Papers

We consider the motion of a charged test particle confined to the round unit sphere in a uniform ambient magnetic field. We use an extension of Noether's theorem for systems with magnetic forces to reduce the problem to a quadrature. The rotation number of a trajectory, the azimuth it gains over one oscillation in latitude, is a complete elliptic integral of the third kind in Legendre form. A trajectory closes if and only if that advance is a rational multiple of a full turn. The criterion holds on every level set of the two first integrals. Then, we differentiate the rotation number with respect to the half-cyclotron frequency and sign that derivative on each of the two branches into which a level set divides. The rotation number sees the charge, the mass, the field and the speed through a single dimensionless ratio, that of the half-cyclotron frequency to the speed. A closed trajectory therefore fixes a value of that ratio. We say how many values carry a given rational winding. The poles are attainable on a single level set, where the motion reduces to a pendulum. The field has no flux through the sphere, so it is globally exact. We compute the Mañé strict critical value of the resulting exact magnetic flow. Above that value the closed trajectories are closed Reeb orbits of an explicit contact form.

We consider a non-interacting Bose gas governed by a general one-particle Hamiltonian in the canonical ensemble. Our main results are large-deviation estimates for the number of particles in the Bose–Einstein condensate. We show that the left- and right-tail probabilities decay at different exponential rates as the number of particles tends to infinity.

Many variational principles for acoustics have appeared in the literature, often introduced in an ad hoc manner by picking and choosing terms in a Lagrangian density to yield a desired governing partial differential equation. In this chapter, we show that such guesswork is unnecessary, as the governing equations of acoustics can be derived systematically from the primitive Lagrangians of classical continuum mechanics. Furthermore, we note that using Eulerian coordinates is not the only way to derive variational principles in acoustics. To this end, we review the construction of an action in the Lagrangian frame for compressible nondissipative fluids. We explore the consequences of material relabeling symmetry and the associated pseudomomentum balance, together with the boundary and jump conditions it implies. Via perturbation expansions, we show how to recover the equations of linear acoustics from each variational principle, the material frame yielding in addition the second-order acoustic energy balance. We show how weakly nonlinear approximations with variational structure emerge in each frame, in both cases for a general barotropic fluid, with the nonlinearity carried by the fluid's parameter $B/A$ and the perfect-gas results recovered as a special case.

We develop a nonperturbative, group-theoretic quantization of the effective guiding center theory of a charged particle drifting in a magnetic field in two spatial dimensions, and compare the resulting quantum theory with a corresponding coarse graining of the underlying microscopic theory. In the classical effective theory, the small gyro motion is not resolved, while the motion of the center of the gyro orbit remains observable. The reduced phase space is the physical space itself, so quantization leads to noncommuting spatial coordinates, and the effective theory loses access to the metric structure of physical space, retaining only its area structure. By formulating a prescription to match between the microscopic and effective quantum theories, we find that the predictions of the quantized effective theory are generally consistent with those of the microscopic theory. However, for closed isomagnetic contours the effective theory predicts a quantization of “radius”, a spatial discreteness absent from the microscopic theory. This illustrates that quantization of an effective theory may yield spurious nonperturbative predictions, even if that quantum theory shows no internal signs of breakdown.

We prove that $(1-o(1))$-fidelity $N$-qubit Greenberger–Horne–Zeilinger (GHZ) encoding can be performed in time $O(\log N/N)$ using all-to-all 2-local Hamiltonians with bounded 2-qubit interaction terms. This saturates the theoretical lower bound $\Omega(\log N/N)$, and by rescaling, also saturates the lower bound on signaling time for Hamiltonians with power-law interaction strengths $1/r^\gamma$, $\gamma<d$ in dimension $d$. The protocol uses spin-squeezing dynamics combined with quantum signal processing.

Macroscopic hydrogen transport codes model material interfaces with local thermodynamic equilibrium (LTE), imposing continuity of chemical potential as a per-species constraint. Three assumptions hide in that condition: fast interfacial equilibration, a single exchange pathway between the two sides, and a carrier species known in advance on each side. The literature scrutinises the first, but the other two are the more consequential failures, and neither survives at a metal/molten-salt interface. We replace the constraint with reversible reaction channels at the interface that obey mass action, with detailed balance fixing each ratio of rate constants from the thermodynamic data that already parameterise LTE, and we implement the framework in festim. LTE is recovered as the fast-kinetics limit of a single channel, in both its Sieverts/Sieverts and its Sieverts/Henry form, so the framework generalises LTE and does not compete with it. A Damköhler number delimits validity within a channel, a branching ratio between channels. In a representative nickel/FLiBe system, hydrogen partitions kinetically between molecular and fluoride carriers. The apparent interfacial law then drifts between Sieverts and Henry with loading and salt redox state, and an LTE condition underestimates the steady permeating flux. The measured pressure exponent is set by the branching ratio, not by any fixed property of the salt; a redox sweep at fixed temperature should continuously shift it between 0.5 and 1. With two isotopes, two metal-side species feed five salt-side carriers, and a per-species LTE condition is ill-posed.

We present a family of second-order, linear, unconditionally stable implicit-explicit (IMEX) methods for the Cahn-Hilliard-Navier-Stokes (CHNS) equations modeling matched-density two-phase flows. The proposed semi-discrete scheme combines extrapolation of the nonlinear terms with an auxiliary-variable formulation of the nonlinear free-energy term and a temporal-curvature regularization controlled by a parameter $\epsilon$. We establish a discrete energy estimate showing unconditional long-time stability of the method for $\theta\in(1/2,1]$ and $\epsilon\geq0$. The resulting scheme requires only linear solves at each time step. Numerical experiments demonstrate approximately second-order temporal convergence and examine mass conservation, energy dissipation, numerical robustness, and several representative interfacial-flow problems, including spinodal decomposition, droplet shape relaxation, two-phase lid-driven cavity flow, and Rayleigh-Taylor instability.

For matrices drawn from the standard Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE), we prove that the normalized zero counting measure of the permanental characteristic polynomial $Per(zI_N-H_N)$ converges almost surely to the standard Wigner semicircle law on $[-2,2]$, rotated by $\pi/2$ onto the imaginary axis. This proves the conjecture proposed by Fyodorov in \cite{Fyodorov2006}.

Traditional plasticity theory is commonly organized through an elastic law, a yield condition, a flow rule, hardening relations, and loading–unloading conditions. This paper formulates these relations through directional stationarity of one scalar functional evaluated over one-sided admissible plastic paths. The first variation determines stress, internal-variable forces, and activity resistance. Restriction to an admissible plastic tangent defines a reduced directional force for each mechanism; its sign and one-sided stationarity give the elastic inequality, complementarity, loading–unloading conditions, and active consistency. Associated response corresponds to alignment between the admissible tangent and the normal to the resulting yield boundary, whereas a nonparallel tangent represents non-associated flow. A self-similarity analysis identifies positively homogeneous stress gauges as a broad associated family and separates the roles of yield-surface shape, isotropic expansion, and kinematic translation. The storage and resistance terms cover isotropic, kinematic, coupled, and gradient hardening, while multiple activity variables describe independently activated mechanisms. A viscous potential extends the construction to rate-dependent evolution. Four closed-form solutions illustrate multi-activity regions, kinematic-hardening fields, pressure-sensitive limit states, and the distinct displacement fields associated with normal and non-normal flow. Time-discrete constitutive integration and dimensional checks are collected in the appendices.

Since Hastings' proof of superadditivity of classical communication over quantum channels, considerable effort has been devoted to finding a structural explanation of this phenomenon that was originally established by concentration of measure for Haar random unitaries. The main observation of this work is that Haar randomness can be replaced by random permutations without changing the limiting geometry responsible for nonadditivity. This replacement turns a continuous problem over unitary matrices into a discrete combinatorial problem over zero–one permutation matrices, and thereby opens a path toward derandomization. The theorem of Bordenave and Collins shows that random permutations have the required limiting behavior and the algorithm of O'Donnell and Wu then provides a deterministic asymptotic construction, running in polynomial time in the size when the channel parameters and accuracy are fixed. Thus the random construction can be derandomized in an asymptotic algorithmic sense. Finally, a quantitative random permutation estimate by Chen, Garza-Vargas, Tropp and van Handel gives a fully numerical estimate: there exists a tuple of 57,836,025 permutations acting on a set of size \[ N \le 5.422\times 10^{116216}\] such that the associated finite dimensional channel exhibits nonadditivity. This enormous value remains an obstacle to a practical construction.

$X$ is a Nakajima quiver variety. In this note we report the observation that the space parameterizing quasimaps from $\mathbb{P}^1$ to $X$, sending infinity to a given point, may be presented globally as the critical locus of an explicit function.

We investigate the statistics of the time-integrated current in an infinite one-dimensional gas of independent active Ornstein–Uhlenbeck particles, as a model system for studying an active generalisation of the corresponding passive (diffusive) phenomenology. Unlike the latter, where current fluctuations exhibit universal sub-diffusive scaling, active systems display diffusive, super-diffusive, and sub-diffusive regimes over different time scales. We fully characterise the distribution of current in terms of large-deviation asymptotics, showing that all of these scaling regimes are described by a single scaled cumulant generating function. Moreover, the statistics retain a dependence on the initial condition even at large times, revealing a persistent memory of the initial state. We further obtain the joint large-deviation statistics of currents measured at two distinct times, characterising temporal correlations. Our analytical predictions are verified using rare-event importance sampling, which resolves probabilities as small as $10^{-1000}$.

The defining feature of one-sided device-independent quantum key distribution is its asymmetric trust model in which only one party is characterized. This scenario sets an interesting middle ground between high key rates achievable by characterizing devices and the security of full device-independence. Here, we provide new tools, methods, and benchmarks for calculating key rates in this setting. To achieve this, we develop and compare two extensions of the NPA hierarchy and derive finite-size security bounds against general attacks with arbitrary device memory. The latter is based on the Generalized Entropy Accumulation Theorem. We then investigate the performance of various protocols: the BB84 protocol both with and without losses, a qutrit mutually unbiased bases protocol, and protocols based on Bell inequalties such as CHSH and $I_{3322}$. We find that the choice of which party is characterized can strongly affects the key rate, surprisingly without a universal ordering. Our work thus offers a general toolbox for calculating key rates for one-sided device-independent QKD protocols.

Adapting ideas of \cite{akutagawa2007perelman}, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in \cite{wei2009comparison}. Using this we show that low dimensional GRS are diffeomorphic to $S^3 / \Gamma$ or $S^3 \times S^1 / \Gamma$. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion $G_2$, strong torsion $\mathrm{Spin}(7)$-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered $5$-manifolds over complex orbifolds \cite{kollar2006circle}.

Boson sampling is one of the most prominent methods to verify quantum advantage, harnessing the computational complexity of the permanent of a network matrix. Similar quantum sampling problems utilize other expressions for the output probabilities, such as the fermion-based analog applying determinants. In this work, we study sampling setups of interacting light-matter systems, combining fermion and boson sampling. The mixed network input consists of fermionic and bosonic excitations. Sampling the output probability of the modes then yields contributions that are neither purely determinant nor permanent but general immanants. Specifically, we prove that these immanant contributions are linearly independent from determinants and permanents, rendering our mixed sampling network an ideal candidate for light-matter quantum application beyond pure fermion and boson sampling. Furthermore, quantum sampling with statistics that are neither bosonic nor fermionic results in expressions which even exceed immanants, further extending the functionality of quantum samplers.

Lower bounds on the pointwise spatial decay of the ground state of the Pauli-Fierz model in non-relativistic quantum electrodynamics are studied. Two models are considered: the full Pauli-Fierz model and the Pauli-Fierz model under the dipole approximation. For the full model, a lower bound is derived by means of a probabilistic approach based on the Feynman-Kac formula. The application of the Agmon distance method to the full model encounters a serious difficulty arising from the path dependence of a double stochastic integral. In contrast, under the dipole approximation, the corresponding integrand becomes deterministic, which allows a lower bound to be obtained in terms of the Agmon distance.

Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlevé equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlevé equations $\rm{VI},\rm{IV},\rm{II},\rm{I}$. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlevé equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlevé equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.

We study a two-particle lattice Schrödinger operator describing a fermion-boson pair on the one-dimensional lattice $\mathbb Z$ with zero-range on-site interaction of strength $\mu\in\mathbb R$ and mass ratio $\gamma>0$. Using relative coordinates and fiber decomposition with respect to the total quasi-momentum $k\in\mathbb T:=(-\pi,\pi]$, we obtain a real symmetric fiber Hamiltonian $H_{\mu,\gamma}(k)$ in $\ell^2(\mathbb Z)$ with essential spectrum $$ [\,2(1+\gamma)-2a_\gamma(k),\,2(1+\gamma)+2a_\gamma(k)\,], \qquad a_\gamma(k)=\sqrt{1+2\gamma\cos k+\gamma^2}. $$ For $a_\gamma(k)>0$ and $\mu\ne0$, the operator has a unique simple discrete eigenvalue $$ E_\gamma(k,\mu)=2(1+\gamma)+\operatorname{sgn}(\mu)\sqrt{\mu^2+4a_\gamma(k)^2}, $$ lying below the band for $\mu<0$ and above it for $\mu>0$, with an exponentially decaying eigenfunction. At the exceptional fiber $(\gamma,k)=(1,\pi)$, the band collapses to ${4}$ and the unique simple eigenvalue is $E_1(\pi,\mu)=4+\mu$. At the critical coupling $\mu=0$ and $a_\gamma(k)>0$, both spectral edges are threshold resonances with bounded non-square-integrable solutions. As $\mu\to0^\pm$, the discrete eigenvalue approaches the corresponding threshold with $$ |E_\gamma(k,\mu)-E_{\mathrm{thr}}^\pm(k)| =\frac{\mu^2}{4a_\gamma(k)}+O(\mu^4), $$ and the normalized eigenfunction converges to the resonant solution. For strong coupling, $E_\gamma(k,\mu)=\mu+2(1+\gamma)+O(|\mu|^{-1})$, while the eigenfunction localizes at the interaction site. We also classify the joint limit $\mu\to0^\pm$, $(\gamma,k)\to(1,\pi)$ according to the relative scale of $|\mu|$ and $a_\gamma(k)$, revealing the nonuniformity of the weak-coupling threshold asymptotics.

We study pair creation for quantum fields of spin $0$, $1/2$, and $1$ in the presence of weak gravitational and gauge backgrounds. Using resummed heat-kernel techniques, we obtain closed expressions for the pair creation probability up to quadratic order in the “generalized curvatures”. In addition, for gravitational backgrounds, we recast the pair production probability in terms of the matter distribution required for consistency with the Einstein equation. Finally, to illustrate the powerfulness of our methods, we analyze the effect of pair creation in a binary star system.

We prove the leading low-temperature free-energy asymptotics of the nearest-neighbour quantum Heisenberg ferromagnet on $\mathbb Z^2$. For every fixed $S\in\{\frac12,1,\frac32,\ldots\}$, the free energy per site satisfies \[ \lim_{\beta\to\infty}\beta^2S f_2(\beta,S)=-\frac{\pi}{24}. \] This formula was predicted by Takahashi's modified spin-wave theory [15]. Napiórkowski and Seiringer proved the corresponding upper bound but left the matching lower bound in two dimensions open [13], a gap later highlighted again by Seiringer [14]. We prove this missing bound through a new comparison between the magnon exclusion dynamics and free bosons, based on extending wave functions to collision configurations with controlled kinetic cost.

Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the Hölder conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\geq5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\geq6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\leq p<4/3$ and an $\ell^{4/3}$-to-$\ell^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman–Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.

This paper summarises our previous work on a fully $3$D Summation-by-Parts scheme, derived for a class of linear wave equations on hyperboloidal slices on a fixed Minkowski background. The scheme is derived in spherical polar coordinates, and allows having grid points at the origin and on the $z$-axis, despite coordinate singularities, and at infinity, by introducing compactification followed by rescaling, and is proved to be stable. Reducing it to the standard Cauchy problem, or to finite spacelike slices with an outer boundary, will follow a similarly. Second-order accurate finite-difference methods are used to implement this scheme numerically, but higher-order finite-difference or spectral methods could also be used. Kreiss-Oliger dissipation operators are generalized to curvilinear coordinates and are defined everywhere in the domain, including at the boundary points, such that they satisfy the dissipative property in the energy norms. We also propose new norm convergence tests that include all the grid points at all resolutions and produce more accurate results. Promising results are obtained, giving hope for application to fully nonlinear systems, like the Einstein Field Equations, and extracting the resulting gravitational waves free of systematic errors or gauge ambiguities.

Paradigmatic interaction models explain how collective behaviors can emerge in complex systems from interactions among the constituent agents. In bio-inspired swarms, however, interactions alone may not suffice to bring the population to a desired aggregate configuration within a prescribed time horizon, as needed in applications ranging from targeted therapy to collective transport and emergency evacuation. In the present work, we consider finite-horizon minimum-energy collective steering for inertial swarms that are subject to stochastic disturbances. We focus on the mean-field representations of these multi-agent systems driven by Cucker–Smale alignment or Morse attraction–repulsion interactions. Our objective is to steer the swarm between prescribed endpoint distributions using a state-feedback control, where the endpoint specifications can be full phase-space distributions (positions and velocities) or position marginals alone. Our formalism is rooted in the theory of Schrödinger bridges, which has inspired contemporary developments spanning statistical inference, biological modeling, stochastic control, and generative learning. Within the bridges framework, the uncontrolled interacting stochastic dynamics are viewed as a prior model, and the optimal control as the minimum-energy corrective drift needed to realize the prescribed distributions. We derive nonlinear, coupled necessary optimality systems with a time-symmetric structure reminiscent of classical Schrödinger bridges, and propose nested fixed-point schemes to numerically solve them. Numerical examples show that the obtained optimal control (corrective drift) can dynamically exploit or counteract the interaction forces, depending on whether the latter are favorable or adversarial to the steering task.

Let $W$ be a representation of the Weyl relations on a separable Hilbert space and write $\mathcal R_W(K)=\{W(f):f\in K\}''$. If the symplectic pairing of $K_1$ and $K_2$ is nonzero, there is a self-adjoint unitary $B\in\mathcal R_W(K_2)$ such that, for every normal state $\rho$, one can choose a self-adjoint unitary $A_\rho\in\mathcal R_W(K_1)$ with $\operatorname{Var}_\rho(-i[A_\rho,B])=4$. Hence the optimized mean-input-energy-constrained coefficient equals $4$ at every admissible energy threshold. For the fixed trigonometric Weyl witness $h$, an exact identity for $4-h^2$ reduces the deficit to a phase-fixed problem for commuting squared Weyl translations. For the one-mode quadratic energy $G_M=\frac12 R^TMR-\frac12\sqrt{\det M}$, with $M>0$ and $u^T\Omega v=\pi$, we prove $4-\gamma_{G_M,E}(h)=\frac{u^TMu+v^TMv+2\pi\sqrt{\det M}}{4E}+O(E^{-2})$. The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For $G=d\Gamma(H_1)$ and witness directions $u,v\in\operatorname{dom}H_1^{1/2}$, the fixed-witness deficit is $\Theta(E^{-1})$ whenever at least one direction lies outside $\ker H_1$; if both directions are zero modes, the constrained supremum equals the endpoint at every positive threshold.

Two independent attempts, separated by two decades, to encode turbulent dissipation with a fractional Laplacian both converge on the operator order as the signature of the fully developedinertial range: the additive, fixed order model, and the dynamically deforming operator of the Adaptive Fractional Navier Stokes. We show that, despite sharing this fixed point, the two constructions belongto distinct universality classes: Chen operator is a superposition of two fixed order kernels whose relative weight drifts with scale, while AFNS posits a single operator whose order itself flows continuously with the local Reynolds number. We derive a scale resolved observable, the effective spectral order, whose logarithmic anomaly term discriminates between these two mechanisms in principle, and then obtain a sharper, purely analytic verdict from the two dimensional, enstrophy conserving limit: the generative mechanism matching a stable process index to the empirical pair dispersion law has no solution once vortex stretching is kinematically switched off. Whether this dichotomy leaves a residual signature in real quasi two dimensional flows is left as an open experimental challenge.

This article arose out of notes for a CIME summer school mini-course in Cetraro. In it we develop differential-geometric constructions on the moduli space ${\cal M}_{g,n}^{\rm spin}$ of smooth spin curves, with particular emphasis on their interpretation in hyperbolic geometry. We relate the locally constant sheaf arising from a hyperbolic spin structure to the corresponding holomorphic sheaf, extending this known relationship to spin curves with Ramond marked points. We use the hyperbolic description to construct characteristic forms and spin measures on moduli spaces of hyperbolic surfaces with geodesic boundary, providing a framework for applying Teichmüller-theoretic methods to their volumes and volume recursions. Finally, we study separately the even and odd components of spin moduli space and find symmetries between their volume contributions which are not apparent from the contributions of individual boundary strata.

We provide a spectral geometric description of the KdV variational equation around cnoidal waves in terms of two interacting spectral curves.

Understanding how (multimodal) large language models perform on physics problems requires benchmarks that reflect the difficulty and breadth of expert-level physical reasoning. Existing physics benchmarks remain limited in the following two important ways: (1) short of high-difficulty datasets, and (2) lack of comprehensive coverage of visual forms, knowledge points, and step-by-step solution processes. As a result, model performance on current datasets may not be fully representative of their ability to solve complex physics problems. To address these issues, we present PhysElite, a large-scale bilingual multimodal benchmark for Olympiad-level physics reasoning. PhysElite contains 11,586 Olympiad-tier problems. For each problem, we provide corresponding visual diagrams, step-by-step bilingual Chinese-English solution derivations, and the final answer. We benchmark 18 open-source and closed-source MLLMs, and find that even the strongest model reaches only 33.7% answer accuracy. We additionally conduct step-level process evaluation to diagnose where models fail in the reasoning chain. Our datasets are released at https://huggingface.co/datasets/physelite/PhysElite.

The oscillator (or metaplectic) semigroup is classically defined as the two-fold cover of the semigroup of positive complex symplectic matrices. Although geometrically precise, this definition does not provide an intrinsic operator-theoretic characterization of the bounded operators corresponding to positive symplectic matrices. This is in contrast with the real metaplectic group, whose relation with the symplectic group can be expressed directly in terms of the Schrödinger representation of the Heisenberg group through its intertwining property. In the complex setting, such a relation appears in the existing literature mainly in infinitesimal form or on suitable classes of Gaussian functions. In this work we prove that the (complexified) Schrödinger intertwining relation holds for every function in $L^2(\mathbb{R}^d)$. The main point is the converse statement: if an arbitrary complex symplectic matrix $S$ admits a nonzero bounded operator on $L^2(\mathbb{R}^d)$ satisfying this intertwining relation, then $S$ is necessarily positive and the operator coincides, up to a nonzero scalar, with the corresponding element of the oscillator semigroup. Consequently, positive complex symplectic matrices are exactly those admitting a nonzero bounded Schrödinger intertwiner, and the associated intertwining space is one-dimensional. This provides a Schrödinger-representation characterization of the oscillator semigroup, parallel to the classical one for the real metaplectic group.

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