulting rate is the ordinary sphere-covering rate over an alphabet of size $q^t$. This extends the recent $t=2$ result of Elimelech and Schwartz for codes without a linearity constraint and the classical $t=1$ result of Cohen and Frankl for linear codes, thereby resolving both open problems posed by Elimelech and Schwartz. Our proofs are probabilistic and combine tools from information theory and probabilistic combinatorics, including the method of types, Janson's inequality, the second-moment method, and a structured alteration argument. Direct applications of Janson's inequality and the second-moment method are obstructed by highly dependent pairs of candidate error matrices. We overcome this obstruction by restricting the errors to a balanced exact-type class of optimal exponential size. Standard type-class estimates, together with Shearer's inequality, then give the required bounds on the number of error-matrix pairs whose selected rows have a prescribed difference.