Papers

We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ random matrices has growth $\rho$ is at least inverse quasi-polynomial: $\Omega(\exp(-c \log^2 (\rho)\log(n)))$ for some constant $c > 0$. This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination. To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices.

In this contribution, we address the exact linearization of forward- and backward-flat discrete-time nonlinear systems with an arbitrary number of inputs. Building on structural properties of the flat parameterization, we show that forward-flat systems can be rendered static feedback linearizable by prolongations of suitably transformed inputs. Analogously, for backward-flat systems, we derive a constructive procedure showing that an exact linearization can be achieved by prelongations of suitably chosen functions of the system variables. In this way, we extend previously established results for two-input systems to the general m-input case. Finally, we illustrate the proposed linearization procedures by an academic example and a discrete-time model of a quadrotor.

Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $|G|$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $|G:G'|\le M$, and that equality holds only for abelian groups if $|G|$ is odd. We prove that $|G:G'|\le M/p$, unless $p=3$ and $G$ is the direct product of an abelian group and a group isomorphic to $\Gamma(2^3)$ or to $\Gamma(2^3)\times C_3$ acting on $V$ in a specific way, in which case $|G:G'|=3M/7$. In particular, $|G:G'|\le 3M/7$ for every nonabelian group of odd order. If $|V|$ is odd, then $|G:G'|=M/p$ holds if and only if $|G'|=p$; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy $|G:G'|\le 3M/13$, and even $|G:G'|<M/(2p)$ if $p\ge5$. The bound $3M/13$ is best possible.

We show that if an open smooth $6$-manifold $M^6$ is the interior of a compact, contractible manifold with boundary $X^6$, such that $\pi_3(X,\partial X)=0$, and $M$ supports a complete metric with uniform positive BiRicci curvature and Ricci curvature bounded below, then $M$ is diffeomorphic to $\mathbb{R}^6$. Additionally, we show that if an open smooth $n$-manifold $M^n$, $n\in\{6,7\}$, is the interior of a compact, contractible manifold with boundary $X^n$, such that $\pi_3(X,\partial X)=0$, and $M$ supports a complete metric with positive BiRicci curvature with at most $C$-quadratic decay at infinity for some $C>C_n$ and nonnegative Ricci curvature, then $M$ is diffeomorphic to $\mathbb{R}^n$. Lastly, for each $n\geq6$, we construct a compact, contractible Newman $n$-manifold, $N^n$, based on a presentation of the Higman group such that $\pi_i(N,\partial N)=0$, $3\leq i \leq n-3$, and $\mathrm{int}(N)$ admits a complete metric with uniform positive scalar curvature and Ricci curvature bounded below.

Motivated by observations in atmospheric modelling, we investigate the effect of “stochastic inertia”: in certain dynamical systems, small noise induces trajectories to stay longer in a dynamically critical subset of state space than in the purely deterministic situation. Under mild general assumptions on the stochastic differential equation, we derive an integral formula along deterministic trajectories whose sign provides a sufficient condition for the occurrence of stochastic inertia. Excluding situations with open sets of infinite escape times and assuming uniformly elliptic diffusion, we additionally identify this integral formula as the zero noise derivative of the average escape time. The proofs make crucial use of Dynkin's formula and suitable adaptations of Freidlin–Wentzell theory. We present an algorithmic approach to compute the integral formula and, by examples of saddle points and relaxation oscillators, demonstrate the following heuristic: stochastic inertia appears in the presence of an unstable direction that predominantly leads the escape dynamics.

We solve a longstanding problem in Bernoulli percolation: we derive sharp Ornstein–Zernike asymptotics for the probability that two distant points are connected by a finite cluster in the whole supercritical regime. Our proof has two main ingredients. First, the renewal structure from Fridbergh and Hammond (2024) yields an a priori bound on cluster volume. Second, we adapt the coarse-graining and irreducible-cluster framework of Companion, Ioffe, Velenik (2008) to finite supercritical connections. The main distinction from the subcritical analysis is that our argument does not require a separation of masses between irreducible and unrestricted finite connections. The volume bound and a modified coarse-graining construction instead provide stretched-exponential tails for the effective random-walk increments, which suffice to obtain the sharp asymptotics.

We first prove a rigidity result for uniformly rotating vortex patch solutions to Euler equations in bounded simply-connected planar domains. Namely, if a bounded simply-connected domain with real-analytic boundary contains a nontrivial vortex patch that remains strictly inside the domain while undergoing a uniform rigid rotation, then the ambient domain must be a disc centered at the rotation center. The proof combines the rotating-frame formulation with harmonic unique continuation and the generalized Schwarz's reflection principle across analytic boundary arcs. These tools imply that, unless the outer domain is rotationally invariant, the velocity must vanish in an open region surrounding the patch, contradicting the nonzero circulation generated by the patch. In the same spirit, we show that only circular domains possess a circular point vortex orbit. As a second result, we construct quasi-periodic vortex patch solutions to the Euler equations in domains that are small perturbations of the unit disc $\mathbb{D}$. The amplitudes of the solutions are linked to the size of the domain deformation, allowing the classical Rankine vortices $b\cdot\mathbb{D}$, $b\in(0,1)$, to be regarded as equilibrium states. Our analysis is perturbative with respect to the construction developed in the undeformed case, and the resulting quasi-periodic solutions exist for most values of the inner radius $b$. This provides the first construction of quasi-periodic vortex patch solutions to the Euler equations in non-radial domains. Moreover, combined with the rigidity result established above, this shows that, in the periodic case, these solutions cannot arise from a purely rigid rotation.

In this paper we consider matrices whose entries are rational functions of several parameters over a Moore-Penrose field, that is, a field with an involutory automorphism over which every matrix has Moore-Penrose inverse. We prove that over any such field the Penrose conditions can be solved with linear algebra alone, even though they form a polynomial system of degree two. We then bound the degrees of the numerator and of the denominator of the pseudoinverse in terms of the degree of the entries and of the rank, and not of the dimensions of the matrix. Specialization is the thread: the closed form is computed once over the field of rational functions, and the question is at which parameter values it still returns the pseudoinverse of the specialized matrix. We determine the values where it does not, from the matrix alone and before computing the pseudoinverse. They are the zeros of a polynomial built from the maximal minors, and they are the values at which the rank decreases. This turns previous sufficient conditions into an exact characterization. The theory also gives symbolic algorithms for the real, the complex and the parametric case, which we implement in Maple and test on 972 timed replications. We also apply them to a Leontief economic model, where the excluded value is the point at which the economy stops being viable.

In this paper, we prove the strict Faber-Krahn inequality for non-negative-mass Dirac operators on simply connected domains with infinite-mass boundary condition. Denoting the first positive eigenvalue of the corresponding Dirac operator with mass $m\geq 0$ as $\lambda_1^{+}(\Omega;m)$, our main result shows that \begin{equation*} \lambda_1^{+}(\Omega;m)\geq \lambda_1^{+}(B_{|\Omega|};m), \end{equation*} where $B_{|\Omega|}$ is the disk with the same area as the domain $\Omega$, and the equality is strictly valid if and only if $\Omega=B_{|\Omega|}$. The key observation leading to the proof is that the upper component of the Dirac spinor associated with $\lambda_1^{+}(\Omega;m)$ is non-vanishing over $\Omega$, which is proved by exploiting its Dirac current and the associated stream function. Based on this global non-vanishing property, we construct a sphere-valued spinor map of degree one, which allows a sharp comparison with the radial disk profile using the planar isoperimetry.

We give an algorithm that samples uniformly random linear codes of any hull dimension and type over finite fields of odd characteristic, implying the first algorithm for sampling uniformly random self-orthogonal codes of any rate, including self-dual codes. Our algorithm is a re-visitation of the algorithm given by Albrecht, Benčina and Lai (EC'25), using the mass formulae proven by Li, Shi and Ling (IEEE Trans. Inf. Theory 71(1)). This allows us to instantiate code-based cryptographic schemes that rely on the hardness of Permutation Code Equivalence (PCE) on `random' self-dual codes for security and that were previously unable to sample them. Building on the observation by Bardet, Otmani and Saeed-Taha (ISIT'19) that Euclidean orthogonality is insufficient when considering PCE over finite extension fields due to non-trivial Galois automorphisms, we study the behaviour of what we call total orthogonality, that is orthogonality with respect to all induced Galois geometries simultaneously. We characterise total orthogonality of vectors and codes, and give an algorithm that samples linear codes with a total hull of a prescribed dimension; a subcode that acts as the hull in all Galois geometries of the ambient space. The algorithm incurs a rate decrease by a factor equal to the extension degree, however, we argue why this may be necessary in the context of PCE and explore how it limits the practicality of our algorithm. We consider the notion of Galois type of a linear code when Galois hulls are symmetric and show that all linear codes have constant Hermitian type.

We establish a global large-data existence theory for the Bhatnagar–Gross–Krook equation in a slab with the physical collision frequency given by the local density. We consider both inflow and Maxwell boundary conditions and assume only finite mass, kinetic energy, and entropy of the initial data, together with the corresponding inflow flux bounds. The data need not be small or pointwise bounded, and we impose neither positive lower bounds on the density or temperature nor prescribed velocity moments above order two. To our knowledge, this is the first nonperturbative global existence result for this initial and boundary value problem under these assumptions. The central obstacle is the simultaneous loss of energy compactness and integrability of the unrenormalized collision operator. Finite energy alone does not give uniform integrability of the second velocity moment, while the density dependent collision frequency prevents separate local $L^1$ control of the gain and loss terms. We overcome these difficulties through a compensated estimate uniform in the truncation of the collision frequency, a de la Vallée Poussin weight determined by the data, and two successive renormalizations using entropy production. This yields the compactness of the macroscopic energy, local Maxwellians, and nonlinear collision term, and hence global renormalized solutions.

Multiscale, parameter-dependent partial differential equations (PDEs) pose severe computational challenges due to strong coefficient heterogeneity and high-dimensional parameter spaces. We develop a geometric interpolation approach within the multiscale generalized finite element method (MS-GFEM) that targets the most expensive component: computing parameter-dependent optimal local approximation spaces. Leveraging the spatial localization of MS-GFEM and assuming local parameter dependence, we decompose the global problem into parametrically low-dimensional local subproblems. The optimal subspaces for each parameter are identified as points on a Grassmann manifold and approximated via Grassmann interpolation on sparse grids, which preserves the geometric structure of these spaces while efficiently handling high-dimensional parameter spaces. The resulting localized model reduction method inherits the nearly exponential spatial convergence of MS-GFEM and the parametric convergence rates of sparse grids. Numerical experiments for elliptic problems confirm the theoretical convergence results.

Anisotropic Voronoi diagrams, in which each cell is the region first reached by an ellipse growing from its site, generate non-convex cells whose integral geometry can be described using Minkowski tensors of arbitrary order. In this work, we derive semi-analytical formulas for Minkowski tensors of arbitrary order of the 0th, 1st and 2nd kind for planar anisotropic Voronoi cells that are star convex at their nucleation point. The resulting formulas involve integrals of contact functions, which describe the cell boundary through a diffeomorphic transformation. This transformation renders the geometry amenable to analytical treatment, enabling us to obtain expressions for the desired Minkowski tensors. Although most of the integrals lack closed-form solutions, they can be efficiently approximated numerically once the contact function and its derivatives are known along the boundary of the cell of interest.

Let $S_E$ be the surface in Xiao's degree-four family identified by Polizzi as a smooth divisor in $E(3)$. Starting from Polizzi's branch model and Xiao's classification of the thirteen singular fibers, we derive the local spherical braids and their Picard–Lefschetz lifts, including the colored $3+3$ partitions at the seven reducible fibers. We identify the normalization of each bisection with $E/\{\pm1\}$ and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove $\pi_1(S_E)\cong\mathbb Z^2$, compute the elementary-divisor-$4$ Albanese kernel of a regular genus-two fiber, and show that the six nonseparating Picard–Lefschetz transformations represent the six cusps of $\Gamma(4)$. We also prove that the natural product tori near an elliptic section are nullhomologous and cannot lower $b_1$ below $2$ by torus surgery. As a separate application of the degree-three monodromy, we give a twisted-double construction of an exotic $\mathbb CP^2\#7\overline{\mathbb CP}^{\,2}$.

This paper presents a model-free method for turbine repositioning control in floating offshore wind farms. The method uses a distributed online optimization framework to mitigate the wake effect. Conventional approaches rely on centralized, model-based control architectures, which suffer from poor practical feasibility. To overcome this limitation and address nonconvexity and inaccessible gradient information, we incorporate physics-informed knowledge of wake dynamics into the control system design. We demonstrate the effectiveness of the proposed method by comparing it with a centralized, model-based baseline in mid-fidelity simulations. Results indicate that the proposed method achieves slightly better performance while significantly reducing computational cost.

We determine the complete relative isoperimetric profile of the three-dimensional unit cube and classify all minimizing sets, including the equality cases at both transition volumes. Up to cube isometries and complementation, the minimizers are corner eighth-balls, edge quarter-cylinders, and coordinate slabs, with transitions at $4\pi/81$ and $1/\pi$. Building on this classification, we prove that the perimeter deficit at each fixed volume controls the squared $L^1$ distance to the full minimizing family, with optimal exponent two. Near either transition, a uniform estimate combines the distance to each adjacent phase with its excess perimeter and identifies the linear degeneration, at neighboring volumes, of the best constant for distance to the actual minimizers. As consequences, we resolve the isoperimetric conjecture in Milman's formulation (Conjecture~1.1) for all cuboids $(0,\beta)\times(0,1)^2$, $0<\beta\leq1$, and the three-dimensional case of Brezis's Open Problem~10.1.

We prove a sharp $L^\infty$ bound for the density of an infinite convolution in terms of $L^2$ norms of convolution powers of the summand densities. The bound may be finite even when every summand density is unbounded and the products arising from Young's inequality diverge.

The Akutsu-Deguchi-Ohtsuki (ADO) invariants of links are a non-semisimple version of the colored Jones polynomials. We prove that the top coefficients of the ADO invariants of links in $S^3$ are multiplicative under Murasugi-sum along connected minimal genus Seifert surfaces. More generally, for any simple complex Lie algebra $\mathfrak{g}$ of type ADE and any $r\geq 3$, we prove that the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant is multiplicative under Murasugi-sum. As a corollary, we get that all the $\mathfrak{g}$-ADO invariants of fibred links are $q$-monic, generalizing a previous result of the authors for $\mathfrak{sl}_2$. Moreover, whenever $\mathfrak{u}_q(\mathfrak{g})$ is ribbon, we find an expression for the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant in terms of Hopf algebra integrals of universal $\mathfrak{u}_q(\mathfrak{g})$-invariants of bottom tangles.

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

Broadly studied origami crease patterns such as the Miura-ori are capable of rigid folding that maintains their spatial periodicity. However, periodic crease patterns generically require additional creases and fold into quasi-cylindrical shapes. Here, using group theory, we show that the folding properties of such sheets are largely determined by their spatial symmetries, known as layer groups. Within each symmetry class, the allowed linear shape-periodic modes of sheets are constrained and classified by the irreducible representation they belong to. Such classification further gives a sufficient condition, containing a symmetry constraint and a topological constraint, for the existence of a nonlinear strain mode. In this way, familiar special cases are placed within a unified framework that connects symmetry, network topology, and deployability. Finally, we turn our classification results into a predictive design principle that enables the rapid design of novel sheets with prescribed properties.

We study the problem of recovering low-dimensional latent geometry from a random geometric graph generated by noisy, high-dimensional data. Specifically, we analyze the performance of a spectral embedding algorithm on the Signal+Noise Graph Model, in which vertices are associated to points perturbed by Gaussian noise, and edges are included for pairs whose inner product exceeds a specified alignment threshold. In the high-dimensional regime where the number $n$ of points and the ambient dimension $d$ both tend to infinity, we show that under a spectral gap condition, the top eigenvectors and eigenvalues of the graph's adjacency matrix can be used to approximately recover the point cloud up to an orthogonal transformation. We illustrate our results on point clouds sampled from nested spheres and high-dimensional sinusoid curves.

Recently, Girsch-Kurinczuk have obtained new formulas for local newforms in depth-zero and minimax integral-depth cuspidal representations of $p$-adic $\mathrm{GL}(n)$. One of their main results is an averaging formula which provides an integral representation of the Whittaker newform in terms of the Paškūnas-Stevens Whittaker function. In the present paper, we first prove a reverse averaging formula by establishing an integral representation of the Paškūnas-Stevens Whittaker function in terms of the Whittaker newform. Together, these formulas give an explicit two-way passage between these two distinguished vectors. Finally, we use our averaging formula and earlier work of Kurinczuk-Matringe to give new explicit test vectors in terms of Whittaker newforms for such cuspidal $\mathrm{GL}(n)\times\mathrm{GL}(n)$ pairs, whenever their Rankin-Selberg $L$-factor is not identically equal to $1$.

We investigate a continuous-time financial market where the asset price exhibits weak (sublinear) mean reversion and has a nonzero drift. Complementing earlier work on strong (superlinear) mean reversion, we show that, for an investor maximizing expected exponential utility, the certainty equivalent grows as $O(T^{2\beta+1})$ where $0<\beta<1$ is the strength of mean reversion. An explicit asymptotically optimal strategy is also given.

We denote by $f_k(\Gamma)$ the largest integer with the property that every subset of a finite abelian group $\Gamma$ of size at least $|\Gamma| - f_k(\Gamma)$ is a $k$-fold sumset. Extending a recent result of Alon and Pham, we prove that $$ f_k(\Gamma) \leq \widetilde{O} \left(n^{(2k-1)/(4k-3)}\right) $$ holds for all finite abelian groups $\Gamma$ and integers $k \geq 2$, where $n = |\Gamma|$. Additionally, we also show that the lower bound $f_k(\Gamma) \geq \widetilde{\Omega} \left(n^{1/k}\right)$ holds if $\Gamma$ has no nontrivial element of order dividing $k$. Our upper bound improves a previous result of Balogh, Liu, and Sharifzadeh, and recovers the bound of Alon and Pham in the case $k = 2$. The proof relies on a new upper bound for the independence number of random Cayley sum hypergraphs, which may be of independent interest.

We study the critical points of a quadratic functional depending on the gradient of the Bach tensor on Riemannian four-manifolds, which generalize the Bach-flat condition. We show that, on every closed four-manifold, there exists a weak Bach-parallel metric, i.e. a critical point for this functional with respect to conformal variations: in particular, we prove that there exist infinitely many conformal classes which contain a unique minimizer for the functional, up to constant positive rescaling. Next, we analyze the global minima of the functional, i.e. metrics with parallel Bach tensor, relating these metrics to well-known variational problems. Using a version of de Rham's splitting theorem on complete four-manifolds, we provide a classification result for products of surfaces, exploiting the theory of conformal gradient solitons; we also construct a new explicit example of a Bach-flat metric which is neither locally conformally flat nor conformally Einstein and we characterize HCMU metrics on complete surfaces. Finally, we prove an equivalence between the Bach-parallel condition on 4D cylinders and the existence of critical metrics for a well-known quadratic curvature functional in dimension three: in this direction, we also prove a characterization of flat three-manifolds, under some curvature and finite energy assumptions.

We establish a counterpart, for individual finite open covers, of the global entropy-mean-dimension comparison of Lindenstrauss and Weiss. More precisely, the mean dimension of a finite open cover is quantitatively bounded in terms of its topological entropy and cardinality. In particular, a finite open cover with zero entropy has zero mean dimension. The same comparison holds for actions of countable amenable groups and in the sofic setting. It follows that every mean dimension pair is an entropy pair and, in particular, that uniformly positive mean dimension (UPMD) implies uniformly positive entropy (UPE). The cover and pair implications answer Questions 2.27 and 4.9 of García-Ramos and Gutman, respectively. The proof associates a polytope to a minimal subcover and combines the Lindenstrauss-Weiss compression argument with Maurey's empirical method to estimate the volumes of its coordinate projections.

Using differential Galois theory—specifically, Magid's theory of the complete Picard-Vessiot closure—we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\ Amer.\ Math.\ Soc.\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.

A Banach space $X$ is $L$-embedded in its bidual if there is a projection $P\colon X^{**}\to X$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in X^{**}$. We show that, as long as $X$ has dimension at least $2$, its Lipschitz-free space, denoted by $\mathcal{F}(X)$, is not $L$-embedded in its bidual. Our methods have applications to the problem of when the $L$-embeddability of $\mathcal{F}(M)$ passes to $\mathcal{F}(A)$ for a metric space $M$ and $A\subseteq M$. This is the case when $A$ is compact or when $A$ is geodesically closed.

Extending the well-known Riemannian positive mass theorem to low-regularity metrics is an ongoing effort, starting from the early 2000s, that has recently seen renewed interest. Contrary to many other approaches we do not wish to assume smoothness at infinity nor to introduce a new definition of mass tailored to our approach, but rather go back to the coordinate isoperimetric mass originally proposed by Huisken. We establish new asymptotic expressions for the coordinate isoperimetric mass of continuous asymptotically flat metrics, which constitute the main ingredient in proving a finiteness criterion, coordinate invariance and equality to the generalized ADM mass of Lee and LeFloch under reasonable assumptions. In particular, for $W^{1,p}_{-\tau}$ asymptotically flat metrics, with $p>n$, $\tau>(n-2)/2$, and nonnegative distributional scalar curvature, the coordinate isoperimetric mass is indeed coordinate invariant, and we establish a positive mass theorem in this regularity without any topological assumptions aside from asymptotic flatness. Our proof relies on a density theorem similar to that of Lee, Lesourd, and Unger, but under a weaker regularity and decay assumption, which may be of independent interest.

We study entropy production for stationary diffusions on separable Hilbert spaces with possibly degenerate, state-dependent trace-class covariance. We first define the reversible–irreversible drift decomposition to provide a candidate drift for the reversed dynamics. Then by working directly with the invariant measure, we establish a lower bound in terms of the extended stationary Cameron–Martin energy of the irreversible drift, without requiring a diffusion representation of the stationary reversal. The bound implies infinite entropy production when the energy is infinite and we establish sufficient conditions for equality in the finite-energy regime. We also develop complementary criteria for identifying the stationary reversal as a Hilbert-space diffusion with constant or continuous state-dependent diffusion coefficients, respectively. Two nonlinear infinite-rank examples are provided to show that finite entropy production does not require the irreversible drift to lie in the covariance range, whereas entropy production may be infinite even with globally Lipschitz coefficients and injective covariance.