Papers

This paper establishes an interior $C^2$ estimate for solutions of \[ \sigma_2(D^2u)+\alpha(x)\sigma_1(D^2u)=f(x,u,Du)>0, \] on the positive branch $\Delta u>0$, under the assumptions $\alpha\ge0$ and $D^2\alpha=\beta I$. Using one extra variable and a quartic correction, the equation is reduced to a quadratic Hessian equation in one higher dimension. The terms involving $D^2u$ cancel, so the interior estimate of Li–Wu \cite{LW} applies. The gradient-independent case is also treated. Finally, an example shows that an $L^\infty$ bound on $\alpha$ alone is not enough.

We provide a geometric description of Lee-Yang tensors, building upon several recent results of Bravyi, Gosset, Liu, and Wong. By applying tools from complex Perron-Frobenius theory, we generalize one of their key lemmas to bound the spectral ratio of non-normal operators with Lee-Yang radius greater than one. As a direct consequence, we provide a local-to-global principle for bounding spectral gaps of Suzuki-Fisher Hamiltonians, a family of $2$-local Hamiltonians, using only the Lee-Yang radius of each local operator to bound the gap from below. On the other hand, we develop a simple perturbative technique to show that the Gibbs states of $k$-local Hamiltonians, $k \geq 3$, cannot have Lee-Yang radius at least one for all inverse temperatures $\beta \geq 0$. We apply this same method to conclude that Suzuki-Fisher Hamiltonians are precisely those Hamiltonians whose Gibbs states are Lee-Yang at all inverse temperatures $\beta \geq 0$.

For a graph $G$, let $f_t(G)$ denote the minimum of the maximum degree of an induced subgraph with $\alpha(G)+t$ vertices, where $\alpha(G)$ is the independence number, and write $f(G)=f_1(G)$. Huang's theorem gives $f(Q_k)\ge\lceil\sqrt{k}\rceil$ for the $k$-dimensional hypercube $Q_k$. We extend this lower bound to Cartesian products of $k$ bipartite graphs with perfect matchings and prove that equality holds when the factors are connected and each has a matched bridge. In particular, we prove that $f(T_1\Box\cdots\Box T_k)=\lceil\sqrt{k}\rceil$ whenever each $T_i$ is a tree with a perfect matching. We determine the sensitivity of every Cartesian product of paths, settling the even-path case left open by Zeng and Hou [J. Graph Theory 107 (2024), 169–180]. For these tree products, with $D=\lceil\sqrt{k}\rceil$, we also prove that $f_t(T_1\Box\cdots\Box T_k)=D$ whenever $1\le t\le 2^{D-\lceil\log_2D\rceil-1}$. When $t=2$, this equality holds for all $k\ge 2$, provided that at least one factor is not $K_2$. Matching cuts give an additional exact range for products of even-order paths. Finally, we prove that $f_2(Q_k)=\lceil\sqrt{k}\rceil$ for every $k\ge 2$ with $k\not\in \{4,9\}$, whereas $f_2(Q_4)=3$ and $3\le f_2(Q_9)\le 4$.

A classical theorem by Mañé states that compact metric spaces admitting expansive homeomorphisms must be finite-dimensional. It is a natural question, raised by Morales and Sirvent, whether this dimensional constraint extends to the measure-theoretic notion of expansiveness. In this paper, we answer this question in the negative. We prove that the shift map on the Hilbert cube is expansive with respect to the product Lebesgue measure. Furthermore, we establish a general sufficient condition: any homeomorphism admitting a full-support expansive measure is densely measure-expansive. As a consequence, we show that the set of expansive and topologically stable measures for the shift map contains a dense $G_\delta$ subset, revealing a rich measure-theoretic structure despite the infinite dimensionality of the phase space.

We construct regular integral models for a class of Shimura varieties of symplectic and orthogonal type with maximal parahoric level at an odd prime $p$. These models are defined over the ring of integers of the reflex field and have special fiber with normal crossings and irreducible components of multiplicity one or two. Our construction uses explicit equivariant modifications of the corresponding canonical local models for symplectic and split even orthogonal similitude groups, with the minuscule cocharacters corresponding to maximal isotropic Grassmannians. These modifications are obtained by successively blowing up Schubert varieties in the special fiber. This proves the equivariant modification conjecture of Pappas at maximal parahoric level in the above cases.

How do algebraic relations among unit directions influence the chromatic number of the plane? We introduce monomial Cayley graphs as a framework for investigating this question, beginning with graphs whose allowed unit steps are signed powers of a single complex parameter. Transcendental parameters give bipartite graphs, while for roots of unity we determine both the chromatic and circular chromatic numbers. As a geometric consequence, every graph admitting a unit-distance embedding in the plane in which the angle between any two edge directions is a rational multiple of $\pi$ is three-colorable. For algebraic noninteger parameters, four colors suffice, and three suffice when the primitive integer minimal polynomial has leading coefficient at least three. We conjecture that three colors suffice for every algebraic noninteger direction parameter and that four suffice for every direction parameter.

Image-size characterization connects coding constraints to the probability carried by source cells and their output images. At second order, this connection must retain the joint fluctuations of all active constraints and distinguish encoder-observable source variation from unobserved channel noise. We extend the single-output posterior characterization to simultaneous posterior and conditional-composition image bounds. The bounds preserve the actual coupling of multiple outputs, quantify conditional likelihood losses for irregular cells, and admit actual-distortion marks. Two further extensions replace disjoint counting by weighted reconstruction-policy images and replace output cardinality by posterior-mass constraints for secrecy. Direct applications yield matching second-order regions for compatible multi-source helper intersections, one lossy reconstruction with several informed lossless reconstructions, Gaussian common/private descriptions, and privacy amplification from a prescribed conditional type. The Gaussian description result also gives joint rate–distortion offsets. For general helper sections, genuinely lossy informed reconstruction, Gaussian Wyner–Ziv coding, and reconciliation followed by extraction, the same image methods give explicit rate bounds with their remaining losses identified. The resulting rate corrections reflect both the image constraints and the observations available to each terminal.

Conditional density estimation describes how the entire distribution of a response changes with covariates, and in imaging studies also with location. Logistic Gaussian processes (LGP) give a flexible prior for such densities. However, the normalizing constant of an LGP has no closed form, so existing methods approximate or replace the likelihood. We propose the exact likelihood finite-rank LGP (ExFR-LGP), which writes the log density as the sum of two bivariate functions, one of the response and a location-varying linear index of the covariates, and one of the response and the location. Each function is assigned a finite-rank Gaussian process prior that is piecewise linear on a regular grid, enabling the normalizing constant, the conditional mean and the quantiles to enjoy closed form representations. Posterior samples are drawn from the exact posterior under this prior via a Gibbs sampler that updates the Gaussian components by elliptical slice sampling. When the true log density is the sum of two such bivariate functions, we show that the posterior contracts at the minimax rate of an $\alpha$-smooth bivariate density up to a logarithmic factor. Numerical experiments using synthetic data demonstrate the effectiveness of ExFR-LGP in conditional density estimation. Furthermore, application to fractional anisotropy responses along the corpus callosum in the Alzheimer's Disease Neuroimaging Initiative data yields covariate-adjusted percentile bands with uncertainty, elucidating how diagnosis changes the distribution along the tract.

A $k$-bend path is a non-self-intersecting polyline that lies on a grid and consists of at most $k+1$ axis-parallel line segments. An $\llcorner$-EPG graph is a graph whose vertices can be represented by 1-bend paths on a grid, where each path is either $\llcorner$, $\shortmid$, or $\text{-}$, such that two vertices are adjacent if and only if the corresponding paths share at least one grid edge. Characterizing chordal $\llcorner$-EPG graphs was explicitly posed as an open problem by Cameron, Chaplick, and Hoàng. We resolve this by giving two equivalent characterizations of connected chordal $\llcorner$-EPG graphs. The first assigns to every connected chordal $\llcorner$-EPG graph an ordered clique partition tree of its vertex set, where the underlying tree is the bipartite incidence graph of the horizontal and vertical edge-intersection components. The second is stated purely in terms of maximal cliques: a connected chordal graph is $\llcorner$-EPG if and only if some clique tree admits an admissible partial orientation. A reduction lemma replaces local clique labels by maximal cliques, linking the two characterizations. For a prescribed clique tree, the existence of an admissible partial orientation reduces to 2-SAT and is decided in $O(nM^2)$ time, where $n$ and $M$ are the numbers of vertices and maximal cliques. Finally, we show that strong chordality is not sufficient: we exhibit a strongly chordal graph that is a minimal forbidden induced subgraph for $\llcorner$-EPG, and a family of split graphs with a unique clique tree for which membership in $\llcorner$-EPG reduces to a neighborhood condition.

Restorative control seeks lasting recovery through an intervention that ends in finite time. We formulate this objective as a reach-avoid problem in which the input must bring the system safely into a desired basin of attraction before withdrawal. Two candidates share bistable baseline dynamics but accumulate irreversible damage through different allocations of drive-generated exposure. We derive a lower bound on the exposure required for basin transfer and a sufficient construction for safe restoration. When the lower bound exceeds the summed tolerances, every common open-loop input that restores either model causes failure under at least one. With sufficient tolerance, proportional allocation restores both without identification. For feedback from output samples with Gaussian noise, we bound safe-restoration probability in terms of failure risk under the alternative model. A numerical example illustrates the certificates under two sensing regimes.

In this paper, we prove that $\mathbb{Q}$-Gorenstein projective complex varieties admitting int-amplified endomorphisms are of Calabi–Yau type.

Thermo-mechanical fracture couples transient heat conduction on a cracked domain with a crack that grows as the temperature and the displacement evolve. Neural energy solvers have been proposed for phase-field fracture and later extended to represent a sharp crack through the network input, but heat conduction on the cracked domain and crack propagation under the resulting thermal stresses have not yet been treated together in these solvers. We present an extended deep energy method for thermo-mechanical crack propagation in which the crack remains a sharp polyline. Two networks represent the temperature and the displacement and receive the crack through a scalar embedding function, discontinuous across the crack and smooth elsewhere, so that both fields can jump across it without a regularization length, and the displacement is enriched near the tip by the Williams expansion with trainable amplitudes. The two fields are obtained by minimizing an incremental conduction functional and the thermoelastic potential energy in a staggered sequence, with Monte Carlo integration on points stratified over background elements, densified near the tip and redrawn during training. The stress intensity factors are extracted by the interaction integral with the area term of Wilson and Yu and checked by a sweep of the contour radius, and the crack advances at the maximum hoop stress angle when the energy release rate of the kink reaches the critical value at the crack-tip temperature. On a stationary thermal edge crack the extracted stress intensity factor agrees with the published value to 0.11%, in a functionally graded shear test initiation agrees with an independent sharp-crack finite element solution to within one load step, and on a notched cruciform specimen the crack paths follow the published solutions under mechanical, thermal and combined loading.

$\mathbb{Z}_p$-folded surfaces are a class of oriented folded surfaces with a fundamental $H_2(\mathbb{Z}/p\mathbb{Z})$ class. In this paper we study the $\mathbb{Z}_p$-folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the $\mathbb{Z}_p$-Thurston norm, which generalises the $\mathbb{Z}_2$-Thurston norm of Jaco, Rubinstein and Tillmann and adapts Turaev's definition of $\theta$. We define canonical $\mathbb{Z}_p$-folded surfaces, which connect the theory of $\mathbb{Z}_p$-folded surfaces to the study of one-vertex $3$-manifold triangulations, and we relate their Euler characteristic to the $\mathbb{Z}_p$-Thurston norm. We generalise a classical result of Bredon and Wood by proving that the maximum Euler characteristic of a non-separating non-empty $\mathbb{Z}_3$-folded surface embedded in $L(3n,1)$ is $2-n$ for $n\in\mathbb{Z}^{>0}$, determining the $\mathbb{Z}_3$-Thurston norm in these cases. We conclude by demonstrating the motivating application of this theory to the study of generalised triangulations by proving that any generalised triangulation of the lens space $L(3n,1)$ must have at least $n$ tetrahedra. This final result is a preview of joint work with Spreer in which we determine the exact complexity of $L(3n,1)$ using more sophisticated techniques.

Rydberg-atom quantum uniform linear arrays (RAQ-ULAs) offer a promising sensing architecture for direction-of-arrival (DOA) estimation in terahertz (THz) beam alignment. Existing studies often model the quantum receiver as a macroscopic linear block and rely on the Cramer-Rao bound (CRB) for array evaluation or power allocation. However, as a local variance bound, the CRB cannot capture threshold breakdown caused by spatial ambiguities in low-signal-to-noise-ratio (SNR) regimes. This paper proposes a physics-aware power optimization framework for RAQ-ULAs based on stochastic Barankin bound minimization. We derive a closed-form stochastic multipoint Barankin bound (BRB) matrix under a low-SNR integrability condition and introduce an arcsine-prior calibration to characterize bounded-domain error saturation. By incorporating a Lindblad-guided electromagnetically induced transparency (EIT) readout model, we couple the ambiguity-sensitive BRB with photon shot noise, power broadening, and laser Rabi frequencies. A CRB-optimized analytical baseline is further derived to expose the limitation of local-SNR-based allocation. To solve the resulting nonconvex problem, we develop a backtracking majorization-minimization (MM) algorithm with a Lipschitz-based quadratic surrogate. The algorithm ensures monotonic decrease and converges to a feasible stationary point. Simulations show that the proposed framework predicts threshold breakdown more accurately than the CRB and enlarges the reliable operating region under severe THz attenuation.

Cancer is driven by an estimated two to nine gene mutations, known as multi-hit combinations. Due to the large number of genes, identifying these multi-hit combinations presents a computationally challenging problem, which has previously required supercomputers to enumerate all or most gene combinations. Recent work formulated this classification task as the Multi-Hit Cancer Driver Set Cover Problem (MHCDSCP) and solved it via column generation on a single commodity CPU, where a pricing problem identifies promising gene combinations (arXiv:2602.22551). The main bottleneck is the pricing problem that has to be solved repeatedly. To solve the pricing problem to optimality, we propose a dedicated depth first search method that enumerates all possible gene combinations. Although enumerating gene combinations with depth first search have been performed on supercomputers with around 150,000 CPU cores (arXiv:2603.16721), our depth first search algorithm solves the pricing problem to optimality within seconds on a single CPU by leveraging dual information to prune large parts of the search space. By eliminating this bottleneck, our approach yields the first exact method for the MHCDSCP, which is guaranteed to find a global optimal solution.

Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X)^4\le648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X)^4\le7332$. Under the additional assumption that a primitive Weil polarization is Cartier outside finitely many points, the latter bound improves to the sharp bound $1024$; if its non-Cartier locus has dimension at most one, we obtain $6084$.

We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov–Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets $A,B\subset\mathbb{R}^d$ of positive measure with finite upper codimension-one Minkowski boundary content, the number $\Lambda_\varepsilon(cA,B)$ of eigenvalues in $(\varepsilon,1-\varepsilon)$ satisfies $\Lambda_\varepsilon(cA,B)\le C(d,A,B,\beta)c^{d-1}D(1+\log_+(c/D))$, where $D=\log(1/(\varepsilon(1-\varepsilon)))$, for every fixed $\beta>0$, $c\ge2$, $0<\varepsilon<1/2$, and $D\le\beta c$, with $\log_+t=\max(0,\log t)$. The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant $d-1$ for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For $C^1$ convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is $2$. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.

In this paper, we establish an $L^2$ maximal inequality for partial sums of Gaussian-subordinated random variables. More precisely, let $f:\mathbb{R}\to\mathbb{R}$ belong to $L^2(\gamma)$ and have Hermite rank at least $d$, where $\gamma$ denotes the standard Gaussian measure. If the covariance matrix of the underlying Gaussian family, with standard normal marginals, satisfies a uniform $d$th-power row bound $K$, then the $L^2$ norm of the maximal partial sum is bounded by $C_{d,K}\sqrt n\,\|f\|_{L^2(\gamma)}$. The constant $C_{d,K}$ is independent of $(f,n)$ and depends on the covariance structure only through $K$. As a consequence, our $L^2$ maximal inequality controls Hermite tails uniformly in the path norm. In discrete time, this yields the Breuer–Major–Donsker principle under the sole finite-variance assumption, removing both the additional $L^{2+}$-integrability assumption of Nourdin and Nualart (Probab. Theory Related Fields, 2020) and the prediction-theoretic or decimation assumptions of Mansanarez, Poly, and Zheng (arXiv:2607.11469). In continuous time, we likewise obtain the Breuer–Major–Donsker principle under finite variance alone, removing the additional moment assumption of Campese, Nourdin, and Nualart (Ann. Probab., 2020). For the nonstationary self-similar setting considered there, we remove the same extra moment assumption in the subcritical regime, while a separate leading-chaos argument yields the corresponding logarithmically normalized critical limit. [Abstract shorten to meet arXiv requirement]

Optimal control of open quantum systems governed by the Lindblad master equation requires the repeated solution of forward and adjoint evolution equations. For large Hilbert spaces, these computations demand numerical methods that are structure-preserving, accurate, and computationally efficient, while remaining compatible with nonsmooth optimization techniques. In this work, a unified framework for optimal control of Lindblad equations is presented. The continuous optimality system is derived from the Pontryagin maximum principle and solved by a sequential quadratic Hamiltonian (SQH) method. Its numerical realization is based on second-order exponential midpoint propagators for the forward and adjoint Lindblad equations with time-dependent Hamiltonians. Full-rank schemes preserving the Hermitian and positive-semidefinite structure of the density matrix are developed together with low-rank formulations that substantially reduce storage requirements and computational cost. Rigorous error estimates are established for the full- and low-rank forward and adjoint propagators. Numerical experiments confirm the predicted convergence rates, demonstrate the effectiveness of the low-rank approximations, and illustrate the performance of the proposed FREM-SQH and LREM-SQH algorithms for optimal control problems with smooth and nonsmooth control costs.

We prove Z-stability for all simple, separable, pure, nuclear C*-algebras. Combining this with very recent work of Thiel completes the proof of the Toms-Winter conjecture. We do this through the framework of II_1 factorial tracially complete C*-algebras. We characterise regularity for these algebras in various terms. In particular they are regular if and only if they have real rank zero and the Murray-von Neumann equivalence classes of projections correspond to continuous affine functions from the designated traces to [0,1]. As an application of our analysis of ultrapowers of tracial completions, we show simple pure C*-algebras whose quasitraces are traces are K-stable. In the amenable setting we give a local quantisation result, a la Popa, for II_1 factorial amenable tracially complete C*-algebras, which obtains hyperfiniteness from regularity conditions. Our Z-stability results follow from this together with a theorem of Matui and Sato. An initial version of an essential lemma in the Bauer trace simplex case was obtained from ChatGPT Astra.

We study the uniqueness of optimal $1$-embeddings on closed surfaces up to weak equivalence. Answering a question of Suzuki in the negative, we show that for every closed surface other than the sphere, the projective plane and the Klein bottle, there exists an optimal $1$-embedded graph admitting two weakly inequivalent optimal $1$-embeddings. Our construction is given explicitly on the torus and is then extended to other surfaces through a suitable connected-sum construction that preserves weak inequivalence. As a consequence, we also obtain optimal $1$-embedded graphs with exponentially many pairwise weakly inequivalent optimal $1$-embeddings as the genus increases.

In our previous papers [33, 34], we established the affine Bernstein theorem and proved the interior regularity for the affine Plateau problem in two dimensions. In this paper, we prove the interior regularity for the affine Plateau problem in all dimensions and present a smooth counterexample to the affine Bernstein problem in dimensions ten and higher. We also prove an affine Bernstein theorem in all dimensions under a growth condition.

Let $G_{n,d}$ be a uniformly random simple $d$-regular graph on $n$ vertices, and let $A_n$ be its adjacency matrix. For every fixed $\lambda\in(0,1/2)$, we prove that for any $\lambda(n-1)\le d\le(1-\lambda)(n-1)$, $\mathbb P(A_n\text{ is singular})\le e^{-cn}$, where $c>0$ depends only on $\lambda$, and $n$ is sufficiently large with $nd$ even.

Quasi-nonsingular relative orbital elements (qnsROE) are among the most widely used state representations for formation flying and rendezvous in near-circular inclined orbits. They admit closed-form perturbed state transition matrices (STMs) and a direct geometric mapping, which underpins eccentricity/inclination (E/I)-vector separation for passive safety. Existing eccentric extensions of the qnsROE resolve this geometry in the radial–tangential–normal frame, where a residual semimajor-axis difference makes the E/I separation margin phase dependent. This paper introduces eccentric ROE that reduce to the qnsROE as \(e\to0\) and whose first-order geometry is resolved in the velocity–normal–binormal (VNB) frame, where the secular drift is confined to the velocity direction. Their first-order position and velocity maps yield a closed-form, phase-independent binormal–normal separation bound with an explicit margin for semimajor-axis residuals. Closed-form Keplerian and secular-\(J_2\) STMs and a decoupled impulsive control input matrix are derived for the same elements, and the elements are recovered as the physically normalized integration constants of the Cartesian equations of relative motion in the VNB frame. Together, these results provide a unified ROE framework for formation flying in general elliptic orbits.

We establish sharp local-in-time dispersive estimates and global Strichartz estimates for the semiclassical Schrödinger equation in a three-dimensional cylindrical domain $\Omega\subset\mathbb{R}^{3}$ with homogeneous Neumann boundary conditions. The partially flat boundary leads to a degeneracy of curvature in the longitudinal direction and produces persistent glancing interactions that distinguish the cylindrical setting from strictly convex domains. At the spectral level, the Neumann condition is reflected in the appearance of the zeros of the Airy derivative in place of the Airy zeros associated with Dirichlet boundary conditions. Using semiclassical microlocal analysis, oscillatory integral estimates, and an Airy–Poisson summation formula adapted to Neumann reflection, we obtain uniform dispersive bounds for the localized Schrödinger propagator. The quadratic structure of the Schrödinger phase yields a uniform treatment of the low-frequency regime and, combined with the boundary-layer analysis, leads to global Strichartz estimates with derivative loss $$ \rho(q)=\frac{3}{2}\left(\frac12-\frac1q\right). $$ As an application, we prove local well-posedness in $H^{s}(\Omega)$, $s>1$, for the cubic nonlinear Schrödinger equation with either focusing or defocusing sign and homogeneous Neumann boundary conditions. These results quantify the effect of partially flat boundary geometry on Schrödinger dispersion and provide a sharp linear framework for the corresponding nonlinear dynamics.

We develop an end-to-end framework for estimating canonical thermal properties of particle-number-conserving second-quantized Hamiltonians using quantum simulations. The framework integrates initial state preparation with mixed Dicke states, particle-number-preserving Trotterized simulation, and generalized quantum signal processing to estimate canonical partition functions and observables. Building on this canonical scheme, we introduce a grand-canonical reconstruction scheme in which the same quantum measurement data can be reused for arbitrary chemical potentials through classical reweighting, thereby avoiding repeated quantum circuit executions as the chemical potential is varied. We prove the existence of system-size-independent temperature regimes in which both the canonical and reconstructed grand-canonical schemes have lower asymptotic time complexity than classical Lanczos methods, up to polynomial factors, for physically relevant Hamiltonians. We numerically validate our theoretical results for the two-dimensional Hubbard model in both weak- and intermediate-coupling regimes.

Let $p\ge 7$ and suppose that $p$ and $p-2$ are prime. We study $E_p:y^2=x(x-2)(x-p)$ using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give $2^\infty$-Selmer corank one for $p\equiv 3,5\pmod{8}$ and corank zero for $p\equiv 7\pmod{8}$. Assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release, we deduce equality of the analytic and algebraic ranks in these cases, finiteness of the full Tate-Shafarevich group, and the exact BSD leading-term formula. The $2$-primary Tate-Shafarevich group is trivial, and the remaining factor has odd square order. We identify precisely the obstruction left in the class $p\equiv 1\pmod{8}$. An appendix gives a local descent proof of the known Selmer dimensions in the coordinates used here.

Motivated by an observation concerning the $S_3$-conjecture, we prove that every abelian composition factor of a finite rational group has order $2$, $3$, or $5$. Thompson previously restricted these orders to $\{2,3,5,7,11\}$; we exclude $7$ and $11$, establishing the bound that he proposed. Together with the work of Feit and Seitz on nonabelian composition factors, this completes the determination of the possible composition factors of finite rational groups.

Network coding lets the nodes of a network combine messages rather than merely forward them. In undirected networks, where the two directions of an edge share its capacity, Li and Li conjectured in 2004 that coding offers no advantage over fractional routing; the conjecture has been confirmed for many special classes but never settled in general. In this paper, we disprove it by constructing a finite connected simple undirected network with unit shared edge capacities, maximum degree three, and distinct leaf terminals, on which a binary linear block code achieves a common rate strictly above the maximum fractional routing rate. The construction turns a short completion-time code into a reversible circuit whose registers all carry independent messages, and a temporal metric on its wires yields the strict routing bound. The underlying integer-coefficient construction works over every finite field and every nontrivial finite abelian group; for deterministic fixed-schedule codes the common-rate supremum is one and the closure of the rate region is the unit cube. Amplifying the gap with a degree-preserving tensor construction and an even subdivision gives, for every $0<\beta<1$, an unbounded family of connected, simple, subcubic, bipartite networks with girth at least $n^\beta$ on which coding approaches rate one while routing is bounded by $K_\beta/(\log n)^c$, with one positive exponent $c$ independent of $\beta$. The finite counterexample and the family theorem are formalized in Lean.

Kühn and Osthus proved that, for every positive integer $\ell$, every $2^{16}\ell^2$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\geq \ell$ for every $v\in S$. And they asked whether the quadratic bound can be replaced by a linear bound. In this paper, we answer this question in the affirmative by proving that every $641\ell$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\ge \ell$ for every $v\in S$. The proof integrates the Moser-Tardos resampling algorithm.