Papers

How do two agents invent a shared language from scratch? In a Lewis signaling game, a sender and receiver must coordinate on a code using only their interaction history. We study five memory architectures across varying channel configurations with LLM agents and find that memory architecture matters more than channel capacity. Agents with a persistent private notebook benefit from surplus channel capacity and avoid the high-capacity collapse seen in stateless agents, achieving the most reliable coordination ($0.867 \pm 0.023$ at capacity = 25). Stateless agents peak at moderate capacity and then degrade as the vocabulary grows beyond what a rolling context window can track The notebook externalizes learned conventions, freeing agents from having to re-derive codes each round. An information bottleneck-inspired argument predicts an optimal capacity equal to the number of objects. Instead, the bottleneck (capacity = 8) proves to be a fragility point, and surplus capacity is generally better. We show that channel capacity alone cannot predict coordination; memory architecture determines whether agents turn interaction history into stable conventions, and both dimensions are needed to understand how signals become language.

In this paper, we prove that the étale fundamental group of the Néron model of an abelian variety over a number field $K$ is the semidirect product of a finite group with the étale fundamental group of the ring of integers of $K.$ We prove this by studying how the Faltings height of an abelian variety changes under covers that spread out to finite étale covers of its Néron model. We then strengthen this result for elliptic curves. Using Merel's torsion theorem, we show the size of this finite group can be uniformly bounded for a fixed number field. We conclude by giving the list of all possible étale fundamental groups for the Néron model of an elliptic curve over $\mathbb{Q}.$

In previous work \cite{GW}, we developed a theory of modulated \(2j-k\) bi-orthogonal polynomial systems \(\{P_n(z;r),Q_n(z;r)\}\) and \(j-2k\) bi-orthogonal polynomial systems \(\{R_n(z;r),S_n(z;r)\}\), which generalize the classical \(j-k\) Toeplitz systems. In the present paper, we further develop this theory in several directions. We derive simplified and unified recurrence relations for both families of polynomials, prove a more transparent Christoffel–Darboux formula, and give Riemann–Hilbert characterizations of the \(2j-k\) and \(j-2k\) systems.

In many service systems, an estimation of customers' waiting times for the service can assist in decision making focused on enhancing the operational efficiency, improving the customers' experience, and ensuring efficient resource allocation. In this paper, we study the customers' waiting times in a finite-capacity service system with a finite number of parallel servers and a shared waiting area. We consider two customer types, Type 1 and Type 2, with dynamic admission priorities, dynamically evolving customer type, and abandonment. We model the system under such assumptions using a continuous-time Markov chain (CTMC) and develop a methodology based on Krylov subspace approximation methods to evaluate the conditional waiting time distributions of Type 1 and Type 2 customers in the system. This methodology (CTMC-Krylov) offers a scalable computational approach that is well suited for analysing large complex systems. Next, we model this system using a quasi-birth-and-death (QBD) process and derive analytical expressions building on matrix-analytic methods to evaluate the conditional and long-run waiting time distributions using recursion. We illustrate the practical applicability of our models in a hospital system through a suite of numerical examples based on a large dataset obtained from a tertiary referral hospital in Australia, considering two types of patients, complex (Type 1) and other (Type 2).

Can two implementation processes with the same mean lag give opposite stability predictions for adaptive competition? We show that they can, and quantify the discrepancy. The motivating problem is technological coadaptation: decisions are revised while earlier changes are still being implemented and the competing system continues to change. Configuration updates in adaptive cyber defence and activation of revised algorithmic trading strategies provide concrete settings for this modelling question. A coupled replicator–mutator model separates intended portfolios from those actually in use. Near a uniform interior equilibrium, the interaction geometry determines the feedback branches; under strict antagonism, the largest singular value identifies the first unstable mode. For continuous implementation, hard observation and implementation lags enter through one delay budget. Replacing one smooth implementation filter by periodic partial updates, with zero hard lags and a uniform eligibility-phase calibration, preserves the mean waiting time but lowers the first gain boundary by an explicit positive quadratic correction for short release periods. A reproducible three-strategy benchmark exhibits decay under the continuous model and growth under periodic implementation at the same mean lag and interaction gain. Balanced performance observables can also understate the underlying portfolio oscillations. The results identify a concrete failure of mean-only timing descriptions and provide a benchmark for modelling adaptation and implementation separately.

We study weak solutions of electrodiffusion systems coupling the Nernst–Planck equations with fluid models. First, for the three-dimensional Nernst–Planck–Euler system, we establish an Onsager-type criterion for the validity of the coupled kinetic-electrostatic energy balance. The energy equality is shown to hold for weak solutions whose velocity satisfies critical Besov regularity and a vanishing dyadic flux condition. Furthermore, assuming the corresponding Onsager-type regularity for the ionic concentrations, we also prove parabolic regularity, preservation of non-negativity of the concentrations, and the associated charge-density energy identity. Second, for the three-dimensional Nernst–Planck–Navier–Stokes system, we prove a Prodi–Serrin-type uniqueness criterion for Leray–Hopf solutions: uniqueness in the Leray–Hopf class holds whenever the velocity field lies in the Ladyzhenskaya–Prodi–Serrin class $L^p_tL^q_x$ with $2/p+3/q=1$ and $q>3$. These results extend energy-equality and weak–strong uniqueness principles from incompressible fluid dynamics to electrodiffusion models involving convection, diffusion, and self-consistent electrostatic forcing.

Watermarking promises statistical traceability of large language model (LLM) uses, but real documents rarely arrive as purely human-written or purely LLM-generated. This motivates a quantitative question beyond detection: what proportion of a document is generated from a pre-specified watermarked LLM? We study this watermark proportion estimation problem under the Gumbel–max watermarking mechanism, treating the next-token prediction distributions as unknown and arbitrary nuisance parameters subject to a non-degeneracy condition. We compare two observation regimes: in the full observation regime, the estimator observes the pseudorandom vector and the selected token at each position; in the more prevalent setting of pivotal reduction, it observes only a scalar pivot, which follows a one-dimensional Uniform–Beta mixture distribution. Under pivotal reduction, we develop a Laguerre-polynomial estimator and establish a matching information-theoretic lower bound for the sample complexity. For full observation, we introduce an event-counting estimator and show a matching lower bound, yielding a substantially smaller sample complexity. As our results imply, although reducing to pivotal statistics is an elegant and prevalent choice, it is not always sample-efficient for estimating the proportion of watermarks.

In longitudinal studies, outcomes of interest are often truncated by death, meaning that they are only observed or well-defined conditional on intercurrent events such as survival. Existing strategies face a trade-off: causally interpretable estimands, such as survivor average causal effects, target a latent subgroup, whereas while-alive and composite summaries apply to the full population but are difficult to interpret as causal effects on the non-mortality outcome. We address these challenges by introducing methodology for a new set of estimands that (i) concern the entire population, (ii) remain causally interpretable, and (iii) leverage the longitudinal data commonly available in studies with outcomes truncated by death. The set of estimands includes single-world marginal separable effects that generalize conditional separable effects to full-population summaries. We develop identification and estimation results for these estimands and apply the methodology in a reanalysis of a prostate cancer trial, highlighting how different estimands can yield different treatment conclusions.

Artificial intelligence (AI) systems are routinely modified after deployment through retraining and changes in their environments. These transformations raise a metaphysical question: under what conditions does an AI system remain the same system over time or across deployments? Earlier work formulates synchronic and diachronic identity propositionally, by relating identity within a fixed AI system type to equality of trustworthiness levels. Such criteria specify when identity statements are true, but leave implicit the structure of the states compared, the transformations connecting them, and the temporal organization of persistence. We develop a category-theoretic formalization of AI identity. An AI system type is specified by a datum consisting of a techno-function, a trustworthiness profile, and a trustworthiness-level function. Profile-relative states are connected by admissible lifecycle paths, which are restricted to trustworthiness-level-preserving transformations and quotiented to obtain a reachability category. Temporally admissible functors represent AI system histories, while time-synchronous natural transformations compare realized histories. The formalization yields two categorical interpretations of the earlier AI identity criteria. A weak interpretation recovers identity as equality of trustworthiness level. A strong interpretation requires mutual trustworthiness-preserving reachability, expressed through state isomorphism or natural isomorphism of realized histories. Category theory therefore replaces a single AI identity relation with a structured hierarchy of diachronic and synchronic criteria. The resulting framework identifies identity-related preconditions for transferring responsible-AI claims, evidence, and governance procedures across versions or deployments, without treating categorical identity as sufficient by itself for such transfer.

This short review examines the primary approaches for estimating the predictive distribution of Laplace-approximated Bayesian neural networks, with particular focus on the Generalized Linear Model (GLM) formulation. We survey the landscape of estimation strategies, from exact GLM computations requiring full Jacobian evaluations to Monte Carlo approximations that trade computational cost for statistical efficiency. The review covers the theoretical foundations of the Laplace approximation, the Kronecker-factored approximate curvature (KFAC) method for scalable posterior inference, and the various predictive estimation techniques developed in the literature. We provide a unified presentation that clarifies the relationships between methods and highlights their respective computational and statistical trade-offs.

A strong majority edge-coloring of a graph is an edge-coloring in which, for every edge $e$ and every color $i$, at most half of the edges adjacent to $e$ have color $i$. Such a coloring exists only for graphs with no pendant path of length two, which, following Kalinowski, Kamyczura, Pilśniak, and Woźniak, we call admissible. They proved that every admissible graph admits such a coloring with at most eight colors and conjectured that four colors always suffice. We improve the upper bound from eight to five.

We develop a framework for analyzing the learning dynamics of $\ell_2$-adversarial training of single-index models on Gaussian mixtures in the high-dimensional limit under streaming stochastic gradient descent (SGD). We derive deterministic equivalents for a broad class of statistics of the SGD iterates, including the adversarial risk and distance to adversarial optimality, in terms of the solution to a system of ODEs. We use them to study two idealized learning rate schedules: the Polyak stepsize and exact line search. In the case of $\ell_2$-adversarial least squares with a single class, we show that, unlike noiseless standard least squares, no constant learning rate guarantees monotone descent of SGD towards a minimizer of the adversarial risk. We identify anisotropic covariance and a mismatch in ridge parameters as the main sources of suboptimality of exact line search relative to the Polyak stepsize. We also introduce a stochastic differential equation (SDE), called adversarial homogenized SGD, that captures the evolution of statistics of the iterates of SGD. For $\ell_2$-adversarial least squares, using this SDE, we show the evolution of the risk is equivalent, up to dimension-free constants, to that of SGD on standard least squares with an adaptive learning rate and adaptive $\ell_2$-regularization. When the dynamics converge, the limiting adversarial risk and SGD iterate are determined by a fixed-point equation, with the limiting iterate being equivalent to the solution of a ridge regression problem whose regularization parameter is the limiting effective regularization of SGD.

We establish the exact exponential growth rate of the $\rho$-th moment of the constrained guesswork $G_{\mathrm{coset}}$ – the rank of the true noise vector within its syndrome coset of a random binary linear code under i.i.d.\ Bernoulli$(p)$ noise: \( \lim_{n\to\infty} \frac{1}{n}\log_2\Eb\!\left[G_{\mathrm{coset}}^{\rho}\right] = \rho\,h_{\frac{1}{1+\rho}}(p)\;+\;\rho(R-1), \, \rho>0, \) where $h_\alpha(p)$ is the binary Rényi entropy and $R=k/n$ is the code rate. The exponent shifts down by exactly $\rho(1-R)$ relative to the unconstrained Ar{\i}kan–Merhav exponent, with each of the $n(1-R)$ parity checks contributing equally. Finite-length simulations confirm convergence from below. We further establish: (i)~a transfer theorem expressing the partition-function exponent in terms of an arbitrary weight-enumerator growth rate $g(\delta)$; (ii)~the exact exponent for $L_n$-list (“$k$-th”) constrained guesswork; and (iii)~a sharp second-order refinement of order $\rho\log_2 n$. Beyond the binary i.i.d.\ setting, we prove a universality theorem: for any code ensemble $\mathcal{E}$ whose weight enumerator concentrates at rate $g_{\mathcal{E}}(\delta)$, the guesswork exponent equals $(1+\rho)\psi_{1/(1+\rho)}(g_{\mathcal{E}})-\rho\,\psi_1(g_{\mathcal{E}})$, where $\psi_\alpha(g)=\sup_\delta[g(\delta)+\alpha\ell(\delta)]$. As concrete applications, we instantiate this theorem for the $q$-ary extension, $\Lambda_q(\rho)=\rho\,h^{(q)}_{1/(1+\rho)}(P)+\rho(R-1)\log_2 q$, and for Gallager's regular LDPC ensemble, obtaining a closed-form guesswork exponent via an exact finite-length identity for the ensemble-average weight enumerator.

We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interest in their boundedness on the associated noncommutative $L_p$-spaces for $1<p<\infty$. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit $L_p$-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide a complete characterization of a compactness problem posed by Ozawa for graph products of finite groups and right-angled Hecke von Neumann algebras.

We show that Bose–Einstein condensation produces a cusp in the time-of-arrival (TOA) statistics of a harmonically trapped gas released into free fall. In the semiclassical long time-of-flight regime, with $\epsilon=\sigma_V/\sqrt{2gH}\ll1$, both the mean and standard deviation of the arrival time distribution, which are governed by the longitudinal velocity variance, remain continuous, but acquire a cusp whose one-sided slope ratio is universal within the ideal-gas far-field limit, $\mathcal{R}_\infty=2.5556\ldots$, and equals the trapped-gas specific-heat ratio $C(Tc^-)/C(Tc^+)$. Finite atom number rounds the cusp and weak interactions perturb it only weakly, leaving a measurable time-domain signature of condensation.

For all Dyer groups, we find an algorithm to determine when two parabolic subgroups are conjugate. Given two conjugate standard parabolic subgroup, we fully describe the conjugating elements in terms of ribbons, showing that the ribbon conjecture holds true. In particular we give a description of the normaliser of a parabolic subgroup using ribbons. We prove the standardisation property for parabolic subgroups and deduce that an arbitrary intersection of parabolic subgroups is a parabolic subgroup.

We show that many of the standard nuclearity properties considered in the literature for the hierarchy of operator system tensor products can be expressed as approximate factorization properties, generalizing the well-known Completely Positive Approximation Property for nuclear C*-algebras due to Choi and Effros and its generalization to nuclear operator systems due to Han and Paulsen.

Many modern technologies improve through use. Each unit deployed generates data that trains the next generation, so deployment is both production and an investment in a shared learning stock. We study how the architecture of this learning, whether pooled across firms or fragmented within them, interacts with firms' deployment decisions and with product-market competition. In a two-period model, symmetric firms make irreversible capacity choices, and capacity in use feeds a learning curve that raises future productivity. We call this learning-by-deploying, replacing the production experience of the classic learning-by-doing tradition with deployment-generated data. With exogenous prices, pooling raises welfare but firms underinvest in early deployment. Downstream Cournot competition overturns this: pooling depresses the price, so the private value of sharing falls with competition and can turn negative. We characterize a sustainability threshold governed, under general demand, by the elasticity of industry demand over the output range pooling induces, and confirm the patterns numerically.

Generalizing the Falconer distance problem, the authors of this paper recently established the first non-trivial dimensional threshold for any distance graph in high enough of a dimension. The methods developed were flexible enough to generalize from the Euclidean distance to any two point configuration, conditional on results on $k$-stars for the two point configuration. A natural question emerges on what happens to configurations that take in more than two points. In this paper we consider a classic three point variant of the Falconer distance problem, namely that on areas of triangles and its generalizations to volumes of simplices. In this model case we develop two methods. One we call the Jacobian method which allows us, through Heron's formula, to leverage earlier results on distance graphs and obtains non-trivial thresholds for volume vectors determined by a wide range of hypergraphs of simplices. Even in the classic case of the volume of a single simplex this method yields the best known dimensional thresholds if the dimension is considerably bigger than the size of the simplex. We develop a conjecture that has connections to rigidity theory. The Jacobian method works best in high dimensions so in the case of areas of triangles in the plane, we refine the work of Shmerkin and Yavicoli, who recently resolved a conjecture for areas of triangles in the plane, and obtain building blocks from which we can get abundance of area vectors determined by certain hypergraphs of triangles, such as chains of triangles connected on edges or vertices. The results improve and extend existing results of Galo and McDonald as well as of Greenleaf, Iosevich and Taylor.

We study uniform-in-time propagation-of-chaos for continuous-time Stein Variational Gradient Descent (SVGD). Classical finite-time propagation-of-chaos estimates for mean-field systems typically deteriorate rapidly with time and therefore do not directly explain the long-time relation between the finite-particle system and its mean-field limit. We obtain two complementary classes of uniform-in-time propagation-of-chaos results. For broad distributional metrics, we introduce a cutoff strategy which combines finite-time propagation-of-chaos estimates up to an $N$-dependent horizon with independent quantitative long-time convergence estimates for the finite-particle and mean-field SVGD flows. This yields uniform-in-averaging-time propagation-of-chaos bounds in Langevin kernel Stein discrepancy, Wasserstein-1 distance, and Wasserstein-2 distance, with logarithmic or iterated-logarithmic rates depending on the metric, target and kernel class. We also develop a finite-dimensional theory for matrix-valued finite-rank kernels. For Gaussian targets with bilinear kernels, the SVGD dynamics close exactly on first and second moments, yielding genuine uniform-in-physical-time parametric propagation-of-chaos rates in finite-dimensional Stein-feature metrics. We then prove a conjugacy principle showing that these feature-level estimates transfer to conjugate target-kernel pairs under orientation-preserving diffeomorphisms, thereby extending the theory to broad classes of nonlinear, including multimodal, targets. Together, these results highlight the contrast between generic distributional metrics, for which our general approach yields logarithmic rates, and closed finite-dimensional Stein observables, for which parametric $N^{-1/2}$ propagation-of-chaos rates persist uniformly in time.

The symplectic geometry of Coulomb branches is complicated and it is particularly difficult to determine their Fukaya categories. Relative Fukaya categories present an approach to circumvent these difficulties by first computing the Fukaya category of the complement of a divisor and then solving a deformation problem. In this paper, we apply this approach to the specific case of horizontal Hilbert schemes by removing their matter divisor and narrowing down the set of possible deformations through an additional $ \mathbb{Z}^2 $-grading. We utilize an existing description of the Fukaya category after removal of the matter divisor, in particular we use a specific generating Lagrangian and the identification between its endomorphism algebra and the NilHecke algebra. The core of this paper consists of solving the deformation problem, after which we recover the result of Aganagic et al.

Let $\xi$ be an unkilled real-valued Lévy process which drifts to $+\infty$ and has positive exponential moments of all orders, and define $I_\xi=\int_0^\infty e^{-\xi_t},dt$, and its reciprocal $X_\xi=1/I_\xi$. Bertoin and Yor proved that $X_\xi$ is moment-determinate when $\xi$ has no positive jumps, and conjectured that this condition is also necessary. We prove the latter. The proof is based on a lower bound near zero for the law of $I_\xi$. We show that a group of sufficiently many positive jumps near the origin puts $I_\xi$ on a suitable small scale. The first selected jump time is used as a one-dimensional smooth coordinate, yielding an absolutely continuous subcomponent of the law of $I_\xi$. After the change of variables, the resulting subdensity of $X_\xi$ satisfies a Krein moment indeterminacy criterion.

A Nikulin surface is the minimal resolution of the quotient of a $K3$ surface $S$ by a symplectic involution $\iota_S$. Equivalently, it is the $2$-dimensional component of the fixed locus of the involution induced by $\iota_S$ on the Hilbert scheme $S^{[2]}$. We study $K3$ surfaces $F$ that are the $2$-dimensional component of the fixed locus of a symplectic involution $\iota$ on hyper-Kähler manifolds $X$ of $K3^{[2]}$-type; we call them generalized Nikulin surfaces. We show that a projective $K3$ surface is a generalized Nikulin surface if and only if its Néron-Severi lattice contains primitively the lattice $E_7(-2)$. Moreover, we show that the transcendental lattices $T_F$ and $T_{\widetilde{X/ \iota}}$, where $\widetilde{X/ \iota}$ is the terminalization of the quotient $X/\iota$, are Hodge isometric. Finally, we describe projective models of generalized Nikulin surfaces of small degrees.

A classical result states that the Hardy–Littlewood maximal operator is bounded on an Orlicz space $L^A(\mathbb{R}^n)$ if and only if its conjugate Young function $\tilde{A}$ satisfies the $\Delta_2$-condition. The same condition also characterizes the boundedness on $L^A(0,\infty)$ of the Hardy averaging operator. We consider a discrete analogue of the problem, extended to a general interpolation framework. We offer several characterizing conditions for the boundedness of discrete maximal and average operators on Orlicz spaces. Although the principal result is as expected, for its proof some new techniques have to be developed. To this end, we introduce a new notion of the so-called $\Delta_2$-fundamental sequence, and give its interesting characterization by a simple condition involving only a limes superior of the ratio of two subsequent terms. We also prove a dual statement concerning operators of Copson type.

This work considers a Differential Reflecting Modulation (DRM) scheme for Reconfigurable Intelligent Surfaces (RIS) not requiring channel state information (CSI). When operating over time-varying fading channels, such schemes with Conventional Differential Demodulation (CDD) receivers experience high error floors and performance degradation. To address these issues, we propose a Decision Feedback Differential Detection (DFDD) technique for DRM. We explore the application of DFDD for RIS DRM and conduct extensive Monte-Carlo simulations to analyze performance. Results demonstrate the viability of our DFDD technique across various RIS scenarios and highlight the importance of proper parameter selection to achieve good performance. The DFDD scheme is also compared with uncoded and Differential Space-Time Modulation (DSTM) coded DRM using CDD based receivers. We observe that at low SNR, the DFDD scheme performs almost as well as the DRM with CDD scheme, but worse than the DSTM coded DRM. As the SNR increases however, both CDD-detected systems encounter high error floors while the error rate of DFDD based scheme continues to improve until it reaches a relatively low error floor. It is shown that the chief merits of employing DFDD receivers in such RIS systems is the low error floors they provide over time varying fading channels, albeit at expense of a small increased complexity.

We show that $J_{\mu + \nu}(r)^2 < J_{\nu-1/2}(r)^2 + J_{\nu+1/2}(r)^2$ holds whenever $\mu \in (-1/2, 1/2)$, $\nu \in [0, \infty)$, and $r \in (0, \infty)$. In fact, we prove a stronger version for any fixed non-trivial linear combination of the Bessel functions of the first and second kinds. This inequality can be regarded as a kind of interpolation with respect to order. As an application, we establish a dimension-comparison result for optimal constants of smoothing estimates for the free Schrödinger equation. Briefly, the optimal constant on $\mathbb{R}^{d+1}$ is at most twice that on $\mathbb{R}^d$ for each $d \geq 2$.

We study constant mean curvature hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$ invariant under a double horocyclic action. We show that the CMC condition reduces to a single autonomous ordinary differential equation for an angular function. From this reduction, we obtain three distinct regimes and solve the ODE explicitly in each case, obtaining an existence and uniqueness result for double horocyclic CMC hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$. Finally, we classify the equilibrium solutions and identify the corresponding homogeneous models: $\mathbb{H}^3$, $\mathbb{H}^2\times\mathbb{R}$, $\mathrm{Sol}_3$, and left-invariant metrics on semidirect product Lie groups.

A pole of order $m \in \mathbb{N}$ at $\beta \in \mathbb{C}$ of a regular operator valued function $Q : \mathcal{D}(Q) \to \mathcal{L}(\mathcal{H})$ is investigated. We provide a characterization of pole cancellation functions $\boldsymbol{\psi}(z)$ of $Q(z)$ of order $k \le m$ at $\beta$ in terms of the coefficients of the Laurent expansion of $Q$. This characterization yields practical and explicit constructions of pole cancellation functions $\boldsymbol{\psi}(z)$. Moreover, it leads to an explicit formula for the associated functions $\boldsymbol{\hat{\varphi}}(z) := Q(z)\boldsymbol{\psi}(z)$, which are root functions of order $k$ at the zero $\beta$ of $Q^{-1}$. The results are illustrated by an example.

We present a counterexample to Ostrogradsky's famous "no go" theorem as usually interpreted in quantum field theory (QFT), namely a four-derivative, UV-complete QFT with a consistent perturbative expansion which describes high energy scattering processes. We carefully quantize the theory on an $\textit{indefinite}$ space of states - a Krein space - using covariant methods which ensure perturbative causality and unitarity (in the form of the optical theorem) to all orders. We generalize the Born rule to Krein spaces and prove that all tree level transition probabilities are positive in spite of the presence of ghosts. A key role in the proof is played by a hidden "ghost parity" symmetry which becomes explicit when the theory is embedded in a two-derivative, two-field $O(1,1)$-symmetric perturbative field theory.

This paper studies a Chambolle-type minimizing movement scheme for mean curvature flow with prescribed contact angle in a smooth bounded domain. The scheme is based on the capillary functional and the geodesic signed distance relative to the container, and yields a time-discrete level-set approximation. The main result asserts that, for every Lipschitz-continuous boundary function prescribing a strictly nondegenerate contact angle, the approximate solutions converge locally uniformly to the unique viscosity solution of the corresponding level-set mean curvature equation with oblique derivative boundary condition. This improves a previous convergence theorem, where the container was assumed to be convex and a curvature-type condition relating the tangential derivative of the prescribed contact-angle function to the principal curvatures of the container boundary was imposed. The main new ingredient is a uniform Lipschitz estimate for the solutions of the variational problems defining the scheme. This estimate is derived by applying a Bernstein-type argument to a suitable weighted gradient, rather than to the gradient itself, which rules out boundary maxima without relying on the previous curvature-type condition.