Papers

We introduce a microlocal formulation of the Schrödinger-Klein-Gordon system describing the interaction between a non-relativistic quantum particle and a Klein-Gordon field through Yukawa coupling. Instead of working directly with the Schrödinger wave function, we represent the quantum component by its Fourier-Wigner transform, and we rewrite the Klein-Gordon equation in terms of a complex Fourier variable. Eliminating the field variable yields a closed nonlinear Fourier-Moyal equation on phase space whose unknown is the Fourier-Wigner distribution associated with the quantum component. The resulting formulation provides a refined description of the particle-field interaction at the microlocal level. In particular, the original cubic coupling is transformed into a quadratic self-interaction governed by an explicit bilinear operator. This new representation permits the use of techniques from time-frequency analysis. Building on this, we develop a well-posedness theory for the Fourier-Moyal equation in anisotropic spaces involving Wiener-type norms and prescribed moduli of continuity. Local existence and uniqueness are established under general assumptions on the ultraviolet cutoff, together with propagation of microlocal regularity. We then introduce a new construction of weak L 2 -solutions by exploiting compactness properties of Fourier-Wigner transforms and obtain global existence for prepared data. The aim is to provide an alternative analytical framework for the study of Yukawa-type interactions and to establish a bridge between nonlinear dispersive equations and phase-space methods.

In this work, we are concerned with a class of multi-dimensional time-inhomogeneous stochastic differential equations (SDEs) on $\R^d$ driven by pure-jump Lévy processes, where the drift coefficient $b(t,x)$ satisfies $\lim_{t\to \infty}b(t,x) =0$ for every $x\in \R^d$. On account of three regimes associated with the index $\alpha$ of large jumps corresponding to the driven Lévy noise, we investigate the quantitative asymptotics of the corresponding rescaled processes. More precisely, for $\alpha\in (0,2)$, we prove that the rescaled processes, governed by time-inhomogeneous SDEs subject to additive processes, converge with respect to suitably chosen Wasserstein distances to time-homogeneous SDEs driven by symmetric $\alpha$-stable processes. Notably, the driven noise in the limiting SDEs depends only on large jumps of the underlying additive processes. In case of $\alpha\ge2$, a phase transition occurs and a diffusive phenomenon arises. In particular, in the setting $\alpha>2$, we establish the ergodicity of the rescaled process by means of asymptotic pseudotrajectories. The resulting time-homogeneous limiting SDEs are driven by Brownian motions, even though the transformed time-inhomogeneous SDEs are driven by (discontinuous) additive processes, and the effective noise intensity is determined by the entire Lévy measure of the original pure-jump process. As far as the critical case $\alpha=2 $ is concerned, we demonstrate that the noise intensity of the limiting SDEs driven by Brownian motions relies merely on the large-jump part of the Lévy measure.

We prove that for $n \geq 6$ there exists $\varepsilon (n)>0$ such that every closed orientable Riemannian manifold with $\left( \lceil \frac{n}{2} \rceil+ \varepsilon\right)$-positive curvature operator is a real homology sphere. For $n=6$ we show that the same result holds for manifolds with $4$-positive curvature operators.

We classify entire stable solutions with quadratic energy growth in the $4$-dimensional abelian Higgs model. More precisely, we prove that such solutions are holomorphic with respect to some complex structure, namely they satisfy the first order equations introduced by Bradlow, and consequently their nodal sets are holomorphic curves. Conversely, we construct solutions whose nodal set is any prescribed holomorphic curve with quadratic area growth. As a corollary, we show that all stable solutions with linear energy growth in $\mathbb{R}^3$ are two-dimensional Bogomolnyi vortices.

We investigate the persistence of most probable paths through the Onsager–Machlup functional for multidimensional stochastic differential equations driven by fractional Brownian motion with time-dependent diffusion coefficients and Hurst parameter $H\in(1/4,1)$. Under suitable structural and variational conditions, deterministic trajectories remain most probable paths for sufficiently small noise in both the fixed-endpoint transition problem and the free-endpoint evolution problem, whereas sufficiently large noise destroys their local minimality. More generally, when exact persistence does not hold, global most probable paths converge to the corresponding trajectories of the noise-free system in both the uniform and Hölder topologies at the rate $O(\epsilon)$. We further analyze the second variation along periodic deterministic trajectories over time intervals of length $NT$. For $H>1/2$, positive definiteness, and hence local minimality, is lost on sufficiently long intervals. For $H\in(1/4,1/2]$, long-time positive definiteness holds for the fixed-endpoint problem, but this conclusion does not directly extend to the free-endpoint setting. We also establish the persistence of KAM tori in nearly integrable Hamiltonian systems in the sense of most probable evolution paths. Finally, a two-dimensional numerical example illustrates the persistence of deterministic trajectories under small noise and their pronounced deviation under large noise.

We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, $\|e^{-it\Delta/2}f\|_{L_t^qL_{\boldsymbol{x}}^r(\mathbb{R}^{1+d})}\le C_{q,r}\lVert f\rVert_{L^2(\mathbb{R}^d)}$, with $2/q+d/r=d/2$, $q,r\geq 2$, and thus $r\leq 2d/(d-2)$ if $d\geq 3$. We show that, in low dimensions $1\leq d\leq 5$, the thresholds $\rho_1=10$, $\rho_2=6$, $\rho_3=4\sqrt{7}-6\approx 4.583$, $\rho_4=2\sqrt{15}-4\approx 3.746$, and $\rho_5=10/3\approx 3.333$ are such that gaussians are stable local maximizers for $2<r<\rho_d$, and fail to be local maximizers for $\rho_d<r\leq 2d/(d-2)$ (with the conventions there is no upper bound on $r$ when $d\in\{1,2\}$ and that $r=\infty$ is excluded when $d=2$). In the cases $(q,r,d)\in\{(6,6,1),(8,4,1),(4,4,2)\}$, we establish global stability inequalities with effective stability constants. Both proofs hinge on spectral gaps which we compute exactly.

We prove a delocalisation bound for eigenvectors of Wigner matrices with the precise relationship between the size of the largest entry and the decay exponent of the probability. We also prove that the largest entry of an individual eigenvector and the largest entry of all eigenvectors are both Gumbel distributed. The proof is based on the inclusion-exclusion principle, the small probability comparison of Erdős–Xu, and partial diagonalisation of Gaussian divisible matrices.

We prove that a homeomorphism of the two-torus in a Dehn twist homotopy class is a topological extension of an irrational circle rotation if and only if its vertical rotation set reduces to a single irrational number and the homeomorphism exhibits uniformly bounded rotational deviations in the vertical direction. This result has been previously obtained by the second author under an additional hypothesis about the geometry of the non-wandering set. We show that this hypothesis can be entirely dropped: the large-scale shear intrinsic to a Dehn twist eliminates the obstructions coming from the wandering set, which cannot be removed in the homotopy class of the identity. Finally, we prove that the factor is unique up to a circle rotation and is locally constant on the wandering set. Consequently, every connected component of the wandering set is contained in a single fiber, is itself a wandering domain and is either inessential or annular.

We prove global reverse Hölder estimates in \(\mathrm L^2\) for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is real, symmetric, and periodic in every spatial variable \(X=(\lambda,x)\), while no periodicity in time is assumed. If its boundary trace \(A^0=A^0(x,t)\) has a quantitative uniform modulus of continuity in time and its convergence to \(A^0\) in the transverse variable is controlled by a square-Dini modulus, then the Poisson kernel satisfies a uniform \(\mathrm{RH}_2\) estimate at scales below the period. Full spatial periodicity is then used to propagate this estimate to every scale. The main new ingredient in the large-scale argument is a positive adjoint height coordinate \(\Phi^*=\lambda+O(1)\). Boundary comparison with \(\Phi^*\) yields the required large-scale decay of the Green function and hence the global \(\mathrm{RH}_2\) estimate. Consequently, the \(\mathrm L^2\) Dirichlet problem is uniquely solvable among weak solutions \(u\) satisfying \(N_\ast u\in\mathrm L^2\). We also establish, for fully spatially periodic coefficients, a nonsymmetric \(A_\infty\) theorem under square-Dini convergence to a transverse-independent boundary trace, with no temporal regularity beyond measurability. We explain how the results transfer to time-independent Lipschitz graph domains under the stated coefficient and periodicity-compatibility hypotheses, and we derive scale-uniform boundary estimates for the periodic homogenization families considered below. The condition \(D_t^{1/2}A^0\in\mathrm L^\infty_x(\mathrm{BMO}_t)\) provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.

We study the spectral complexes arising from the de Rham complex on the Engel group $G$ viewed as a truncated multicomplex. We describe the two resulting complexes, analyse their compactly supported counterparts, and define the corresponding spectral currents by duality. We prove that the smooth spectral complexes embed naturally into the current complexes. Finally, we show that the Pansu pullback of a $W_{\mathrm{loc}}^{1,q}$ map $\varphi\colon G\to G$ with $q>7$ induces a morphism into the corresponding complexes of spectral currents.

Let $L$ be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra $H$ containing $L$ as a Hopf subalgebra and satisfying $H_0=L_0$ and $\dim H=2\dim L$ is a normalized quadratic extension of $L$. For a fixed support $(g,h)$, such extensions are described by a coalgebra Hochschild $2$-cocycle together with Ore-type data $(\sigma,\delta,u,v)$ satisfying explicit compatibility conditions. Their isomorphism classes are parameterized by the orbits of gauge transformations and Hopf automorphisms of $L$. As an application, we classify non-connected pointed Hopf algebras of dimension $16$ in characteristic $2$ with one-dimensional infinitesimal braiding whose diagram is not a Nichols algebra.

We investigate the mixed moments of products of irreducible $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions on the critical line, and establish sharp moment bounds for these quantities. In particular this establishes the order of these mixed moments in certain ranges. This determines unconditionally, for the first time, the order of magnitude of the first moment of cubic Dedekind zeta functions with negative discriminant. The proof of the lower bound introduces a new method for computing mixed moments, that is a mix of the classical method of Heath–Brown with the modern machinery of Harper, Heap, Radziwill, and Soundararajan.

Inspired by constructions of entire curves in Oka geometry, we construct rational curves on complex Fano manifolds analytically, by alternately deforming a holomorphic disk to reduce its area and enlarging its source. Positive Ricci curvature yields an area-decreasing deformation, while an affine-lift perturbation enlarges the source at a controlled area cost. The two operations change the area and the weighted derivative by amounts we estimate explicitly, and analytic compactness passes from the resulting disks to a holomorphic sphere. The main result produces a sphere that meets a prescribed compact fiber without being contained in it, with an explicit area bound.

A novel, non-parametric clustering algorithm is developed, based on flows in the space of soft assignments $P_k^i$, representing the probability that point $x^i$ belongs to class $k$. These flows have two drivers: a nonlinear reaction component that relaxes $P$ to an evolving Bayesian-like posterior, and a diffusive component that relaxes $P$ to its local average, with a diffusivity $\nu$ that adapts dynamically so as to balance the two components. The methodology extends to semi-supervised classification, where a subset of the labels are known. In the limit of infinitely many observations, it yields a set of non-standard reaction-diffusion equations, which produces sharp boundaries between species that separate spontaneously into well-balanced domains. Including more than one diffusive network opens the way to a broader class of applications, including the detection of regime changes in time series and a clustering procedure based on numerous features that defeats the curse of dimensionality. In the continuous limit, this extension results in a novel, non-local class of diffusive operators.

Identifiability of tensor decompositions is often established through linear-algebraic conditions on the factor families. For nonnegative decompositions, however, positivity provides additional information that is not captured by dimension and independence alone: nonnegative terms cannot cancel, and their supports constrain competing decompositions. We introduce a positive scattering term that quantifies this additional source of identifiability and combine it with the dimension budget underlying the Lovitz–Petrov generalization of Kruskal's theorem. For every subset of components, we obtain two sufficient conditions: a threshold of $2|S|-2$ guarantees minimality and nonnegative rank, while the stronger threshold $2|S|-1$ guarantees uniqueness among nonnegative decompositions of the same length. The key result is a positive splitting inequality for irreducible exchanges of nonnegative rank-one tensors, which combines the dimension constraint with support-induced geometric rigidity. Although the scattering term is defined through an optimization over intermediate factor spaces, we show that its mode costs are exactly $0$, $1$, or $+\infty$, yielding an exact activation characterization in terms of graph connectivity. The resulting criterion can strictly certify sparse nonnegative tensor decompositions beyond the reach of Kruskal and Lovitz–Petrov conditions, including examples for which those conditions fail even after reshaping. In the matrix case, the two criteria reduce respectively to full-rank factorization and two-sided separability.

We study a random Schrödinger operator with a potential distributed according to a Poisson process. Asymptotic expansions for traces of resolvents in the limit of small disorder are derived. Explicit estimates for the expansion coefficients are given and we show that their infinite volume limits are finite as the spectral parameter approaches the spectrum of the free Laplacian. As an application we derive bounds on the integrated density of states.

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

We investigate the class of Liouville regular Jordan curves, which was introduced in our previous paper under the phrase "Jordan curves that admit continuous Legendrian lifts". We prove that a Jordan curve satisfying certain variation condition in a neighborhood of each point is in this class. We also prove that the boundary of a Jordan domain is Liouville regular if the images of the radial tails under a conformal mapping have uniformly vanishing length. Together with our previous result, these criteria imply the existence of an inscribed rectangle in every prescribed similarity class.

Triple Aeppli–Bott–Chern–Massey products, shortly, triple ABC-Massey products are higher-order operations on the Bott–Chern and Aeppli cohomologies of a compact complex manifold, and their non-vanishing is an invariant of its pluripotential homotopy type. We prove that every non-Abelian complex unimodular solvable Lie algebra admits a non-vanishing triple ABC–Massey product, and we deduce that so does every compact complex parallelisable solvmanifold.

For a nonzero real polynomial $p$, let $\W(p)$ denote the numerator of its ordinary generating function. We prove that if the coefficients of both $\W(p)$ and $\W(q)$ are nonnegative and log-concave with no internal zeros, then so are the coefficients of $\W(pq)$. This provides an affirmative answer to a question of Brändén, Ferroni, and Jochemko. As applications, we derive corresponding results for finite products and for Cartesian products of lattice polytopes, answering a question of Ferroni and Higashitani.

Direct evaluation of the defining product of the Cayley-Dickson algebra $A_n$ of rank $N:=2^n$ over a unital commutative ring $R$ in which $2$ is invertible requires a quadratic number of arithmetic operations. This paper gives a uniform bilinear algorithm for multiplication in this algebra costing $O(N \log N)$ arithmetic operations in $R$ and $O(N)$ auxiliary storage when the doubling parameters of $A_n$ are units whose square roots lie in a fixed commutative coefficient algebra $S$ finitely generated as an $R$-module and admitting an $R$-linear map $\pi\colon S \to R$ with $\pi(1) = 1$. In particular, over $\mathbb{R}$ and every finite field $\mathbb{F}_{q}$ of odd characteristic, a quadratic field extension contains a square root of every possible doubling parameter, so the algorithm applies to every Cayley-Dickson tower over these fields, and the tensor rank of multiplication in $A_n$ over them is at most $(9n-21)2^n+28$ when $n\ge2$. When the top doubling parameter is a non-square, as in the standard real tower, the bound is $(9n-15)2^{n-1}+10$. To the author's knowledge, the algorithm presented in this paper is the first uniform subquadratic algorithm for multiplication in $A_n$. When $R=\mathbb{R}$ with every doubling parameter equal to $-1$, the real quasilinear schedule uses fewer total arithmetic operations than both direct multiplication and a uniform algorithm of Cariow-Cariowa for $n\ge4$. An implementation for the standard real tower is available in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface. In single-core benchmarks against direct multiplication and the uniform Cariow-Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was over $17$ times faster than direct multiplication at $N=1024$.

We prove that the skew stickiness ratio converges to two at short maturity under local volatility models. This appears to be the first rigorous proof of this limit for a general time-dependent local volatility function. As a by-product, we strengthen the one-half rule of the implied volatility skew by removing uniform ellipticity and global bounds on spatial derivatives of order at least two. The proof uses a first-order Watanabe expansion.

This paper investigates finite element approximations of the one-dimensional viscous stochastic Burgers equation with additive trace-class noise. For the $P_2$ finite element spatial semi-discretization, we derive strong error estimates that are optimal with respect to regularity in \(L^p([0,T] \times \Omega;H_{D}^{\alpha,q})\) for \(p,q\in[2,\infty)\) and \(\alpha\in[-1,0]\), as well as an almost regularity-optimal estimate in \(L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))\). Furthermore, we derive weak error estimates for moments of both terminal $L^q$-norms and space-time $L^p(0,T;L^q(\mathcal{O}))$-norms, with weak convergence rates (nearly) twice the corresponding strong ones. For the fully discrete scheme, which combines the \(P_2\) finite element method in space with a drift-implicit Euler–Maruyama scheme in time, we establish a strong temporal convergence rate of order \(\tau^{1/2-\varepsilon}\) in a discrete analogue of \(L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))\), under the condition \(\tau \leqslant h^2\). Numerical experiments are presented to illustrate the theoretical convergence rates.

Many classes of two-player zero-sum stochastic games have the orderfield property: if all payoffs and transition probabilities lie in a subfield of $\mathbb{R}$, so does the undiscounted value. Absorbing games fail this property, and Oliu-Barton and Vigeral [Absorbing games with irrational values, Oper. Res. Lett. 51 (2023) 555–559] conjectured the precise extent of the failure: every real algebraic number $\alpha$ of degree $m\geq 1$ over $\mathbb{Q}$ is the undiscounted value of a rational $m\times m$ absorbing game. We prove this conjecture, and in fact within a special subclass of absorbing games: for every such $\alpha$, the game realizing it is strictly absorbing and recursive, i.e., every action pair is absorbing with positive probability and all non-absorbing stage payoffs are zero; when $\alpha>0$ it can moreover be taken positive recursive, with positive absorbing payoffs. As a corollary, the set of undiscounted values of rational $m\times m$ absorbing games is exactly the set of real algebraic numbers of degree at most $m$.

The random-cluster model with cluster parameter \(0<q<1\) is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first establish a reduction showing that the pairwise conjecture is equivalent to its restriction to regular graph sequences of diverging degrees $(d_n)$, even for edge pairs at distance \(o(d_n/\log d_n)\). We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed \(p\in(0,1)\), all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance \(o(d_n/\log d_n)\), uniformly in \(q\in[0,1)\) and in the choice of the two edges. In particular, the result includes the \(q=0\) endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.

This project began as an attempt to prove that Medvedev logic is decidable with the help of generative AI systems. The author (as well as the generative AI systems, or at least they claim to be since I have asked them) was surprised by its eventual conclusion. We prove that Medvedev logic ML, the intermediate logic of finite problems, is Pi-01-complete under computable many-one reductions. Consequently, ML is not recursively enumerable, a fortiori undecidable, and admits no recursively enumerable sound and complete proof calculus. The proof connects the periodic domino problem with intuitionistic formulas through a shared intermediate structure that we call a Wang-Medvedev pair. Such a pair consists of a finite partially ordered set of roles together with demands. Demands define the interaction between roles. A realization labels nonempty subsets of a finite set with these roles, respecting the order and satisfying the demands. We associate a pair with each finite Wang system and show that it has a realization iff the system tiles a finite torus. We then construct an intuitionistic formula that fails on some finite Medvedev frame iff the same pair is realizable. Realizability thus provides the link between periodic tilings and the countermodels.

In derivative-free optimization (DFO), one minimizes functions for which the gradient is unavailable or expensive to compute. In many applications, objective function values and gradients are noisy due to simulations or system randomness. A class of standard direct-search methods for DFO accept a trial point when it decreases the objective function by an amount proportional to the squared stepsize. However, when applied to complex landscapes, such a requirement may trap the algorithm in a neighborhood of sub-optimal solutions. We study a non-monotone direct-search alternative where the trial function value is compared with the largest objective function obtained through the $M$ most recent distinct iterates. This max-$M$ non-monotone condition permits temporary increases in the objective function and can help navigate narrow curved valleys; however, its theoretical analysis is significantly more challenging due to the lack of monotonic decrease. In this paper, we develop a comprehensive complexity theory for the max-$M$ non-monotone direct-search in both deterministic and stochastic DFO problems. For deterministic objectives, we establish a worst-case iteration bound for a complete poll based on a positive spanning set and an expected iteration bound for a probabilistic-descent poll. We then analyze a stochastic variant using independent function estimates and show the expected iteration complexity under tail-bound assumptions of the stochastic errors. All three results have the standard complexity of $\mathcal{O}(\epsilon^{-2})$, which matches the iteration complexity of monotone direct-search methods. Our theory is enabled by a new family of merit functions that correct the stored objective values by ordered multiples of the squared stepsize, together with a renewal-reward stopping-time argument for the probabilistic methods.

We prove that the system of Favre-filtered compressible Navier–Stokes equations with subgrid-scale closure terms, written in the coordinates $(Lt,Lx_1,Lx_2,Lx_3)$ for an arbitrary parameter $L>0$, is obtained from the system written in standard coordinates $(t,x_1,x_2,x_3)$ by the action of a smooth diffeomorphism of the underlying spacetime manifold. The proof begins with a self-contained treatment of infinite-dimensional manifolds and the Fréchet manifold of smooth field configurations. We then define the one-parameter dilation group $\{\phi_L\}_{L>0}$, verify that each $\phi_L$ is a diffeomorphism, and compute its tangent and cotangent actions. The induced pullback action on sections of the configuration bundle is shown to commute with the total derivative operator, which implies that the first-jet prolongation of $\phi_L$ maps the equation submanifold of the system onto itself. Equation-by-equation verification—continuity, momentum, and energy—confirms that every residual is multiplied by the common nonzero factor $L^{-1}$ (or, equivalently, is identically zero in the transformed coordinates), so that the solution set is preserved. We conclude that the family of systems parameterised by $L>0$ satisfies the diffeomorphism transformation expression and that solution spaces are canonically isomorphic for all $L$.

We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\operatorname{Re}s=1/2$. More precisely, for every $2<p<\infty$, there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies $$ \sup_{\theta\in\mathbb{R}}\int_{\theta}^{\theta+1}\left|P\left(\frac12+it\right)\right|^p\,\mathrm{d}t\le C_p\left\lVert P\right\rVert_{\mathscr{H}^p}^{p}, $$ with $C_p$ independent of the number and choice of prime variables on which $P$ depends. Before the present work, the embedding was known at $p=2$ and, by taking integer powers, at the even exponents $p=2k$; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for $0<p<2$, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for $p\ge2$. Thus, the true threshold is $p=2$, rather than even integrality. The proof passes to the dual exponent $q=p/(p-1)\in(1,2)$, where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For $1\le p<\infty$, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon–Hedenmalm composition operators.

We initiate a geometric theory of $p$-Brownian motion, the nonlinear Markov process associated with the $p$-Laplacian introduced by Barbu-Rehmeier-Röckner. More precisely, we analyze its radial processes thoroughly, relative to an arbitrary center and in every dimension. On the one hand, we show explicit Tanaka–Meyer semimartingale formulas; our consequential characterization of nontriviality of the associated local times reveals notable differences to classical Brownian motion. In parallel, we establish sharp exit time estimates as well as scaling estimates for the self-similarly rescaled radial process. We also prove isometry of the corresponding marginal laws in each Wasserstein distance. Our contributions equally cover the Leibenson process - the nonlinear Markov process associated with the "porous medium equation" with $p$-Laplacian - recently introduced by Barbu-Grube-Rehmeier-Röckner.