Papers

We develop mixed finite element methods for the linear stress-gradient elasticity problem based on symmetric Taylor–Hood elements. We establish the stability of the symmetric Taylor–Hood pair on simplicial meshes in both two and three dimensions, thereby resolving the stability left open by Brezzi, Fortin, and Marini in 1993. Combing the Nitsche's method, we further construct finite element schemes for linear stress problem in general boundary condition. Numerical experiments confirm the theoretical results.

Let $q$ be an odd prime power. We revisit Florea's asymptotic formula for the first moment of quadratic Dirichlet $L$-functions over the odd-degree hyperelliptic ensemble $\mathcal H_{2g+1}$, and show that its secondary term is not the complete one. The square-dual generating function has three double poles of the same modulus on the secondary circle: the positive real one reproduces Florea's polynomial, while the two nonreal conjugate poles contribute at the same order. The complete secondary term is $q^{2g/3}(\alpha_{g\bmod3}g+\beta_{g\bmod3})$, with real coefficients whose dependence on the genus has minimal period exactly three, and on at least two of the three residue classes the nonreal poles contribute a term of order $gq^{2g/3}$. We also prove the formula for every odd prime power $q$, whereas Florea assumes $q\equiv1\bmod4$ throughout; this is achieved by a phase-normalized quadratic Gauss sum that restores the multiplicativity and the local evaluations her argument uses. Exact finite-ensemble moments for $q=3$ exhibit the three residue-class trends.

The classification of Brauer graph algebras under derived equivalence was recently given by Opper and Zvonareva. In this paper, we prove that two Brauer graph algebras are derived equivalent if and only if they are stably equivalent of Morita type. As an application, we show that every stable Picard group orbit of simple-images of Morita type contains a liftable representative. Moreover, we give a new proof of the fact that Brauer graph algebras are closed up to semisimple summands under stable equivalence of Morita type.

Let $D=C_1+C_2\subset\mathbb{P}^2$ be a simple-normal-crossing union of smooth plane curves of degrees $1\leqslant d_1\leqslant d_2$. We prove an effective Second Main Theorem on a nonempty Zariski-open set of ordered pairs whenever \[ d_1,d_2\geqslant3, \qquad\text{or}\qquad d_1=2,\ d_2\geqslant5, \qquad\text{or}\qquad d_1=1,\ d_2\geqslant8. \] After intersecting with the standard very-general algebraic-hyperbolicity locus, we obtain Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof uses logarithmic invariant two-jet differentials, the corrected Demailly–El Goul zero-locus argument, and a two-component adaptation of slanted vector fields with base twist $\mathcal{O}_{\mathbb{P}^2}(3)$. The adapted twist three is uniform in $(d_1,d_2)$: simultaneous cancellation of the two root equations requires third Taylor differences, but no seventh-order polarization. For a linear component the third-remainder space vanishes, and ten compatible mixed generators replace the generic Taylor package. A numerical phase diagram shows that exact finite computation is needed only for the six boundary pairs $(3,3),(3,4),(2,6),(2,5),(1,9),(1,8)$. Their key vanishings are certified by full-column-rank matrices over finite fields. For two cubics, a strengthened range reduces to twelve theorem-relevant degree–twist cases; two low-weight self-tests bring the computational total to $168$ character–eigenvalue blocks. The exact source, certificates, and reproduction instructions form an accompanying computational archive. In the two-cubic case the resulting explicit estimate is \[ T_f(r)\leqslant57\bigl(N_f^{[1]}(r,C_1)+N_f^{[1]}(r,C_2)\bigr) +o(T_f(r))\ \|. \] For a conic and a quintic one may take the explicit coefficient $45$, and for a line and an octic one may take $69$.

We study singularities of height functions on singular surfaces in $\mathbb{R}^3$ parameterized by smooth map-germs $\mathcal{A}$-equivalent to $H_k$ in Mond's classification, and the versality of the family of the height functions. We also study relations of the singularities of the height functions with the parabolic locus of these singular surfaces.

This paper studies a continuous-time general-sum non-cooperative game with linear costs, positive linear system dynamics, and elementwise linear input constraints. In the finite-horizon case, we present a verification theorem characterizing feedback Nash equilibria, in terms of absolutely continuous solutions of a coupled system of vector-valued ordinary differential equations, realized by time-varying feedback laws. Unlike linear-quadratic differential games, whose Riccati-based equilibria scale quadratically with the state dimension, this formulation scales linearly. However, the resulting piecewise-constant feedback saturates between its constraint bounds rather than varying smoothly, and additional mathematical challenges arise when characterizing the solutions of the differential equations, which are generally discontinuous due to the switching nature of the feedback gains. In this work, we study the case where switching occurs only at isolated time instants. In the infinite-horizon case, under stabilizability assumptions, the equilibrium is characterized by coupled vector-valued algebraic equations. For this game, we propose iterative methods to compute both finite and infinite-horizon equilibria. The approach is illustrated through a large-scale pollution game.

We study the one-dimensional quasilinear Majda–McLaughlin–Tabak (MMT) equation on a large torus $[0,L]$: \begin{align*} i \partial_t u +2\pi|\nabla|^\sigma u +\lambda^{2}|\nabla|^\beta\left[ \left||\nabla|^\beta u\right|^{2} |\nabla|^\beta u\right]=0, \end{align*} with $\sigma\in(0,1)\cup (1,2]$ and $0<\beta<\sigma/4$. We consider Gaussian random initial data and work in the joint limit $L\to\infty$ with $\alpha=\lambda^2L^{-1}\to0$. The quasilinear structure leads to unavoidable derivative loss, which prevents the direct construction of solutions via iteration of the Duhamel formula. For $\sigma\in(1,2]$, we prove that, with high probability, solutions exist up to time scales ${T} \sim\alpha^{-{\frac43}+} \wedge \alpha^{-\frac1{\sigma/2-\beta}+}$, and that only trivial four-wave resonances occur, leading to a degenerate wave kinetic equation. For $\sigma\in(0,1)$, we prove the existence up to time scales $T\sim \alpha^{-\frac43+}$, and show that the second-order statistics are well approximated by the wave kinetic equation. In both cases, the solutions remain smooth while exhibiting smallness in suitable $L^\infty$-based norms. The proof combines propagation of randomness for a truncated equation with deterministic energy estimates and a bootstrap argument. The main technical difficulties arise in the regime $\sigma\in(0,1)$. At low frequencies, the counting argument requires simultaneous control of the frequency gap and the associated resonant phase gap. In the energy estimate, nontrivial resonances generate quartic terms that prevent the energy estimates from closing the bootstrap on $[0,T]$. To overcome this obstruction, we develop a new normal-form $L^2$ energy estimates, using distinct frequency truncations for the initial data and the nonlinearity.

We study the quantitative fast and slow weak flocking of the phase-spatially extended kinetic Cucker-Smale model in confining potential fields. The confining potential is allowed to be nonconvex. In contrast to the phase-spatially confined case, the communication weight may have no positive lower bounds, while spatial and velocity diameters may remain infinite. Moreover, an indefinite Hessian of confining potential prevents the direct use of convexity-based coercive estimates. To overcome these difficulties, we identify three structural assumptions that are sufficient for the noncompact hypocoercive method: quadratic confinement, a globally Lipschitz force, and a strict virial inequality. The admissible class of confining potentials includes genuinely nonconvex radially symmetric ones and localized oscillatory perturbations of the harmonic potential. Weak flocking analysis combines three key ingredients: a microscopic mechanical-energy estimate, a time-varying effective region, and a macroscopic Lyapunov functional. For this, we first establish global existence for initial data with finite second moments. In the exponential mechanical-energy class, we further show the uniqueness and finite-time Osgood-type stability in 1-Wasserstein distance. For polynomially decaying initial mechanical-energy tails, we derive the optimal algebraic decay exponent of the fluctuation energy throughout the admissible communication-decay regime. The optimality is verified by a symmetric countably infinite particle solution for a fixed nonconvex potential. For exponentially decaying tails, a two-stage localization argument yields exponential weak flocking on an optimal exponential time scale. These results show that well-posedness and weak flocking persist even for fully noncompact data under genuinely nonconvex confining forces.

A companion paper \cite{ItkinDF2026} introduced the Diagonal Frog (DF) positivity-preserving schemes for anisotropic Fokker–Planck equations, advancing each directional substep by a Krylov-computed matrix exponential, which dominates the cost. Replacing that exponential by a rational map $r(\gamma L)$ reduces the substep to a banded solve, but the positivity argument no longer applies. We prove that for eventually exponentially positive generators the entrywise sign of $r(\gamma L)$ at large steps is decided by a single number, the value $r(\infty)$ taken on infinitely stiff modes. Nonnegativity holds above a computable threshold when $0\le r(\infty)<1$, and at most on a bounded interval, empty or vanishingly narrow in all our tests, when $r(\infty)<0$. The criterion rejects the Crank–Nicolson (trapezoidal) method, where $r(\infty)=-1$, and selects the subdiagonal Padé$(0,2)$ method, which is second order, L-stable and provably positive above an explicit threshold. The resulting DF-ADI scheme costs $O(N)$ per step, keeps the implicit factorized mixed derivative unchanged, is second order in space and time, and conserves discrete mass exactly. In the strong cross-diffusion regime, however, the directional factors demand a step larger than the mixed derivative permits, so the composite second-order scheme is only empirically positive there, and the criterion serves to discriminate the well-behaved multiplicative factors from the stabilizing-correction schemes rather than to guarantee positivity. Against the Krylov exponential it runs ten to thirty-two times faster at matched accuracy in our tests with the gain growing with the mesh. We extend the construction to the backward Kolmogorov equation and to jump-diffusion models.

Motivated by Landis's conjecture on unique continuation at infinity for the Laplacian, we study the corresponding property for the Cauchy-Riemann operator. We prove that every weak solution of $\bar\partial u=Vu$ on a neighborhood of infinity, with $V\in L^\infty$, vanishes identically if it decays exponentially at a rate greater than $ 2\|V\|_{L^\infty}$. This conclusion is sharp both in the constant $2\|V\|_{L^\infty}$ and in the order of exponential decay required. More generally, we establish unique continuation at infinity for a broad class of radially decaying bounded potentials, with optimal decay rates determined by the decay of the potential. We also obtain related unique continuation results for $L^2$ potentials and for compactly supported potentials under weaker assumptions at infinity.

Alkaline water electrolysis (AWE) units supplied by insulated gate bipolar transistor rectifiers (IGBT-Rs) may experience oscil-lations caused by coupling between rectifier control and electro-lyzer (ELZ) dynamics. Because this risk varies with unit loading and power allocation, production-oriented dispatch may place a multi-ELZ renewable power-to-hydrogen (ReP2H) plant near or exceed its stability boundary. This paper proposes a stability-oriented framework for control design and plant production dis-patch. A three-port admittance model links the ac port, dc link, and electrolysis stack. Unit-level dc-port analysis quantifies the effects of loading, temperature, Buck bandwidth, and dc-link capacitance, while plant-level aggregation evaluates how unit commitment and power allocation affect stability. Results show that higher loading reduces stability, whereas larger dc-link ca-pacitance and higher Buck bandwidth improve it. Under the same plant loading, different power allocations result in different plant-level stability margins, with balanced allocation generally providing a larger margin than concentrated allocation. The plant-level model thus distinguishes the stability margins of ad-missible schedules. Hardware-in-the-loop (HIL) tests validate these trends and the proposed redistribution rule. The resulting operating regions and dispatch rules can be used to screen unit commitment and power allocation decisions in plant production scheduling.

In random geometry, geodesics often have a tendency to coalesce together and traverse regions highly singular relative to typical environments. In this paper, working with the model of Liouville quantum gravity, we develop a technique which yields zero-one laws for many extremal statistics measured along interior segments of the geodesic. Namely, working with statistics such as the Euclidean dimension, optimal Hölder continuity exponents with respect to the Euclidean metric and the minimal/maximal encountered thickness for the underlying GFF, we obtain zero-one laws for the above for segments in the bulk of geodesics, thereby upgrading the results of Gwynne-Pfeffer-Sheffield '22. The primary technique used, developed in Bhatia-Kavvadias '25, is to lay down a large family of small-scale "typical" and well-behaved geodesics along interior segments of a long geodesic.

We study integro-differential sweeping processes of Volterra type in a separable Hilbert space, in which the velocity is governed by the normal cone to a prox-regular moving set and is driven by an outer set-valued perturbation together with a history-dependent integral term that endows the dynamics with memory. The perturbation is assumed measurable, with closed convex values, of linear growth, and upper semicontinuous from the strong to the weak topology. We study the existence of absolutely continuous solutions under either of two alternative compactness hypotheses: ball-compactness of the moving sets, or a measure-of-noncompactness condition on the perturbation. After a reduction of the constrained dynamics to an unconstrained differential inclusion and uniform a priori bounds on the state and its velocity, existence follows from a fixed-point theorem for set-valued maps with contractible values. The memory of the process makes this contractibility delicate, and we obtain it through a continuation argument that propagates the history of the dynamics. As applications, we solve a quasistatic frictionless viscoelastic contact problem with long memory and a spatially distributed bioeconomic fishery model with ecological memory, both featuring uncertain set-valued forcing and a moving constraint set that is not ball-compact.

We consider families of pseudoconvex domains $X[s]\subset \mathbb C^n$ parametrized by complex manifolds $S$, and the maximal hermitian ellipsoids inscribed in each $X[s]$. The paper investigates how these maximal ellipsoids and their volume vary with $s$.

This article presents a novel algorithmic framework for solving a broad class of nonlinear inclusion problems involving non-monotone and Lipschitz continuous operators. Our primary motivation originates from the study of adjoint mismatch problems, which frequently arise in inverse problems and data science when the adjoint of a linear measurement operator is replaced by an approximation. Because this approximation may inherently destroy classical monotonicity properties, standard convergence guarantees for splitting methods may no longer apply. To address this, we consider a Nonlinear Forward-Backward algorithm and rigorously establish its convergence without assuming monotonicity. By leveraging a warped resolvent formulation and semimonotonicity assumptions, we derive explicit conditions ensuring both weak convergence and, under stronger assumptions, R-linear convergence. Furthermore, our theoretical analysis yields new convergence guarantees for several popular methods, including the Condat-Vũ and Forward-Half-Reflected-Backward algorithms. Finally, we illustrate our theoretical findings and demonstrate the applicability of our approach through numerical experiments in signal recovery.

This work develops a discrete-time framework for identifying probability localization dynamics through finite stochastic representations adapted in space, time, memory, and state information. A compact dynamically relevant set is localized by a finite measurable partition, producing an observable probability state and a relational graph of admissible transitions. Structured stochastic liftings derived from Stochastically Structured Reservoir Computing (SSRC) give lossless polynomial representations of the observable state, while stochastic delay liftings add finite observable memory. These are distinguished from dynamically informed state-space enrichment: refinement of observational fibers containing states with the same present observation but different observable futures, yielding an exact obstruction-to-closure criterion. A route-network toy problem gives a minimal obstruction example, while four numerical laboratories (rotational phase dynamics, the chaotic logistic map, the Van der Pol oscillator, and a synthetic cyclic inventory system) show how spatial scale, temporal scale, polynomial degree, and delay depth interact. The logistic map isolates representation-induced memory in an otherwise Markovian chaotic system, using its exact invariant law as an ergodic benchmark and its zero-mass pseudospectrum to separate relaxation from transient amplification. An exact rotational cycle calibrates pseudospectra as a robustness diagnostic rather than a closure certificate. The inventory example gives a closure-driven enrichment procedure: residence-age hazards trigger age-refined states that improve predictive scores. These results motivate a minimal adequate representation: the least complex representation meeting predictive, structural, and identifiability requirements.

Under channel switching or drift, the posterior used to order Guessing Random Additive Noise Decoding (GRAND) queries can differ from the matched correction posterior, increasing query rank and finite-budget error. We bound log query rank by matched posterior self-information plus positive log-posterior mismatch; exact random-subset collision probabilities bound GRAND with abandonment (GRANDAB) error. Under switching, a state-path mixture achieves vanishing error uniformly over admissible paths below the minimum constituent capacity when log path-class size is sublinear and uniform input is capacity achieving. Under drift, pilot refresh bounds mismatch and yields a continuous refresh-period minimizer. Generalized-Gaussian BPSK simulations evaluate both regimes.

Let $\bar{V}$ be a connected smooth complex projective variety. Let $D \subset \bar{V}$ be a normal crossing divisor. Let $\mathbb{V}$ be a complex polarizable variation of Hodge structure on $V := \bar{V} \setminus D$. Suppose that the period map of $\mathbb{V}$ is immersive at a point of $V$. We prove that for every integer $p$ with $1 \leq p \leq \dim V$, the vector bundle $\Omega_{\bar{V}}^p(\log D)$ is L-big (i.e. the tautological line bundle on $\mathbb{P}\Omega_{\bar{V}}^p(\log D)$ is big). If the local monodromy is quasi-unipotent, we give a method to determine an $m \in \mathbb{N}$ such that if $p > m$, then $\Omega_{\bar{V}}^p(\log D)$ is moreover Viehweg-big. We give the optimal value of $m$ explicitly when $V$ is a locally symmetric variety. We prove that if $V$ is a finite étale cover of the fine moduli space of smooth quintic threefolds, the result holds for $m = 90$ (in this case $\dim V = 101$). The proof of the previous results is based on a study of an augmented base locus associated with $\Omega_{\bar{V}}^p(\log D)$. In the case of locally symmetric varieties, we introduce “higher degree characteristic subvarieties”, generalizing the characteristic subvariety defined by Mok in the case $p = 1$, and prove that they coincide with these augmented base loci.

For a $3$-dimensional compact Kähler manifold $X$ with a pair of Kähler class $(\alpha,\beta)$ which is $J$-semistable, we show that the $J$-null locus of $(\alpha,\beta)$ forms a subvariety of $X$. We also give an analyic characterization of the $J$-null locus using Kähler currents with analytic singularities analogous to the theorem of Collins-Tosatti \cite{CT15}. As an application, we show the smooth convergence of the $J$-flow off the $J$-null locus. It gives higer order partial regularity for weak solutions of semistable $J$-equations obtained in \cite{M26b}.

The uniform car-parking process is the one-dimensional random sequential adsorption of unit cars on a segment of finite length $s$: cars arrive at uniformly random positions and park wherever they fit, until no gap admits another. This paper develops the exact finite-$s$ theory. The joint density of the parked positions is resolved into jamming cells, on each of which it is a rational function, and evaluated by a subset recursion in $O(2^n n)$ operations; the marginal and gap order statistics are obtained as hyperlogarithms whose weight is fixed by the number of coordinates integrated out; and the absorption count and the aggregate quantities are treated through the integral equation descending from Rényi.

Cooperative emission from three-level ensembles is conventionally diagnosed through the radiated intensity alone, which cannot say which channels carry the photons at each instant. We promote every transition of the cooperative-decay graph into a time-dependent edge flux and show that node-wise conservation of this flux is an exact restatement of the Pauli master equation. Suppose the graph carries a depth function that decreases by one along every edge. Then the total cooperative rate is the descent speed of the mean graph depth, and its energy-weighted form gives the radiated intensity as $I(t)=-\dd\langle E\rangle/\dd t$ for arbitrary level spacings. The same ordering triangularizes the generator. This delivers the relaxation spectrum from the state exit rates, together with a generalized Bateman closed form for every population on an arbitrary finite directed acyclic graph, in which all exit-rate degeneracies are absorbed into a polynomial-times-exponential recursion.

For $p:[0,1]\to(0,1)$ a continuous function, we consider the Luxemburg functional \[ \Lambda_{p(\cdot)}(u):=\inf\left\{\delta>0:\int_0^1\left|\frac{u(t)}{\delta}\right|^{p(t)}\,dt\leq 1\right\}. \] We develop a suitable functional analytic framework for $\Lambda_{p(\cdot)}$ so that it can be incorporated into a positive order cone, which can then be used in the analysis of nonlocal differential equations having the general form \[ -A\left(\int_0^1 b(1-s)(u(s))^{p(s)}\,ds\right)u''(t) =\lambda f(t,u(t)),\qquad t\in(0,1). \] By means of a specific numerical example, we demonstrate that, as opposed to a methodology relying only on the supremum norm, our methodology via the Luxemburg functional yields an appreciably sharper localisation of any solutions of the nonlocal equation.

A smooth scalar process with one fixed Gaussian covariance kernel determines any finite graded family of symmetric Hilbert tensors up to a common orthogonal isomorphism of their minimal supports. For two heat tensors, its law determines a closed positive-dimensional Riemannian manifold, a Euclidean bundle, its connection and potential, and the unknown heat time, up to isometry and gauge. The quadratic tensor recovers the scaled generator; the quartic tensor recovers the spatial norm and volume, which fix the time scale. For compact finite-dimensional RCD spaces, the same law determines the measured space with distance divided by the square root of the heat time.

A null mode of the bordered Faddeev–Jackiw matrix is not, in general, a gauge generator. We determine exactly what the residual kernel encodes and identify the additional condition required for genuine gauge symmetry. Writing $f^{(m)}=\begin{pmatrix}f^{(0)}&B\\-B^{\mathsf T}&0\end{pmatrix}$ and $\Gamma=N^{\mathsf T}B$, we prove $\dim\ker f^{(m)}=\dim\ker\Gamma^{\mathsf T}+\dim\ker\Gamma-\operatorname{rank}(\Pi M\Pi)$, where $M=B^{\mathsf T}f^{(0)+}B$ and $\Pi$ projects onto $\ker\Gamma$. The two kernel contributions have distinct meanings: the first consists of null directions of $f^{(0)}$ tangent to the constraint surface, while the second counts independent first-class combinations. We then show that $\Gamma$ is not the Dirac constraint matrix: it is the primary–secondary block of the reduced constraint matrix, with $M$ providing the secondary–secondary block. For a residual null mode with nonzero multiplier component, we derive the associated two-step chain and prove that it closes to an exact gauge symmetry precisely when the chain contraction vanishes in the effective ring modulo the constraint ideal. A four-variable counterexample shows that the Faddeev–Jackiw stopping condition can hold while the system has no gauge freedom, and that the chain criterion rejects the false generators. Mechanical examples then exhibit the criterion and show that parameter strata can carry distinct gauge and dynamical content that is erased by premature cancellation.

Problem F of \cite{FLOS18} asks whether an ideal generated by $r$ generic $\mu$-power forms, where $\mu\vdash d$ is a non-pure partition, has the same Hilbert series $G_{n,d,r}=\big[(1-t^d)^r(1-t)^{-n}\big]_+$ as an ideal generated by $r$ generic forms of degree $d$. We show that the answer is negative, and we determine exactly when the natural obstruction is available. Our main tool is an apolarity inclusion that converts a distinguished large part of $\mu$ into a fat-point linear system in the dual projective space. This yields a single master obstruction (Theorem \ref{thm:master}): a counterexample exists whenever, for $r$ general points of $\PP^{n-1}$, the homogeneous system $\LL(q;m^r)$ is non-empty while $\LL(q;(m-a)^r)$ has non-positive virtual dimension. We complement this with a positive result (Theorem \ref{thm:small-r}): by Stanley's theorem, Problem F has a positive answer for every $\mu$ as soon as $r\le n+1$. For ternary forms, the obstruction produces counterexamples precisely for $5\le r\le 8$, with extremal curves given by the conic through five points and products of $(-1)$-curves on del Pezzo surfaces. It is vacuous unconditionally for $r\le 4$ and $r=9$, and, assuming Nagata's conjecture, for every non-pure $\mu$ when $r\ge 10$. We also obtain counterexamples in four variables for $r=9$. An exact finite-field certificate shows that $d=14$, $\mu=(13,1)$ is the first failure for five ternary generators.

We study how the noncentral zeros of an elliptic-curve $L$-function enter the Birch–Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed $L$-function yields an exact Hadamard decomposition, and the zero statistics $J_E$ and $V_E$ satisfy the positive identity $J_E=V_E/8+\Delta_E$. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average $82.4%$ of $\Delta_{E,20}$, while independent edge values reveal a discrete $35/36$ zero-count split beyond height $20$. For Haar $SO(2N+1)$ we prove a fixed-dimensional hard-edge law $\Pr{J_N>y}\sim C_Ne^{-3y/2}$. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.

We test hyperuniformity from one large realization of a stationary point process. Hyperuniformity is equivalent to the average $A_r$ of the structure factor over $B_r$ vanishing as $r\downarrow0$, so the target is a low-frequency average rather than the value of the structure factor at the origin, which need not exist. We estimate $A_r$ from squared Fourier coefficients at frequencies drawn uniformly from $B_r$; if these coefficients are asymptotically Gaussian at almost every fixed frequency, the squared coefficients converge to independent variables with mean $A_r$. Assuming $A_r=s+c r^\alpha+\varepsilon_r$ with $|\varepsilon_r|\leq Lr^\beta$ and given exponents $0<\alpha<\beta$, a two-radius extrapolation removes the $r^\alpha$ term and estimates $s$ with deterministic error of order $r^\beta$. This gives confidence bounds and a one-sided test of $H_0:s=0$ with asymptotic level at most $\gamma$, consistent against every fixed $s>0$ as $R\to\infty$, then the number of sampled frequencies tends to infinity, and then $r\downarrow0$. Essentially free translation actions with completely positive entropy satisfy the Fourier assumption.

Let $G$ be a finite group and let $p$ be a prime. We prove that if the common-divisor graph on the nontrivial conjugacy-class sizes of $p$-regular elements of $G$ is disconnected, then $G$ is soluble. This settles the remaining case identified by Camina and collaborators. Their structure results provide an abelian normal $p'$-subgroup $M$. We choose a noncentral $r$-element $x\in M$, with $r\ne p$, such that $[G:C_G(x)]$ is coprime to $pr$, and show that $C_G(x)/M$ is a $\{p,r\}$-group. Thus $C_G(x)/M$ is a Hall $\{p,r\}$-subgroup of $G/M$. The separation between the two graph components then implies that every $p$-regular conjugacy-class size in $G/M$ has all its prime divisors in $\{p,r\}$. A theorem of Dolfi and Lucido, together with Burnside's $p^a q^b$-theorem, yields the solubility of $G/M$ and hence of $G$.

We establish Harnack-type inequalities for local minimizers of nonlocal double phase functionals, involving an explicit two-radius positive-part tail and a negative far-field tail. The key difficulty is the mismatch between the natural \((p,q)\)-growth scales in the available modular supremum estimate and the low integrability exponent provided by the weak Harnack inequality. To overcome this obstacle, we establish a concentric modular estimate and derive from it a local upper estimate valid for every integrability exponent. To the best of our knowledge, our results provide the first Harnack-type theory for such nonlocal double phase functionals under the natural structural conditions.

We report a Lean 4 formal audit accompanying the paper "Completing the Arakawa–Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras" (arXiv:2607.25249). In a single Lean source, the development kernel-checks the paper-local deductive architecture used for the new maximal-ideal cases: affine-kernel detection and lifting, extremal PBW-line deductions, finite Ramond–Casimir arithmetic and case enumerations, reduced-simplicity contradiction schemes, and the case-level affine conclusions for the level -1 D_l family and the positive-parameter cases of D4, E6, E7, and E8. It also audits the paper's alternative D_l level -2 argument and the separate rank-reduction application of Section 8. Previously published exceptional n=0 cases are not claimed as newly formalized results of this bundle. The released project builds successfully with the pinned Lean/Mathlib environment and contains live #print axioms queries for the principal final wrappers. The development does not reconstruct affine vertex algebras, minimal W-algebras, BRST/DS reduction, Ramond Zhu theory, or Li spectral flow from foundational definitions in Mathlib. Those ingredients, together with the concrete realization of case-specific representation-theoretic objects, are exposed as theorem parameters and semantic interfaces. Accordingly, the precise verification claim is a kernel-checked deduction of the paper-local proof from explicitly stated VOA/BRST/DS boundary inputs, rather than a from-scratch formalization of the ambient representation theory.