Papers

We give a solution to Polchinski's equation for the Wilsonian effective action in terms of an expansion in binary trees, and prove that this expansion converges in fermionic field theories, provided the fermionic covariance has finite determinant and decay constants. A novel element of the proof are detailed combinatorial estimates for the number of leaf trees associated to binary trees. The method can be used on standard models of fermionic quantum field theory and quantum statistical mechanics.

This paper proposes a nonlinear trajectory shaping guidance strategy for arrival time and angle control of a constant-speed fixed-wing unmanned aerial vehicle. The look angle is parameterized by a fourth-order polynomial. The nonlinear guidance problem is transformed into solving two coupled nonlinear integral equations with respect to two unknown guidance parameters. Directly solving these equations is challenging due to the strong coupling between the parameters. To address this, a two-stage solution procedure is developed. In the first stage, an analytical warm start is constructed. Specifically, by applying approximations, a linear relationship between the two guidance parameters is obtained. With this relationship, a scalar quadratic equation in the first guidance parameter is derived, leading to a good initial guess for the first guidance parameter. In the second stage, this initial guess is refined by solving a one-dimensional nonlinear equation. The obtained solution is finally used as the initial guess for solving the original two-dimensional nonlinear system of equations. Numerical simulations demonstrate that for the test cases, the obtained solution is very close to the open-loop optimal solution, even in highly nonlinear scenarios where some existing methods fail to generate a feasible solution.

In this paper, we investigate which hereditary classes of graphs admit sub-linear (in the number of vertices) bounds on twin-width. By modifying conference graphs, we can show that split, bipartite, and co-bipartite graphs can all have linear twin-width. However, excluding an induced subgraph of each of these three types is equivalent to the class of graphs having bounded VC-dimension, as shown by Bousquet, Lagoutte, Li, Parreau and Thomassé. Graphs of bounded VC-dimension can have unbounded twin-width, but whether it can be linear was an open question. In this paper, we first present a tool for obtaining twin-width bounds in general by contracting a graph based on a partition by distinct neighbourhoods. Then, using this tool, we prove that graphs with bounded VC-dimension have twin-width at most sub-linear. We also obtain a separate, tighter upper bound for the class of interval graphs, as well as a lower bound.

We address a long-standing endpoint problem arising from Zhizhiashvili's logarithmic modulus theorem for multiple Fourier series. We prove an endpoint Dini criterion for almost-everywhere Pringsheim convergence of ordinary rectangular partial sums. In Zhizhiashvili's theorem the logarithmic modulus is assumed with an exponent strictly above the critical value; here this strict power margin is replaced by a summable endpoint Dini condition. As a consequence, one obtains double-logarithmic endpoint classes lying outside the range of the classical theorem. The proof reduces the endpoint smoothness assumption to the Kaczmarz–Kojima product-logarithmic coefficient criterion by weighted translation-difference estimates.

Recently [LT; Theorem 3.1] showed an extension of Walker's inequality [W] based on $N=3$ random variables. In this note we prove that extension is just a particular three-dimensional instance of a general family of second order mixed moment inequalities based on Gram matrices of arbitrary random vectors. We also discuss some implications of these inequalities on Cramer-Rao lower bound for biased estimators.

It is well-known that if $n\times n$ integral matrices $A$ and $B$ of a number field $K$ are similar over all completions of the ring of integers of $K$, then $A$ and $B$ are similar over the ring of integers of a finite extension of $K$. We prove that there is no uniform bound of the degree of extension of $K$ valid for all $n\times n$ matrices. On the other hand, we provide a upper bound of the degree of extension of $K$ for a given separable characteristic polynomial.

We propose and analyze a spherical harmonic pseudo-spectral method for the mean curvature flow of closed surfaces with spherical topology. The evolving surface is represented by a global parametrization over the unit sphere, and the continuous weak formulation underlying Dziuk's method [G. Dziuk, Numer. Math., 1991] is discretized in a finite-dimensional space of spherical harmonics. By explicitly taking into account quadrature errors on the autonomously evolving numerical surface, we prove exponential convergence of the position error under the assumption that the initial global parametrization is analytic. The convergence analysis developed herein is general and could apply to other moving-domain and geometric evolution problems. Numerical experiments confirm the theoretical result.

This paper pertains to the general theory of boundary value problems for general linear differential equations with smooth coefficients in a bounded domain with a smooth boundary and contains new advances in the general theory related to the boundary properties of solutions. Specifically, conditions on the traces of a solution to a general differential equation on the boundary of the domain are found and studied, allowing the solution to be reconstructed from its traces and the right-hand side of the equation. For the case of a general equation with constant coefficients, the resulting conditions on the traces of the solution take the form of a generalized moment problem.

Mathematical search problems present a unique challenge for Reinforcement Learning (RL) due to vast search spaces and sparse rewards. In previous works, the Andrews-Curtis (AC) conjecture was established as an illustrative example of such problems. In this work, we identify a critical structural barrier in the AC landscape: a "Two-Hump" distribution, where problem instances are either trivially solvable or effectively impossible, with a scarcity of intermediate "hard-but-solvable" instances required for effective learning. We tackle this challenge through two primary avenues: novel data generation techniques to populate the difficulty gap, and significant algorithmic enhancements including the introduction of supermoves and Transformer-based architectures. We demonstrate substantial performance improvements over previous baselines, and release new comprehensive benchmark datasets including AC-19 (125,192 AC-trivial presentations of varying difficulty with length at most 19) and AC-1M (1,136,154 hard AC-trivial presentations of length at most 30), the first large-scale, publicly available datasets of this kind.

We give a negative answer to Toponogov's question whether positive mixed sectional curvature on a closed manifold forces the Ferus-Adams dimension bound for a totally geodesic foliation. For every even $N\ge 4$, we construct explicit metrics on $S^{4m-1}\times S^3$ and on totally geodesic fixed point submanifolds $S^{4m-1}\times S^1$. The leaves are closed geodesics forming smooth circle fibrations, but the metrics are not bundle-like. One subfamily has a positive mixed-curvature operator that is parallel along every leaf but nonscalar, with spectrum $\{\kappa,4\kappa\}$; it is a closed realization of Rovenskii's local anisotropic Riccati model. In a second subfamily, $\delta_{\mathrm{mix}}\to 1$ while the curvature operator remains nonparallel. After normalizing $\max K_{\mathrm{mix}}=1$, the leaf lengths tend to $2\pi$, thereby answering the bounded-length question for the local pinched models in the closed setting. Thus neither leafwise parallelism nor pinching by any fixed constant below one replaces scalarity in the Ferus argument; by contrast, the Ferus-Adams bound remains valid in the bundle-like case.

We show there is a translation surface (the golden L) that has unbounded bunching: for every positive integer K there exists a ball B of radius 1 in R^2 that contains at least K vectors that are periods of saddle connections on this surface.

Deep neural networks transform input data into latent representations that support a wide range of downstream tasks. These representations can be characterized along information-theoretic and geometric dimensions, but their relationship remains poorly understood. A central open question is whether low mutual information (MI) between inputs and representations necessarily implies geometrically compressed latent spaces and vice versa. We investigate this question using class-wise clustering as a measure of geometric compression and theoretically sound MI estimation in conditional entropy bottleneck (CEB) networks and continuous dropout networks. We evaluate the interplay between MI, geometric compression, and generalization on classification tasks under controlled noise injection schemes. Our findings show that low MI does not reliably correspond to geometric compression, and that the connection between the two is more nuanced than often assumed. Indeed, our experiments reveal a negative and nonlinear relationship that can reverse when varying training setup. Our results put forward a hypothesis that generalization acts as a potential confounder in this connection rather than being their direct consequence.

In the siblings (or brotherhood) variant of the coupon collector's problem, a main collector draws coupons until her own album is complete and passes every duplicate down a chain of siblings; the $j$th collector is then left with $U_j^N$ empty places, $j\ge 2$. It has been conjectured [stated as an open problem in the work that introduced the model] that, for every fixed number of coupon types $N$ and every $j\ge 2$, the expected deficit $\E[U_j^N]$ is maximized by the equiprobable coupon distribution. We prove this in a sharp, finite-$N$ form: $\E[U_j^N]$ is strictly larger at the uniform vector than at any other probability vector, and indeed strictly increases along every ray running from an arbitrary distribution toward the uniform one. The proof is exact and elementary in its ingredients. An inclusion–exclusion step turns the governing Poissonized integral into a one-dimensional integral with a separable integrand; a single integration by parts then rewrites the radial derivative of $\E[U_j^N]$ as a positively weighted covariance of an increasing function, whose sign is settled by Chebyshev's correlation inequality. We show that $\E[U_j^N]$ is not Schur-concave, so that no majorization or pairwise-smoothing argument can yield the result, and we explain why the recent variance-extremality method of Long~[Long, arXiv:2604.25108, 2026] does not transfer. As by-products we obtain a finite closed form for $\E[U_j^N]$ over subsets of the coupon set and the exact Hessian of $\E[U_j^N]$ at the uniform vector. The argument extends without change to all real $j>1$.

A foundational result in inverse spectral theory due to Ambarzumian (1929) states that the Neumann Laplacian on an interval is not isospectral to the Neumann Laplacian with an additional non-zero potential. In this note, our aim is to investigate Ambarzumian-type theorems for certain classes of Hermitian matrices, including well-known matrices such as the discrete Laplacian on finite graphs. In addition, using different methods, we establish an Ambarzumian-type theorem for matrices with vanishing diagonal, in particular, the adjacency matrix on finite graphs. In this way, we generalize existing results on Ambarzumian-type theorems to general finite discrete graphs.

A finite agent, a machine's digital twin or any bounded reasoner, infers a fixed and noisy world through finite sensors, so its coherent output is a belief: a probability density over states (the Bayes posterior). Such an agent stops short of certainty, and revising a belief carries a cost. We propose a framework for belief costs based on optimal transport, motivated by these facts. We pose two postulates. P0 (the arena): a revision cost is a scalar price on optimal transport, so beliefs live in Wasserstein space. P1 (uniform pricing): one nat of knowledge costs the same metric length everywhere, the eikonal condition. Among conceivable pricing rules we study this one. Under P0 and P1 the cost metric is optimal transport conformally reweighted by Fisher information, $\tilde g_{e,U}=2(e+U)\,g_{W_2}$, and the Fisher family is a characterization: among continuous reliefs, uniform pricing is equivalent to $U=cJ$. Two consequences follow on the conformal class. Certainty sits at infinite cost-distance once the relief dominates the Fisher information, so a well-posed inference has a cost floor diverging at certainty (necessity conjectural beyond power laws). On location-scale leaves the geometry is hyperbolic, and the Stam bound places the Gaussian as the most curved one (at $e=0$). The results are geometric, in nats, and hold up to units: a change of cost unit rescales all distances and preserves every conclusion (boundary, eikonal family, hyperbolicity, Gaussian extremum), a gauge theorem; a global change of state units at $e=0$ is an isometry; the content lies in signs, rankings and ratios. Via Landauer (one nat worth $k_BT$) the cost floor becomes an energy floor: revising toward certainty would demand unbounded energy. Physics anchors the unit and enters no theorem. Removing either postulate leaves the selection open.

Let $D_d(m)=\operatorname{Cay}((\mathbb Z/m\mathbb Z)^d,\{e_1,\ldots,e_d\})$, with all generators oriented positively. We give a second proof that $D_d(m)$ decomposes into $d$ directed Hamilton cycles for every $d\ge 2$ and every odd $m\ge 3$. The combinatorial core is a fixed-row-sum selection theorem for replicated supports: when each indexed support $A$ is repeated in $m$ identical rows, one can select $\lfloor |A|/2\rfloor$ entries from each row so that every column total is a unit modulo $m$. Applied to the Hamilton factors using a chosen coordinate direction, these selections prescribe the voltages in a cyclic lift that splits the direction into two. In fibre coordinates, the lifted successor is $\widehat h_j(x,z)=(h_j(x),z+\mathbf 1_{\{j\in M(x)\}})$. After one traversal of the base Hamilton cycle, the fibre return is translation by the total carry. Since this carry is a unit modulo $m$, the return is a single $m$-cycle and the lifted factor is Hamilton. The new fibres also preserve the direction-constant block structure required for the next split. Iterating from a directed $m$-cycle with $d$ parallel copies of each arc yields the desired decomposition. The proof strategy was proposed with the assistance of OpenAI GPT-5.5 Pro and formally verified in Lean 4.

Muon-type methods have demonstrated potentially superior performance over Adam and its variants, and have shown hyperparameter transferability across model sizes when specific norms are chosen for the LMO in deep architectures. However, while the LMO is solved approximately via iterative algorithms in practice, most convergence analyses consider the ideal case where the search direction is the exact solution to the LMO. Recently, the inexact Muon update was analyzed by Shulgin et al. [2025], which reveals a fundamental coupling between the inexactness and the optimal step size and momentum. However, the convergence is guaranteed for the non-degenerate case only, i.e., the smallest positive singular value of the rescaled momentum is assumed to be bounded below by some positive constant when the spectral norm is used. In this work, we investigate Muon-type methods with inexact LMO in the degenerate case, where the smallest positive singular value of the rescaled momentum can approach zero, for the general non-convex case and the star-convex case with weight decay. Novel assumptions are proposed to address the challenges posed by inexact LMO in such degenerate scenarios, and convergence rates are established under the layer-wise $(L^0, L^1)$-smooth assumption for both cases.

We study the Cauchy problem associated with the higher-order Benjamin-Ono-Schrödinger system \begin{equation*} \begin{cases} \partial_{t}r-a\partial_{x}^{3}r-b\mathcal{H}\partial_{x}^{2}r =cr\partial_{x}r -d\partial_{x}(r\mathcal{H}\partial_{x}r+\mathcal{H}(r\partial_{x}r)) +\beta \partial_{x}(|q|^{2}), \quad x,t\in \mathbb{R},\\ i\partial_{t}q-\alpha \partial_{x}^{2}q=-\beta qr , \end{cases} \end{equation*} where $b,c,d,\alpha,\beta$ are positive constants and $a\neq 0$ is a real constant. This system was introduced by Kairzhan, Kennedy, and Sulem in \cite{Higher-order-Benjamin-Ono-NLS-System-Sulem}. We prove that this system is locally well-posed in the energy space $H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. Furthermore, in the case $a<0$, this result extends globally for initial data $(r_{0},q_0)$ with sufficiently small $H^{1}\times H^1$-norm. The proof combines compactness arguments with energy methods. To provide smooth solutions, we have to deal with the lost of the derivatives phenomenon introduced by higher-order derivatives and the Hilbert transform in the nonlinear terms when performing energy estimates. This is overcome by introducing a modified energy functional that cancels the problematic terms arising in the standard energy estimates. Once this is done, we extend the method put forward by Molinet and Pilod in \cite{HOBOinH1-Didier-Molinet} to study a single higher-order Benjamin-Ono equation. Their procedure includes the use of a gauge transformation of Tao's type \cite{GWP-BO-Tao}, and delicate bilinear estimates in Bourgain type spaces.

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with nonsmooth max-type payoff \(G(x_1,x_2)=x_1\vee\alpha x_2\). The main novelty is a measure-valued variational formulation: the stopping gain \(\Gamma=c+rG-\mathcal LG\) is shown to be a signed Radon measure whose singular component is supported on the kink diagonal \(\{x_1=\alpha x_2\}\), and this component is computed explicitly. We prove that the value admits the killed-resolvent representation \[ V=G-R_r^{\mathcal C}\Gamma, \] where the reflected diffusion is killed upon entry into the stopping set. This corrects the generally invalid unrestricted-resolvent formula. Under explicit monotonicity hypotheses, the stopping set has epigraph form, and the free boundary is characterized by a killed-potential trace condition. A verification theorem certifies locally Lipschitz candidate boundaries as optimal.

The class $M_\alpha$ of $\alpha$-convex functions, introduced by Mocanu in 1969, interpolates between starlike ($\alpha=0$) and convex ($\alpha=1$) mappings and has traditionally been studied only for $\alpha\geq0$. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes $M_\alpha$ for every real $\alpha\neq0$ through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of $\beta$ for which $M_\alpha\subset C_\beta$ when $\alpha\geq1$; we also show that the reciprocal inclusion $C_\beta\subset M_\alpha$ fails for every $\alpha>1$. Building on this, we obtain a sharp Fekete–Szegő inequality, valid for every $\alpha\notin\{0,-1,-1/2\}$, and sharp bounds for the first two logarithmic coefficients of $M_\alpha$, which recover the classical values for starlike and convex functions as special cases. We also determine the order of $M_\alpha$, establish a piecewise sharp bound for its Schwarzian norm together with an explicit quasiconformal extension criterion via the Ahlfors–Weill theorem, and compute the Schwarzian norm of the $\alpha$-Koebe function explicitly for $\alpha=1/n$, $n\in\mathbb N$, verified for $n\leq9$ and numerically supported for $n<50$, leading to a conjecture for the general case. Finally, we propose a conjectural sharp bound for the second Hankel determinant of $M_\alpha$, consistent with the known values $1$ and $1/8$ for the starlike and convex subclasses.

Non-monotone variational inequalities (NMVI) are an important class of problems that generalize non-convex optimization and have various applications in optimization theory, machine learning, game theory, and economics, among others. Most existing work on NMVIs focuses on the asymptotic convergence of algorithms proposed to solve these problems. In this paper, we tackle the problems of exponential and fixed-time stability of the solution set of a class of NMVIs for both unconstrained and constrained problems. We first present novel conditions guaranteeing exponential stability of solutions to unconstrained NMVIs for a uniquely constructed dynamical system under mild assumptions on the gradient of the non-monotone map. Then, under similar assumptions, we construct another novel dynamical system whose equilibrium point is fixed-time stable, i.e., the trajectories reach the equilibrium within a fixed time, independent of the initial conditions. For the case of constrained NMVIs, we employ a continuous-time variant of the Korpelevich method for exponential stability of the solution set, and provide a novel scaling factor in the dynamics to achieve fixed-time stability. We illustrate the efficacy of the proposed modified dynamical systems through numerical simulations and conclude the paper with a brief note on the behavior of the discretized variant of the proposed dynamics and on further work that remains to be done.

The topic of fixed points in digital metric spaces has drawn yet more publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We discuss publications with bad assertions concerning fixed points of self-functions on digital images, as in some of our previous papers

In this paper, we establish a complete classification of minimal translation surfaces within the 3-dimensional Heisenberg group $\mathcal{H}_3$. While translation surfaces in $\mathcal{H}_3$ yield diverse families of solutions depending on the choice of generating curves, we prove that, up to isometries of $\mathcal{H}_3$, there exists a unique class of translation surfaces.

We revisit the topic of false vacuum decay in field theory. We focus on a toy model of a real massive scalar field with an unstable quartic potential. This model has a false vacuum, and decay out of the false vacuum can be described via the method of constrained instantons, which work by introducing a constraint on the path integral. We identify and develop three different asymptotic limits which enable analytic construction of approximate {constrained} solutions. The first, in which the constrained solution is small compared to the inverse mass of the scalar field, is an application of the perturbative methods of Affleck, although we re-derive the main results and identify several terms which were previously neglected. Second, for very large constrained solutions we adapt the thin-wall approximation of Coleman. However, we find that the large instanton limit does not always exist. In this case we identify another useful limit, in which the Lagrange multiplier used to implement the constraint is large. In this limit, the solution's scaling with the parameters may be found via dimensional analysis and an exact solution is obtained with a single numerical computation.

In this work we prove an Aubin-Nitsche Lemma for a positivity-preserving discretisation of an elliptic problem. Due to the nonlinearity of the discretisation, the result requires as a first step the proposal of a linearised adjoint problem that can be linked to the method of choice by appropriately selecting weights. This linearised adjoint problem, together with a regularity result, allow the proof of an optimal-order error estimate in the $L^2$-norm of the error.

Systems in which the network structure and particle states co-evolve in mutual influence are increasingly recognized as essential for modeling complex adaptive systems. However, traditional models of non-exchangeable interacting particle systems frequently assume a fixed network topology, a simplification that fails to capture the dynamical nature of many real-world phenomena. In this paper, we rigorously establish the mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The primary analytical challenge arises from the coupling between the network dynamics and the agents' states, which induces non-Markovian dynamics where the system's evolution depends on its entire history. Consequently, the macroscopic limit is not governed by a classical partial differential equation, but rather by a coupled system of path-dependent McKean-Vlasov SDEs for the particles' states together with a transport equation for the distribution of the weights. A further difficulty stems from the non-linear weight dynamics, which requires an adequate choice for the limiting network structure. To overcome the structural limitations of classical graphon theory, we employ the framework of \(\mathcal{K}\)-graphons (Lovász and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025). To the best of our knowledge, this is the first example in the literature in which the natural topology of probability-graphons is used in the context of mean-field limits, providing a natural and rigorous framework that is fully compatible with non-linear network adaptivity.

Maximum entropy, Bayesian updating, and exponential-family estimation are all instances of a common inference principle: selecting the measure or distribution that minimizes a divergence subject to the available constraints. Which divergence to use is usually decided by analytic convenience, by empirical performance, or by a set of axioms chosen to single it out, leaving open a basic question: why one divergence and not another? We answer it from a single requirement: an inference method should return the same answer whenever the same problem is presented in an equivalent form, for instance, after simply renaming its parts. This requirement alone forces inference to be the minimisation of a classical divergence, and each further reformulation it must respect tightens the admissible family one notch, narrowing the broad f-divergences to the {\alpha}-divergences and finally to the single Kullback-Leibler (KL) divergence. Mathematically, inference is recast from minimising a numerical functional to selecting a least element under a preorder on positive measures, a divergence being merely one numerical scale that reproduces that preorder. The reformulations are the morphisms of a category of inference problems, and the invariance requirement says the inference operator is a covariant functor into the category of statistical models of Cencov, mirroring his characterisation of the Fisher metric. The representation is proved on finite spaces and lifted to general measurable spaces by an elementary closure, covering discrete and continuous spaces alike. Earlier axiomatisations, such as those of Shore-Johnson and Csiszar, postulate their consistency axioms directly and only on finite alphabets; here the axioms follow from reformulation invariance alone.

An $\ell$-cycle decomposition of a graph $G$ is a set of $\ell$-cycles in $G$ whose edge sets partition the edge set of $G$. The $\ell$-cycle decomposition threshold $\delta_{C_\ell}$ is then the least real number such that any $n$-vertex graph $G$ with minimum degree at least $(\delta_{C_\ell}+o(1))n$ has an $\ell$-cycle decomposition if and only if $\ell$ divides $|E(G)|$ and each vertex of $G$ has even degree. Nash-Williams' famous conjecture on triangle decompositions states, asymptotically, that $\delta_{C_3}=\frac{3}{4}$. A very recent breakthrough result of Delcourt and Postle completely resolved this conjecture, however, Glock, Kühn, and Osthus have posed the problem of determining $\delta_{C_\ell}$ for larger odd values of $\ell$ (the behaviour of $\delta_{C_\ell}$ for even $\ell$ is different and well understood). A natural generalisation of Nash-Williams' conjecture implies that $\delta_{C_\ell}=\frac{\ell}{2\ell-2}$ for all odd $\ell \geq 3$. Here we prove that this conjecture holds for all $\ell \geq 73$.

We develop a quantitative partial regularity theory for stable solutions of \[ -\Delta u=f(u), \] where $f:\mathbb R \to [0,+\infty]$ is increasing and convex. The theory is uniform in the nonlinearity and allows for a finite or infinite blow-up level $T_f\in(-\infty,+\infty].$ Our first result is a universal $\varepsilon$-regularity criterion that answers a celebrated question of Brezis: smallness of the scale-invariant mass of the stability potential $f'(u)$ forces Hölder regularity. Moreover, if $T_f<+\infty$, the same smallness condition forces almost quadratic contact between the solution and the blow-up level $T_f$. This result is optimal and, in particular, covers the case of MEMS-type nonlinearities. Our second result identifies a critical exponent $q_f\ge1$, given explicitly in terms of the asymptotic behavior of $f$, $f'$, and $f''$, such that \[ f'(u)\in L^q_{\text{loc}}\text{ for every }q<q_f . \] Combined with our $\varepsilon$-regularity theorem, this yields quantitative bounds for the singular set, in particular \[ \dim_{\mathcal H}\Sigma(u)\le n-2q_f. \] Remarkably, our exponent $q_f$ recovers the sharp thresholds for all standard model nonlinearities, including $f(t)=(1+t)^p$, $e^t$, and $(1-t)^{-p}$. Also, this result provides the first general quantitative singular-set estimates for stable semilinear equations beyond the model nonlinearities. Finally, in the two-dimensional case, we provide a complete picture by proving the universal Hessian estimate \[ \|D^2u\|_{L^\infty(B_{1/2})}\le C\|u\|_{L^1(B_1)}, \] where $C$ depends neither on $u$ nor on $f$. This $C^{1,1}$ regularity is essentially optimal: one cannot expect $C^{2,\alpha}$ estimates for any $\alpha>0$, and in general even $C^2$ regularity should fail.

The energy-based modelling framework proposed by Altmann and Schulze is demonstrated to have two distinct representations as port-Hamiltonian systems. The ambiguity is expressed in the role of the algebraic variable, and is distinct from the usual ambiguity in the geometric representation of coordinate representations of classical port-Hamiltonian systems. A straightforward extension to systems with feedthrough is given.