Papers

Let $\lambda>-1/2,\neq0$. Consider the Bessel operators \begin{align*} \Delta_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}\frac{d}{dx} \end{align*} on $\mathbb{R}_+$. In this paper, we first establish several characterizations for $\widetilde{A}_{\infty,\lambda}$ weights introduced by Li et al. (Math. Z. 309(1) (2025), Paper No. 13, 29pp), which are different from the previous $A_p$ class with respect to the measure $x^{2\lambda}dx$. Among these characterizations, we obtain the sharp reverse Hölder inequality (RHI) for $\widetilde{A}_{\infty,\lambda}$ weights. As applications of RHI, we derive a precise open property of $\widetilde{A}_{p,\lambda}$ weights, and obtain a sharp weighted bounds for the Hardy-Littlewood maximal operator $\mathcal{M}_\lambda$ associated to $\Delta_\lambda$, which was missing in the previous papers. Next, we also establish several extrapolation theorems related to $\widetilde{A}_{p,\lambda}$ weights. Furthermore, we characterize the restricted weighted weak type estimates for $\mathcal{M}_\lambda$ via a new class of weights $\widetilde{A}_{p,\lambda}^{\mathcal{R}}$, which is strictly larger than $\widetilde{A}_{p,\lambda}$. Meanwhile, we obtain quantitative restricted weighted weak type $(p,p)$ bounds for sparse operators. As applications, we derive various kinds of weighted inequalities for Riesz transforms, maximal operators, square functions, variation operators and their commutators associated to $\Delta_\lambda$.

We prove that, for any number field K and and d at least 2, there are at most finitely many monic, centred, post-critically finite polynomials of degree d with coefficients in the extension of K by all roots of unity.

We study closed-set stabilizers in topological full groups of minimal effective ample groupoids with Cantor unit space. For almost finite or purely infinite groupoids, we extend the maximal stabilizer theorem of Grigorchuk and Vorobets to every subgroup containing the commutator subgroup. For a nonempty closed nowhere dense set, maximality of its stabilizer is equivalent to minimality of its stabilizer action, minimality of the regular reduction, and primeness of a canonical wide open subgroupoid. Without assuming almost finiteness or pure infiniteness, we prove spatial rigidity for stabilizers acting minimally on infinite closed nowhere dense sets. Their isomorphism types determine the ambient Kakutani equivalence class and, when Hausdorff, the effective reduction. We also realize every AF groupoid with compact unit space as the full restriction, with its relative topology, of any minimal effective ample groupoid with Cantor unit space to a closed nowhere dense subset. Combining rigidity and realization yields continuum many pairwise nonisomorphic stabilizers in every subgroup containing the commutator subgroup; these stabilizers are maximal when the ambient groupoid is almost finite or purely infinite. The realization also gives explicit AF subquotients of the ambient reduced groupoid $C^*$-algebra, including the CAR algebra, with compatibility on the canonical diagonal subalgebras.

Code conversion changes the parameters of encoded data while avoiding complete decoding and re-encoding. Existing general bounds largely rely on the algebraic structure of linear codes. This paper develops an information-theoretic framework for nonlinear code conversion in the merge regime. Using the entropy/length profile (ELP), we derive upper bounds on the number of unchanged symbols and lower bounds on the number of read symbols for arbitrary $(n,M)_q$ codes, without assuming linearity or requiring $\log_q M$ to be an integer. We also obtain joint bounds for multiple initial codes and consequences involving minimum distance. For linear codes, the proposed bounds reduce to the corresponding generalized Hamming weight (GHW) bounds. The construction based on Hamming codes converts Hamming codes of redundancy two into a Hamming code of any larger redundancy and simultaneously attains the general bounds on unchanged and read symbols. The construction based on quasigroups shows that new bounds remain tight for genuinely nonlinear codes of nonintegral dimension and at a nonintegral entropy threshold, which, to the best of our knowledge, is the first construction of convertible codes for nonlinear codes.

We study the numerical radius \(\omega(T)\) of a bounded linear operator \(T\) on a complex Hilbert space. Write \(T=A+iB\), where \(A\) and \(B\) are the real and imaginary parts of \(T\). A central result is the exact formula \[ \omega(T)=\frac{\pi}{2}\sup_{\|x\|=1}\int_0^1 \left|\left\langle (A\cos \pi t+B\sin \pi t)x,x\right\rangle\right|\,dt . \] For every \(n\in\mathbb N\), we also prove the power form \[ \omega^{2n}(T)=\frac{4^n}{\binom{2n}{n}} \sup_{\|x\|=1}\int_0^1 \left\langle (A\cos \pi t+B\sin \pi t)x,x\right\rangle^{2n}\,dt . \] This yields sharp upper bounds; for \(n=1\), together with the matching integral lower bound, it recovers Kittaneh's classical two-sided inequality \[ \frac14\|T^*T+TT^*\|\le \omega^2(T)\le \frac12\|T^*T+TT^*\|. \] Furthermore, we obtain exact numerical radius formulas for some block operator matrices, including a \(3\times3\) nilpotent matrix and longer alternating paths. The factor \(\pi/2\) is best possible.

Hoffman constants quantify the distance to the solution set of a polyhedral system in terms of its residual; they control error bounds, geometric conditioning, the convergence rate of first-order iterative algorithms, and the stability of regression estimators to target perturbations. Despite their importance, explicit values of Hoffman constants are rarely available due to their computational complexity. In this paper, we derive explicit bounds on the Hoffman constant for a class of structured polyhedral systems arising in sparse regularization. We show that the constants admit a decomposition into spectral, geometric, and interaction components, which together determine their scaling behavior. This framework explains the gap between worst-case dimension-dependent bounds and the favorable conditioning observed empirically in structured problems. As a primary application, we analyze $\ell_1$-regularized least squares, demonstrating that the effective constants depend on low-dimensional active subspaces rather than the ambient dimension. We then extend to the elastic net and adaptive $\ell_1$-regularized formulations, demonstrating how the underlying optimization geometry is altered by the $\ell_2$ penalty and reweighting scheme. Furthermore, we relate our results to classical statistical assumptions and show that the restricted eigenvalue condition bounds only the primal curvature of the active set, leaving the dual interaction geometry unconstrained and is thus insufficient alone for fast theoretical convergence rates. We further show standard $\ell_\infty$ coherence conditions do not yield dimension-independent Euclidean bounds to close this gap. Our theoretical results show that Hoffman constants admit explicit, interpretable forms in structured regression settings and, in practice, provide actionable tools for post-hoc feature pruning and a priori optimal hyperparameter selection.

Consider an $n$-dimensional sum of the form $S_k=\sum{q^{f({\bf z})}}$ over lattice points ${\bf z}$ which lie in $K{\mathcal P}$, the $K$ scaling of an $n$-dimensional rational polytope $\mathcal P$. We formulate a conjecture on the structure of $S_K$ considered for $K$ prime, as a power series in $h=q-1$ modulo the $K^{th}$ cyclotomic polynomial, and in particular the existence of a decomposition in powers of the Gauss sum $\mathbb G$, up to the $n^{th}$, whose coefficients are almost independent of $K$, where the leading coefficient is related to the volume of $\mathcal P$ and the coefficient of ${\mathbb G}^r$ has contributions from the codimension $r$ skeleton of $\mathcal P$. The conjecture is proved for $n=1,2$.

We give a single counterexample to Conjectures 1–3 as stated in Luo's paper: a minimal two-tetrahedron triangulation of the lens space L(3,1) with a null-homotopic loop edge. The proof uses quadrilateral matching equations, an explicit calculation of shape parameters, and known results on volume maximization.

We give a self-contained derivation of the closed form of the power sums $S_m(n)=\sum_{i=1}^{n}(a_0+(i-1)d)^m$ in which the Bernoulli numbers enter as constants of integration rather than as prescribed coefficients. The derivation rests on an elementary fact: a sum whose terms are polynomial in the index is the restriction to the integers of a unique polynomial, which satisfies the defining difference equation identically and may therefore be differentiated and integrated exactly. Differentiating lowers the power by one; integrating back introduces a single constant, equal to the slope $S_m'(0)$ of the polynomial at the origin and fixed by the value $S_m(1)=a_0^m$. This yields a closed form for $S_m(n)$ in terms of the constants and a recurrence that determines them, obtained without induction on the exponent and without generating functions. With the Bernoulli numbers defined by their classical recurrence, we identify the constants as $d^mB_m(a_0/d)$; the progression starting at $0$ reproduces that recurrence exactly. In the normalized case they are $B_m(1)$, so $C_1=B_1(1)=+\tfrac12$, a value that reflects the summation range; a reflection identity shows, without generating functions, that the odd constants beyond the first vanish. The coefficients of the recurrence are scaled binomial coefficients from a single row of Pascal's triangle, and the same approach gives closed forms for sequences with polynomial differences. The integral recursion behind the method is known in the normalized case; each result is related to its classical counterpart, and a verified implementation is provided.

In this paper, we study hypersurfaces $M^4$ in the Minkowski space $\mathbb{E}^5_1$ whose shape operator $S$ is non-diagonalizable, having a triple real eigenvalue $\lambda$ with a one-dimensional eigenspace together with a simple eigenvalue $\mu$ (Case~(iii) in the classification of Magid). Assuming that $\lambda$ and $\mu$ satisfy a functional relation $\lambda=f(\mu)$ and that the eigenvector field of $\mu$ is geodesic, we show that $\lambda$ and $\mu$, as well as the pseudo-orthonormal frame, are intrinsic invariants of $M$, determined by the induced metric up to sign. As a consequence, we prove that such hypersurfaces are unique up to isometries of $\mathbb{E}^5_1$. We also obtain a classification of the warped product hypersurfaces of this type.

In 2012, D. Karp studied a family of polynomials constructed from Polya frequency sequences and conjectured that all their zeros are real and negative. We prove this conjecture for positive sequences and establish further results concerning the location of these zeros.

The Diophantine equation $F_n^x+F_m^y=2^a,$ where $F_k$ denotes the $k$-th Fibonacci number and $n,m,a\in \mathbb{Z}_{\geq0}$, has attracted attention in recent years. The cases $(x,y)=(1,1)$ and $(x,y)=(2,2)$ were completely resolved by Bravo and Luca, and by Jeong and Park, respectively. In this note, we investigate the mixed exponent case $(x,y)=(2,1)$ and we completely determine all nonnegative integer solutions.

We study the uniqueness of bound states for the following scalar field equation $$ -\Delta u+u=-\varepsilon |u|^{q-1}u+|u|^{p-1}u \,\ \ \mbox{in}\,\ \mathbb R^N, \,\ \varepsilon>0,\tag*{$(*)$} $$ where $ N\ge3,\ 1<p<(N+2)/(N-2)$ and $q>1$ satisfies $q \neq p$. For each integer $k \geq 1$, we prove that the equation $(*)$ has, up to translation and reflection, a unique bound state with precisely $k$ nodes in a natural coupling regime, where $\varepsilon>0$ is sufficiently small. Our results address a conjecture of [H. Berestycki and P. L. Lions, ARMA, 1983][M. X. Tang, Invent. Math., 2026] on the uniqueness of fixed-node bound states for the equation $(*)$.

The 1-Meixner random vectors consist of $d$ real-valued random variables, having finite moments of all orders, for which the commutators between their joint semi-quantum operators and the multiplication operators that they generate are linear combinations of these multiplication operators and the identity operator. This property of random variables imposes a system of first-order partial differential equations on their joint Laplace transform on a neighborhood of $0$ in a Euclidean $d$-space. We provide explicit descriptions of all solutions of the system just mentioned in every dimension $d$.

Recent observations that deep learning models train at the edge of stability have renewed interest in understanding stability properties of classical optimization algorithms. For quadratic objectives, standard gradient descent has a sharp stability threshold: it converges from every initialization when $\eta<2/L$, diverges from almost every initialization when $\eta>2/L$, and diverges from every nonzero initialization when $\eta>2/\mu$. For non-Euclidean gradient descent, the corresponding known guarantees are weaker: convergence holds for every initialization when $\eta<2/S$, where $S$ is the generalized sharpness, while every nonzero initialization diverges when $\eta>2/\mu$. The behavior in the intermediate regime $2/S<\eta<2/\mu$ is not fully characterized. We show that Hilbertian norms are the only ones for which $2/S$ is an almost-everywhere sharp threshold for every positive-definite quadratic objective. Using linear conjugacy between non-Euclidean and Euclidean gradient descent, we show that for general norms the intermediate regime can contain arbitrarily many edges of stability associated with distinct attracting directions. We identify a forward-invariant region of exponentially divergent directions beyond the generalized stability threshold. When the directions of minimum curvature are finite and nondegenerate, we prove that, sufficiently close to $2/\mu$ from below, only their invariant directions remain convergent. We give smooth two-dimensional constructions with arbitrarily many distinct thresholds, arbitrarily large relative threshold separation and show these properties to be stable.

We develop an adaptive Galerkin neural-network method for computing minimum-$L^2$-norm interior controls for wave equations with spatially varying coefficients. Within the Hilbert Uniqueness Method, adjoint waves are approximated directly in space-time by least-squares Galerkin projections onto trial spaces enriched by neural-network basis functions selected through residual-based training. Linear Galerkin solves determine the adjoint approximations, and a separate linear system reconstructs the control. The least-squares objective provides an indicator for adaptive enrichment and stopping; the experiments use a Ritz-based terminal-data version whose continuous-error interpretation additionally requires an eigenpair consistency estimate. The main difficulty is that adjoint errors perturb this reconstruction system. Assuming observability of the continuous equation, we derive sufficient stability conditions involving a worst-case factor $\sqrt{n}$ multiplying the maximal adjoint error, where $n$ is the number of retained terminal modes. A fully discrete error estimate separates terminal-space truncation, adjoint approximation, and assembly errors, yielding convergence to the minimum-norm control when the truncation error, the $\sqrt{n}$-weighted adjoint error, and the assembly errors tend to zero as $n$ grows. For spectral terminal bases, additional regularity and uniform contraction assumptions yield quantitative convergence rates and show that $O(\log n)$ accepted enrichment steps per mode suffice to balance the neural and truncation errors. Numerical experiments on a nonconvex Cassini oval, a square containing a low-speed lens, and a three-dimensional ball with a graded coefficient illustrate the method.

Non-intrusive reduced-order modeling is essential for systems simulated with black- or gray-box solvers, yet computing accurate data-driven models remains challenging when the dynamics exhibit large-amplitude transient energy growth. This is because state-of-the-art non-intrusive methods typically rely on orthogonal projection onto high-energy linear or nonlinear manifolds identified by minimizing static reconstruction losses, which truncates low-energy, yet dynamically important, states. Intrusive methods such as Balanced Truncation and its extensions address this issue by using adjoint information to define dynamics-aware oblique projections. Here, we show that balancing-based oblique projections for nonlinear systems can instead be computed non-intrusively from forward simulation data. Specifically, we develop a framework to compute the nonlinear empirical reachability Gramian and balance it against its observability counterpart, projected onto a suitable restriction subspace. This procedure provably recovers the oblique projection of Balanced Truncation when the dynamics are linear and the range of either Gramian is contained in the restriction subspace. Moreover, in the linear case and in the absence of adjoint information, we show that the minimax-optimal restriction subspace spans the dominant eigenspaces of the reachability Gramian, which informs an efficient balancing algorithm with cost comparable to a standard SVD. Once the oblique projection is computed, we use Operator Inference to learn the latent-space dynamics on its range. The method is demonstrated on several examples of increasing complexity, including the three-dimensional incompressible flow past a wing with $8\times 10^7$ degrees of freedom. In all examples, the Balanced Operator Inference models significantly outperform their POD-based counterparts, reducing forecasting error by up to two orders of magnitude.

We construct a sentence of $L_{\omega_1,\omega}$ having exactly $\aleph_1$ many countable models up to isomorphism, but no perfect set of non-isomorphic models. This disproves Vaught's conjecture for infinitary logic. The construction follows a framework devised by Robin Knight, and the proof has been formally verified in Lean.

We study the long-time behavior of the infinite-energy solutions to the white-forced Navier–Stokes equation in an infinite pipe $\mathbb{R}\times (-1,1)$ with no-slip boundary conditions. The equation is globally well-posed in uniformly local spaces satisfying the divergence-free condition. Under suitable degeneracy and spatial decay assumptions on the noise, we prove the uniqueness of the invariant measure and establish polynomial mixing in the dual-Lipschitz metric. This provides an affirmative answer to Open Problem 1 posed by Kuksin and Shirikyan \cite{KS17} in the case of zero cross-sectional flux.

We prove Krzyż's conjecture for the sixth coefficient. Every holomorphic function $f(z)=\sum a_kz^k$ from the unit disk to the punctured disk satisfies $|a_6| \le 2/e$. Equality holds precisely for rotations of $\exp(-(1-z^6)/(1+z^6))$. Every extremal has a Herglotz representation with at most six atoms. By the reduction of Agler and McCarthy and the entropy theorem of Lei and Zhang, it suffices to exclude extremals with fewer than six atoms. We do this by a rigorous computer-assisted branch-and-bound, leaving the equality case entirely to the analytic argument. The main ingredients are explicit first-order conditions for extremals with fewer than the maximal number of atoms, which reduce to Chebyshev relations when one atom is missing, and a bound for the total mass obtained by convexity of the Poisson integral. The same method independently establishes the full-degree property for the known cases $n=3,4,5$.

We study stochastic convex optimization, where each sampled function can be evaluated at two arbitrary points. Our accelerated method chooses step sizes, batch sizes and smoothing radii from function values. A forward difference between smoothing scales controls the radius; we account for changes in the smoothed objective. Under the stated regularity and second-moment assumptions, exact function evaluations give \begin{multline*} \mathbb{E}\bigl[f(x_N)-f(x^\star)\bigr] \le \widetilde{O}\biggl( \frac{\sqrt{d}\,M_2 D}{\sqrt{T}} +\min\biggl( \frac{L D^2}{N^2},\; \frac{d^{1/4} M D}{N}+\frac{\sqrt{d}\,M D^2}{\gamma_{\max} N^2} \biggr)\biggr). \end{multline*} Here $N,T,d,D$ denote the numbers of optimization steps and paired evaluations, the dimension and a feasible-set diameter bound, respectively. Evaluations are available within Euclidean distance $\gamma_{\max}$ of the feasible set. For the population objective $f(x)=\mathbb{E}_{\xi} F(x,\xi)$, $M$ satisfies $|f(x)-f(y)|\le M\|x-y\|_2$, and $M_2$ satisfies $\mathbb{E}_{\xi}\|\nabla F(x,\xi)\|_2^2\le M_2^2$ for sample weak gradients in the interior of the evaluation domain. The method requires no $L,M,M_2$, where $L$ is a Lipschitz constant of $\nabla f$; set $L=+\infty$ without smoothness. The method uses exactly $2T$ values in $3N$ sequential sampling rounds with predetermined batches, or $5N$ with batches chosen from observations. An extension selects a search radius without a supplied distance bound. Fixed radius parameters can leave nonvanishing error; suitable choices depending on the budgets yield convergence.

We study the effect of boundary conditions on the selection of wavenumbers in pattern-forming systems. Prior work established that, for a large class of boundary conditions, equilibria only within a narrow band of wavenumbers are compatible with the boundary. We demonstrate here that for a large class of initial conditions, solutions evolve towards an extremal member of this narrow band of equilibria. In fact, this extremal member is neutrally stable, nonlinearly unstable, and the convergence is realized through a series of snapping events at the boundary. We observe this behavior in simulations of the Swift-Hohenberg equation when the pattern forming region is confined to a large spatial interval through a parameter quench. We prove the convergence in a phase-diffusion approximation using comparison principles and astutely chosen subsolutions that mimic the snapping behavior.

Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed $0<\alpha<1$. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error $\epsilon$ across a planar cut of cross-section $A$ scales as $\exp[\tilde{\Theta}((A\log\epsilon^{-1})^{2/3})]$ when $\log\epsilon^{-1}\gtrsim A$. Thus $\alpha=1$ is the sharp threshold for a universal Rényi area law within this adiabatically connected class.

The $N$-dimensional Euclidean Onofri inequality was proved by Del Pino and Dolbeault for smooth compactly supported functions in $\mathbb{R}^N$ with $N \ge 2$. In this paper, we consider the singular version introduced by Lam and Lu, and extend it to a natural weighted Sobolev space. Next, we establish a sharp singular Carleson-Chang type inequality and prove its equivalence with the singular $N$-dimensional Euclidean Onofri inequality. Furthermore, we prove the existence of nontrivial, non-constant optimizers for the functional associated with the singular Euclidean Onofri inequality, answering open questions left by Del Pino and Dolbeault in the non-singular case, and by Lam and Lu in the case with singularities. To this end, we exploit the classification result obtained for a singular Liouville-type equation involving the $N$-Laplacian. We also prove the conformal invariance of the functional, establish the sharpness of the singular $N$-dimensional Euclidean Onofri inequality in terms of optimal constants, and state several consequences, including a sharp Poincaré inequality.

The practicality of realizing high-directivity sources is studied using a generalized sensitivity metrics. These measures are based on the error model adopted in previous studies on array tolerance, but account for heterogeneous array characteristics in an arbitrary source-expansion basis, making it applicable to the study of continuous sources, ideal uncoupled arrays, and heterogeneous arrays with mutual coupling. The formulation provides the ensemble average of realized directivity under the selected error model, making it a fast predictor of anticipated directivity reduction due to source perturbation. The data required to compute these predictions are straightforward to obtain via simulation or measurement, with a significant simplification when the system can be approximated as lossless. The directivity and the proposed tolerance measures allow for the formulation of optimization problems illustrating the tradeoff between these measures. Several examples demonstrate these tradeoffs, special cases, comparison to previous estimates of maximum directivity, and the accuracy of sensitivity as a predictor of statistical performance over large ensembles of design realizations.

Royen's Gaussian Correlation Inequality (GCI) is a fundamental result in statistics and probability, but unfortunately, at least by now, the result is not very well known in the statistical community. In this paper, we demonstrate the potential of GCI in high-dimensional testing problems that involve multivariate Gaussian variables. Leveraging the GCI, we establish sharp Type I error bounds under arbitrary Gaussian covariance structures. We also formulate a conjecture and relate it to the well-known Spherical Cap Covering Conjecture in discrete geometry and Weak Simplex Conjecture in information theory.

Sparse near-field arrays phase-cohere $M$ widely separated sub-arrays of $N$ elements and underpin distributed MIMO, satellite formations, and integrated sensing and communication. We show that their interference suppression over a protected region is set by coherence within a single sub-array rather than by the span of the formation: the effective rank of the region covariance can collapse from $MN$ to $M$. Two independent arguments establish this. For canonical line and disk regions each diagonal covariance block is exactly a classical concentration operator, which supplies sharp per-block rank thresholds and makes fewer than one spatial degree of freedom per sub-array a conservative design rule; an orthogonal decomposition with eigenvalue interlacing gives a nonasymptotic sufficient condition that assumes nothing about the coupling between platforms. The consequences are covariance-side and beamformer-independent: the unloaded Capon functional upper-bounds worst-case signal-to-interference ratio, while its regularized counterpart scales very differently for angularly separated and for in-cone targets. An eight-satellite LEO study validates the theory, including realized worst-case SIR.

We prove a variational formula for the Fuglede-Kadison determinant of a self-adjoint matrix semicircle $S_a = a \otimes 1 + \sum_j A_j \otimes s_j$ with deterministic mean $a$. For every real $z$, $\log \Delta(S_a - z) = -1/2 + \inf_b F_z(b)$, where the infimum is over positive definite matrices $b$ such that the map $X \mapsto bXb - \sum_j A_j X A_j$ is positive semidefinite, and $F_z$ is an explicit geodesically convex function. Both sides equal $-\infty$ exactly when $z$ is an atom of the spectral measure. The formula was proposed, without proof, in a recent post of Speicher. In the centered case $a=0$ it gives a new proof of the Mai-Speicher capacity formula, and by hermitization it yields a finite-dimensional variational formula for the logarithmic potential of the Brown measure of an arbitrary matrix-valued semicircular element. The proof identifies the determinant with the value of the Dyson action at its critical point and rewrites this critical-point problem as a saddle problem in three matrix variables. For a spectral parameter $z + i\varepsilon$ in the upper half-plane, eliminating two of the variables gives a geodesically convex minimization whose global minimizer is the positive factor $b_\varepsilon$ in a symmetric unitary factorization of the matrix Cauchy transform. The formula on the real axis follows by letting $\varepsilon \to 0$ in the optimal values, with no compactness assumption on the minimizers. At points where the boundary value of the Cauchy transform is nondegenerate, the scales $b_\varepsilon$ converge and the infimum is attained at their limit; this minimizer lies on the boundary of the feasible set when $z$ is in the support of the spectral measure and in its interior when $z$ is in the resolvent set.

We prove two supercongruence conjectures of Li–Long–Tu concerning truncated very-well-poised ${}_7F_6(1)$ series and Fourier coefficients of modular forms. The proofs are based on a common deformation method. By deforming the original truncated hypergeometric series to suitable terminating series, we are able to apply classical hypergeometric transformations together with Mishev's $W(D_5)$ symmetry. This reduces both problems to congruences for lower-rank hypergeometric sums that can be treated using the $p$-adic methods of Long–Tu–Yui–Zudilin. Although the two cases require different local arithmetic calculations, their global proof structure is the same. The resulting argument gives a unified explanation of these two supercongruences and suggests that the same deformation-and-symmetry method may apply more broadly.

The distribution function $F(y)=\Pr(S^2\le y)$ of the sample variance $S^2$ of four independent observations is elementary on its entire support for every polynomial parent density on $[0,1]$ (without loss of generality), and we give it in finite closed form for the uniform, Beta$(2,1)$ and Beta$(2,2)$ parents. The support splits at $y = 1/6$, $2/9$ and $1/4$, with the ceiling at $1/3$, all fixed by geometry: in coordinates aligned with the cube diagonal the event $S^2\le y$ is a ball of radius $\sqrt{3y}$, the cube projects to a rhombic dodecahedron, and these are the levels at which the ball reaches its faces, edges and two kinds of vertices. The divergence theorem reduces every moment of the clipped ball to integrals along circular arcs, so each regime closes in algebraic functions of $\sqrt{y}$ and inverse trigonometric functions. For the uniform parent the only non-obvious constant is $K_g = 12\sqrt2\arctan\sqrt2$, proportional to the mean width of the cuboctahedron; the Beta constants are rational expressions in $\pi$, $\sqrt2$, $\sqrt3$ and $\arctan\sqrt2$, as in the uniform case. At $n = 4$ the normal-theory $\chi^2$ recipe used in acceptance testing misplaces the $99$th percentile of $S^2$ by $50\%$ for the uniform parent. For three observations, polynomial parents also give elementary closed forms (arXiv:2609.25333), but two elementary cases in a row do not guarantee a third, and we make no claim beyond $n = 4$.