Papers

We study a local graph-based interpolation framework for data defined on graph vertices embedded in a metric space. The resulting interpolants are constructed from variational graph splines defined on small neighborhoods, yielding continuous-domain extensions of the graph data. We analyze their behavior through energy estimates, Lipschitz continuity, and stability properties, and compare them with $k$-Nearest Neighbor ($k$NN) interpolation as a natural baseline due to its simplicity and widespread use. Numerical experiments on the manifold of symmetric positive definite matrices illustrate the relative performance of the proposed local interpolation schemes and $k$NN interpolation across varying levels of smoothness and noise.

For endomorphisms of the complex projective plane of a given degree, we prove that the existence of a special curve of bounded degree, in the sense of the dynamical Manin–Mumford conjecture, is confined to a proper algebraic subvariety of the parameter space. This confirms a conjecture of DeMarco and Mavraki in dimension two. For general regular polynomial endomorphisms of $\mathbb A^k$, we obtain a uniform dynamical Manin–Mumford bound outside an exceptional hypersurface. These results arise from a common strategy, valid in every dimension and codimension, which reduces such questions to the geometry of exceptional loci in the space of pairs formed by subvarieties and maps. We control these loci by combining arithmetic equidistribution, blender constructions, dynamical directions, monodromy arguments, and degenerations at the boundary of parameter space.

For a $C^*$-algebra $A$ let $\mathcal I(A)$ be the set of all elements $a+b$ with $a,b\in A_+$, $ab=0$ and $\|a\|=\|b\|=1$. We study the algebraic, metric and topological properties of $\mathcal I(A)$. We describe $\mathcal I(A\oplus B)$ and show that $\mathcal I(A/J)\subseteq\pi(\mathcal I(A))$ for the quotient map $\pi:A\to A/J$. For finite-dimensional $A$ we prove $\mathcal I(A)=\operatorname{Sph}^+_A(E(A))$, where $E(A)$ is the set of minimal projections of $A$ and $\operatorname{Sph}^+_A(E)$ is the set of positive norm-one elements at distance one from every element of $E$; we also show that the analogous statement with the minimal projections of the bidual fails in general. When $\mathcal I(A)\neq\varnothing$ the distance from $1$ to $\mathcal I(A)$ is $0$ or $1$, according to whether $A$ has a nontrivial projection. Moreover, $\mathcal I(A)$ is norm compact if and only if $A$ is finite-dimensional, closedness of $\mathcal I(A)$ passes to hereditary subalgebras, $\mathcal I(A)$ is norm closed for every von Neumann algebra, and the extreme points of $\mathcal I(A)$ are exactly its projections. For a compact, locally path connected space $X$ the set $\mathcal I(C(X))$ is closed, whereas $\mathcal I(\ell_c(A))$ fails to be closed when $A$ is unital without nontrivial projections.

We show that if $\Pi \subseteq G/P$ is a projected Richardson variety, then the degree map $\mathrm{Pic}(\Pi) \to A_1(\Pi)^\vee$ is injective. Moreover, it is an isomorphism in characteristic zero and in all but finitely many positive characteristics. We conjecture that it is an isomorphism in every characteristic. Our proofs are geometric, using a linear paving, topological properties of klt singularities, and reduction to characteristic zero.

We study two invariants of $C^*$-algebras built from positive zero divisors. First, we associate with a $C^*$-algebra $A$ the graph $\Gamma_+(A)$ whose vertices are the nonzero positive zero divisors of $A$, two vertices being adjacent when they are orthogonal. We show that $\Gamma_+$ of the Calkin algebra is connected of diameter $3$, that $\Gamma_+(B(H))$ is connected of diameter $4$ (for $\dim H\ge3$), that $\Gamma_+(A)$ is connected of diameter at most $4$ for every von Neumann factor $A\neq\mathbb C,M_2(\mathbb C)$, and of diameter at most $3$ if $A$ is not of type~I, whereas $\Gamma_+(M_2(\mathbb C))$ is disconnected. Second, we study the zero-divisor real rank $ZDRR(A)$, obtained by replacing invertibility in the definition of real rank with the weaker condition of being a non-zero-divisor, so that $ZDRR(A)\le RR(A)$. For commutative algebras $C(X)$ we prove $ZDRR(C(X))=RR(C(X))=\dim X$ when $X$ is a compact almost-$P$-space, while $ZDRR(C(X))=0$ for separable $X$. Using Levy's connected, totally ordered almost-$P$-space we construct unital commutative $C^*$-algebras $A_n$ with $ZDRR(A_n)=n$ for every $n\in\mathbb N_0$. On the other hand $ZDRR(C(X\times[0,1]))=0$ for every compact Hausdorff space $X$, so $ZDRR$ and $RR$ can differ arbitrarily.

For integers $k\ge3$ and $0\le \ell<k$, we call a graph $(\ell\bmod k)$-cycle-restricted if every cycle has length congruent to $\ell$ modulo $k$. Akbari, Jamaali, Mahmoody, and Seyed Fakhari [On the size of graphs whose cycles have length divisible by a fixed integer, Australas. J. Combin. 45 (2009), 67–72] determined the maximum number of edges in a $2$-connected $(0\bmod k)$-cycle-restricted graph and characterized the graphs attaining equality. In this paper, we determine the maximum number of edges in a $2$-connected $(\ell\bmod k)$-cycle-restricted graph for every $\ell \neq 0$, and completely characterize all graphs attaining equality. Combined with the result of Akbari et al., our result gives a complete solution for all residue classes modulo $k$.

In a previous work, we defined a quotient graph of the Bruhat-Tits tree associated to the degree two special linear group over a local field whose vertex set is in correspondence with the maximal orders in a certain number of spinor genera. This graph has been applied to the study of the structure of some arithmetic groups, their presentations and their (co)homology groups. The genera of Eichler orders, which, for instance, are related to the problem of finding a basis that simultaneously splits two vector bundles with the same generic fiber, can often be described in terms of simpler genera via quotients. Furthermore, some embedding problems of orders can be studied via quotient graphs. These results provide a geometric framework for the study of maximal orders over function fields. In this article, we extend that construction to a higher dimensional setting through the use of singular chain complexes. Then we describe the set of vertices in this quotient that correspond to split maximal orders and the topology of the corresponding region of the quotient complex. More precisely, we show that these “split vertices” fall into a finite number of quotients of apartments by suitable subgroups of the corresponding Weyl group, the symmetric group. We relate these “quotient apartments” with the cuspidal sectors defined by previous authors who studied the quotient by the group of integral points of the special linear group. Both quotients are indeed related by a ramified cover that is unramified on cusps, as proved here.

The Game of Cycles, invented by Francis Su, is a two-player game in which the players take turns drawing arrows on the edges of a plane graph, never creating a sink or a source. On a tree only the sink-source rule matters, and the player who makes the last move wins. Several authors conjectured that on a tree the second player wins exactly when the number of markable edges is even. We show that this parity conjecture is false, and we explain why. The explanation is a hidden game of Nim: a path segment that is blocked by an arrow at one end and free at the other behaves exactly like a nim heap whose size is its length. Combined with a calculus for cutting and gluing trees at vertices next to leaves, this shows that the conjecture fails as soon as a single tree has Grundy value at least 2. The smallest counterexamples have 15 vertices. Trees with Grundy values 2 and 4 first appear at 17 vertices, and every value up to 63 occurs, which answers a question of Fokkink and Zandee. We also refute a related conjecture of Zandee. We then look closely at spiders. The spiders S(2,3,z) violate the conjecture for every z >= 14. The Grundy values of S(1,2j,z) are given by an exact formula governed by binary carries. For 99 pairs (x,y), the value of S(x,y,z) is eventually the parity of z. On the positive side, the conjecture holds for all trees in which every thread has at most two edges, and this bound is sharp. Along the way we prove a twin-branch reduction, give an algorithm that is polynomial for spiders with a fixed number of legs, and collect the questions that we could not answer. The boldest asks whether every non-negative integer is the Grundy value of some tree.

In deep hierarchies of adaptive mesh refinement, the update work that subcycling avoids is set mainly by the dimension of what is refined, not of the space. Subcycling (local time stepping) does less work than global stepping, advancing each level with the finest step, provided its extra coordination cost stays below an overhead budget $\beta=1-1/\Sid$ as a fraction of the global-stepping work. Here $\Sid$ is the ideal speedup that the cell counts give for equal update costs. When each level follows a feature of box-counting dimension $\delta$ and the counts grow geometrically, the budget of a code that advances every cell tends to $(r-1)/(r^{\delta+1}-1)$ for refinement ratio r. A point leaves a budget near one, while a surface in three dimensions leaves 1/7 at r=2 and 1/21 at r=4, values implied by the planar estimates of Dursi and Zingale. Published comparisons in discontinuous Galerkin and octree codes with local time stepping imply overheads of 0.1 to 5 percent of the work, below 1/7 and comparable with 1/21. In paired runs of one AMReX code on one core, a cost model fixed in advance and fitted on points and filled regions predicted the speedups of curves and surfaces with a median error of 7 percent (22 percent on the saving); post hoc, the update counts alone did as well except for points. The overhead, as defined here, was negative in all 17 configurations, but when computed from speedups like the published values, it ranged from -0.11 to 0.19, and no configuration came near break-even. When an update costs the same under both schedules, one profiled run can show that subcycling pays.

In this paper, it is proved that every sufficiently large odd integer $n$ can be represented as \begin{equation*} n=p_1^2+p_2^2+p_3^4+p_4^4+p_5^4+p_6^5+p_7^7, \end{equation*} where $p_1,\dots,p_7$ are prime variables. This result constitutes an enhancement upon a previous result of Andrade [1].

Denote by $\Pi_\gamma(x)$ the number of primes $p \leqslant x$ such that $[p^{1/\gamma}]$ is also a prime. In this paper, it is proved that the expected asymptotic formula \begin{equation*} \Pi_\gamma(x)=\frac{\gamma x}{(\log x)^2} (1+o(1)) \end{equation*} holds for $\gamma > 15/13$, which constitutes an improvement upon the previous result of Balog [2] who showed that the above asymptotic formula holds for $\gamma > 6/5$.

We continue the study of weak universality for the fractional hyperbolic $\Phi^4_3$-model initiated by Tzvetkov, Wang, and the author (2023). In this work, we establish weak universality in a regime where the associated Gibbs measure is mutually singular with respect to the underlying Gaussian measure. We first investigate the weak convergence of the Gibbs measures associated with macroscopic models to the fractional $\Phi^4_3$-measure constructed by Tzvetkov, Wang, and the author (2023). We then obtain a structured representation for these Gibbs measures via the stochastic quantization method. Finally, we prove dynamical weak universality for the fractional hyperbolic $\Phi^4_3$-model by using an extended second order expansion and exploiting the invariance of the Gibbs measures.

We provide a new rearrangement in Gauss space transforming symmetric sets while decreasing their Gaussian perimeter. We use this rearrangement to analyze the minimizers of the Gaussian perimeter among symmetric sets.

Chromatic choosability is a fundamental notion in list coloring. A graph $G$ is chromatic-choosable if its chromatic number $\chi(G)$ equals its list chromatic number $\chi_{\ell}(G)$. In 1990, Kostochka and Sidorenko introduced the list color function of a graph $G$, denoted by $P_{\ell}(G,m)$, which is the list-coloring analogue of its chromatic polynomial $P(G,m)$. A graph $G$ is enumeratively chromatic-choosable if $P_{\ell}(G,m)=P(G,m)$ for every $m \in \mathbb{N}$. In 2002, Ohba proved that for every graph $G$, there exists an $N \in \mathbb{N}$ such that $K_p \vee G$ is chromatic-choosable whenever $p \geq N$. Allred and Mudrock recently conjectured that an analogous statement holds for enumerative chromatic choosability. In this paper, we prove that $K_1 \vee G$ is enumeratively chromatic-choosable whenever $G$ is a theta graph. Our proof combines classical inequalities, the polynomial method, and ideas from DP (or correspondence) coloring.

The Fibonacci, Lucas, and Pell sequences satisfy certain families of “polynomial” identities. While a few members of these families are well-known, we are not able to find a general statement of these identities in the mathematical literature. We show that such polynomial identities exist for many generalized Fibonacci sequences, and we classify sequences that have these identities obtained by two distinct approaches. One approach uses the Girard-Waring formula, while the other relies on Binet's formula and linear algebra.

We study Nesterov's accelerated gradient method with its standard constant parameters for smooth strongly convex optimization. We characterize the exact worst-case asymptotic factor for the squared distance to the minimizer. To this end, we construct candidate worst-case trajectories by combining planar spirals with a common contraction factor that satisfy a self-similarity condition. Each component rotates by a fixed angle at every iteration. The exact factor is the optimal value of a linear optimization problem over probability measures on a fixed disk, with constraints derived from smooth strongly convex interpolation. Every optimal measure has a common contraction factor, which is also the largest factor for which the angular interpolation conditions are feasible. We also show that this worst-case asymptotic rate can be realized by a single function in a finite-dimensional space whose iterates attain this factor. A matching upper bound holds uniformly over all functions in the class and all finite dimensions. Consequently, allowing the function, dimension, and initial point to vary with the iteration horizon does not increase the worst-case asymptotic factor. We also show that the exact factor is $1-4/(3\sqrt\kappa)+o(\kappa^{-1/2})$ as $\kappa\to\infty$, where $\kappa=L/\mu$ is the ratio of the gradient Lipschitz constant to the strong convexity parameter.

We study the attractor lattice of a class of hybrid dynamical systems on compact metric spaces. Under the trapping guard condition, we prove that hybrid attractors form a bounded distributive lattice isomorphic to the attractor lattice of an associated semiflow. The regularized preimages define a lattice epimorphism from a sublattice of the regular closed, attracting neighborhoods of the semiflow onto those of the hybrid system, and from the whole lattice when the guard set and its reset image have empty interior. Then, using grids on the state space, we construct a convergent cofiltration of outer approximations of a fixed-time map of this semiflow. This construction realizes every finite sublattice of hybrid attractors by a lattice of attracting neighborhoods that are unions of grid elements at sufficiently fine resolutions, and provides index pairs for computing Conley indices. We illustrate these constructions computationally by extracting the Conley–Morse graphs of the bouncing ball, the rimless wheel, the spiking neuron, and exhibiting the attractor lattice for an impacting oscillator.

Motivated by three-dimensional wave diffraction by conical shapes, we consider the two-dimensional problem of wave scattering on the real sphere $\mathcal{S}^2$ resulting from a point-source incidence on a scatterer $\mathbb{S}$. Using the recently introduced notion of plane waves on $\mathcal{S}^2$ as well as recent results on the Laplace–Beltrami Green's function $G$ on $\mathcal{S}^2$, we derive three plane-wave representations for the scattered field $u$. We then explain how $u$ can be analytically continued to the complexified sphere $\mathcal{S}^2_{\mathbb{c}}$. This allows us to formulate a complexified version of Green's third identity, and to provide a characterisation of complex sources. All our results are then used to draw strong analogies between planar and spherical scattering problems, notably through the introduction of directivities.

This paper studies the last-coverage vertex of a nearest-neighbour random walk on a cycle: the vertex most likely to be the final one discovered. We show that the visited vertices always form a single contiguous arc of the ring, a property we call the Connected Arc Property. This lets the walk be replaced by a two-state time-inhomogeneous Markov chain, which we call the interval-expansion representation, in which each expansion step has a probability given by the classical gambler's ruin formula. Within this framework we prove that the last-coverage probability of node k in an n-node cycle is (1-r)r^(n-1-k)/(1-r^(n-1)), where r = (1-p)/p and p is the clockwise transition probability. As p approaches 1/2, we recover the uniform distribution, giving an independent derivation of a classical result. For a walk with a deterministic time-varying bias, we give an exact series representation for the last-coverage probabilities, together with a finite truncation that serves as a computable, error-bounded approximation. All results are validated by Monte Carlo simulation, and we run experiments on sinusoidal time-varying bias. Finally, we compare generalized Petersen graphs with Erdos-Renyi graphs of matching size, finding that structural symmetry is associated with more uniform last-coverage distributions.

Let $H$ be a closed subgroup of a compact connected Lie group $G$ such that $G/H$ is simply connected, the action of $G$ on $G/H$ is almost effective, and $G/H$ is a strongly isotropy irreducible space. The smallest positive eigenvalue of the Laplace-Beltrami operator $\Delta$ associated to the standard metric on $G/H$ was recently computed. In this article we obtain a second non-zero eigenvalue of $\Delta$. As a consequence, we prove that all eigenspaces of the Laplacian are irreducible as $G$-modules if and only if $(G,H)$ is a compact symmetric pair of real rank one, or $(G,H)$ is equal to $(\operatorname{G}_2,\operatorname{SU}(3))$ or $(\operatorname{Spin}(7),\operatorname{G}_2)$.

Data and the governing partial differential equations (PDEs) of continuous-time dynamical systems provide rich structure for statistical learning. A central challenge is how to integrate data with PDE constraints in a principled way to improve generalization. In this paper, we characterize the statistical estimation and generalization properties of learning solutions to the heat equation with a noisy source function from finite, randomly sampled data. We first derive a novel variational formulation, which we characterize as the minimizer of loss function for an energy regularized and ellipticity debiased Sobolev regression. By formulating the learning task as a supervised problem over space-time domains under physically meaningful norms, our approach goes beyond the traditional supervised learning framework by effectively leveraging auxiliary data—samples of the source function of the heat equation—to design a physics-informed energy regularization. Our analysis reveals that learning in linear parabolic PDEs exhibits intrinsic anisotropy between spatial and temporal regularity, in sharp contrast to the static elliptic problems, leading to fundamentally different estimation and generalization behaviors. Specifically, we derive non-asymptotically rate-optimal estimation errors, as well as generalization error, which explicitly capture the interplay among temporal sampling, spatial resolution, and solution smoothness. Our numerical experiments demonstrate substantial orders of magnitude of reduction in testing errors against various physics-informed neural network methods on various parabolic and hyperbolic PDEs.

We prove Kitaoka's conjecture for all real cyclotomic fields, i.e. for fields of the form $\mathbb{Q}(\zeta_k+\zeta_k^{-1})$ where $\zeta_k$ is a primitive $k$th root of unity. More precisely, we show that such a field admits a universal classical ternary quadratic form if and only if it is a real quadratic field.

Latent recursion repeatedly processes internal representations through the same neural network. Recent work has shown this approach to be effective on structured reasoning tasks such as Sudoku and maze solving. However, whether latent recursion can support neural routing remains unclear. We introduce RELAGRAM (Recursive Latent Graph Model), a novel approach that brings latent recursion to graph combinatorial optimization. RELAGRAM encodes a capacitated vehicle routing problem (CVRP) instance as edge tokens; a compact transformer recurrently updates the latent representations of these tokens. An output head maps the answer state to an edge heatmap, from which an autoregressive decoder constructs a feasible solution. On CVRP-100 instances, the 0.53M-parameter RELAGRAM achieves gaps to a PyVRP reference of 13.4%, 9.8%, and 8.3% under greedy decoding, eightfold symmetry augmentation, and 256-candidate search, respectively. Trained only on CVRP-100, RELAGRAM generalizes zero-shot to CVRP-500 and CVRP-1000, achieving gaps of 15.6% and 21.7% under 256-candidate search. Together, these results demonstrate that latent recursion is a promising approach to neural routing.

A central problem in zero-sum theory is to determine the Davenport constant of a finite abelian group. For a finite abelian group $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$, with $1<n_1\mid\cdots\mid n_r$, let $\mathsf D(G)$ be the least integer $t$ such that every sequence of $t$ elements of $G$ contains a nonempty zero-sum subsequence, and denote $\mathsf D^*(G):= 1+\sum_{i=1}^r(n_i-1)$. The standard construction shows that $\mathsf D(G)\ge\mathsf D^*(G)$. Equality holds for several important classes of groups but fails in general, and even the question whether $\mathsf D(G)/\mathsf D^*(G)$ is bounded by an absolute constant remains open. In this paper, we prove the upper bound \[ \mathsf D(G)<4.86\,\exp(G)\frac{\log|G|}{\log\exp(G)} \] for every finite abelian group. When $|G|$ is large compared with $\exp(G)$, it improves the previous classical upper bound by a factor of order $\log\exp(G)$. We also prove that, for almost all finite abelian groups (counting isomorphism classes by order), \[ \mathsf D(G)\sim\exp(G)\frac{\log|G|}{\log\exp(G)}\sim\mathsf D^*(G). \] We also study the difference $\mathsf D(G)-\mathsf D^*(G)$: for every function $L$ with $L(x)\to\infty$, almost all groups satisfy $\mathsf D(G)-\mathsf D^*(G)\le L(\mathsf D^*(G))$. However, for every fixed $C\ge0$, the groups with $\mathsf D(G)-\mathsf D^*(G)>C$ have positive lower density.

I introduce a new class of cell complexes, called CL-complexes, that provide alternative topological realizations of finite posets. Each realization of a given poset has the same homotopy type as the order complex of that, so in some sense these spaces do not provide anything really new. Their value is that they enable short transparent proofs of standard results in poset homotopy. These include Quillen's Theorem A for finite posets, the Nerve Theorem for good covers of a topological space, and Sheehy's multicover generalization of the Nerve Theorem. Order complexes retain a special status: the order complex of a finite poset admits CL-complex structures both for the poset itself and for its opposite poset. The definition of CL-complexes is motivated by an elementary generalization of the Carrier Lemma of J. W. Walker, which in its original form constructs a canonical homotopy class of maps from a simplicial complex to a space covered by a suitable family of contractible sets. In the generalized lemma, the role played by simplicial complexes and the role played by spaces covered by contractible sets can both be fulfilled by CL-complexes. This sets up an equivalence of categories that tacitly underlies our approach.

In this paper, we study finite-velocity random motions subject to a killing mechanism acting at direction changes: at each change of direction, the motion is killed with a fixed probability and remains at its current position thereafter. We first consider the one-dimensional case of a standard telegraph process. We derive the exact distribution of the position at the killing time and show that it is Laplace distributed. We also characterize the law of the process conditional on survival up to a fixed time, proving that it coincides with that of a standard telegraph process driven by a thinned Poisson process with reduced switching intensity. We further derive the full time-dependent distribution of the killed process, including both its absolutely continuous and singular components. We then extend the analysis to a planar finite-velocity random motion with infinitely many possible directions. In this setting, we derive the distribution of the killed process and obtain an explicit representation of its density in terms of modified Bessel functions and a series involving Bessel polynomials.

Among connected, planar sets, the ratio between the isoperimetric deficit and the squared barycentric asymmetry is known to be bounded from below by a strictly positive constant (as proved in recent papers by Bianchini-Croce-Henrot and Gambicchia-Pratelli). This opens the natural question whether the infimum is attained at some set. In this paper, we prove that a minimizer for this ratio actually exists, and moreover it is simply connected.

We study the large-time behavior of the $L^2$-norm of solutions to the free wave equation on $\mathbb{R}^N$ and, more generally, their homogeneous Sobolev norms of negative order. We refine and extend the results of R. Ikehata [J. Hyperbolic Differ. Equ. 20 (2023)] who identified and quantified the infinite-time $L^2$-growth of free waves in dimensions $N=1,2$. For finite-energy data with $u_1\in L^1(\mathbb{R}^N)$, we derive exact first-order asymptotics: when $P=\int_{\mathbb{R}^N}u_1(x)\,dx\neq0$, the squared $L^2$-norm grows linearly for $N=1$ and logarithmically for $N=2$, with a leading coefficient depending only on $P$. In particular, no weighted moment assumption on $u_1$ is required; $u_1\in L^1\cap L^2$ suffices. More generally, we prove that the $L^2$-norm is bounded, and hence convergent, if and only if $u_1\in\dot H^{-1}(\mathbb{R}^N)$. We further establish first-order asymptotic profiles in negative-order homogeneous Sobolev spaces, covering polynomial growth, logarithmic growth, and convergence to a finite limit according to the Sobolev index. These results reveal a unified low-frequency mechanism underlying the low-dimensional $L^2$-growth. We also derive second-order asymptotic expansions in dimension one and at the critical Sobolev index in dimensions $N\ge2$. Finally, in dimension two, the exact $L^2$-asymptotics yield a sharper local energy decay estimate with an explicit upper asymptotic constant independent of the localization radius.

This article is concerned with the problem of nonlinear (stochastic) filter stability for a discrete time hidden Markov model (HMM) with finite state and observation spaces. For this purpose, a comprehensive duality theory is developed whereby the dual control system is a backward stochastic difference equation (BSDE). The duality is expressed in two inter-related forms. In its variational form, the minimum mean squared error of optimal filtering equals the value of a dual optimal control problem. In its structural form, controllability (resp. stabilizability) of the BSDE and observability (resp. detectability) of the HMM are dual properties. The dual optimal control problem supplies the Lyapunov function and the dissipation: the former is the value function (a variance), the latter the optimal running cost accumulated over the horizon. The main result is an invariance principle: under any one of three model properties—ergodicity of the Markov matrix, or observability or detectability of the HMM—zero dissipation over a finite horizon forces the Lyapunov function to vanish. A converse is also obtained: detectability is necessary. Two applications are given. The first is filter stability in chi-squared divergence: the invariance principle is applied in the manner of LaSalle, to limits of trajectories along which the dissipation vanishes asymptotically, the role of the model properties being to render the zero-dissipation set trivial. The second is a quantitative form of the invariance principle, through a Poincare constant, from which convergence rates follow. Both the model properties that govern stability and the Lyapunov function that certifies it are thus obtained from a dual optimal control problem, as in the classical theory for the linear Gaussian model.

We present a typed proof-term calculus for intuitionistic intensional first-order logic with Hilbert's epsilon-operator and prove its soundness and completeness with respect to hyperdoctrinal models with thin bicartesian closed setoid categories (BiCCCs) as fibres and formula-indexed witness selections. The calculus provides an explicit internal language for these epsilon-hyperdoctrines: for each fixed typed signature, its syntactic term model is initial for syntax-indexed, interpretation-preserving model morphisms, with uniqueness up to setoid equality. The nested epsilon syntax is organized by a well-founded dependency relation on formula labels. Concrete countermodels exhibit failure of excluded middle and show that formula conversion need not identify the selected witnesses. We also compare the full calculus with its epsilon-free and epsilon-only fragments. The principal constructions and metatheorems are formalized in Rocq. With setoid BiCCCs, proof irrelevance is a choice of equality, not the erasure of proofs: the proofs and their categorical operations remain available as the basis for a future proof-relevant categorical semantics.