Papers

This paper develops a functional H_infinity filter for nonlinear descriptor systems subject to external disturbances. Conventional H_infinity filtering approaches for descriptor systems impose restrictive regularity assumptions and employ implicit descriptor-form filters, leading to practical implementation difficulties. Moreover, existing approaches mainly target full-or reduced-order state estimation, which is computationally inefficient when only a specific functional of the state is required. To address these limitations, the filter is formulated directly in an explicit state-space framework and can be initialized with arbitrary initial values. The filter order is chosen to be less than or equal to the dimension of the functional vector to be estimated, thereby reducing computational complexity. The considered nonlinearities are characterized using incremental quadratic constraints parameterized by appropriate multiplier matrices, which encompass Lipschitz, one-sided Lipschitz, monotone, and many other nonlinearities. Sufficient criteria for the existence of the proposed filter are established through a rank condition imposed on the system matrices together with a set of linear matrix inequalities (LMIs). Under these conditions, asymptotic stability of the estimation error dynamics is guaranteed, while the influence of external disturbances on the error is bounded within a prescribed L2-performance framework. Finally, numerical simulations demonstrate and validate the effectiveness of our theoretical results.

We study a viscous one-dimensional conservation law, coupled to a fast ordinary differential equation. For a vanishing viscosity, the system has an infinite-dimensional family of time-periodic solutions, corresponding to relaxation oscillations. We show that for symmetric initial conditions, a small positive viscosity selects a unique periodic solution, which we prove to be linearly stable. The proof exploits the slow-fast structure through an averaging strategy, as well as spectral-theoretic methods. The results are illustrated by numerical simulations.

Observables in quantum mechanics are generally complementary, that is, they reveal mutually incompatible pieces of information about a given system. This property is not only a fundamental tenet of the quantum formalism, but also a key component in quantum cryptographic protocols. As such it has fuelled much research into finding sets of highly complementary observables. In its strongest form – the one we consider in this work – the information between complementary observables is not merely incompatible but mutually exclusive: maximal information about one observable implies no information about the other, and vice versa. Maximal sets of non-degenerate complementary observables are known to exist in systems of prime power dimension such as n-qubit systems. Here, we study complementarity of degenerate observables, specifically we prove that observables associated with the n-qubit Pauli group also exhibit complementarity under this restriction: first, we obtain a criterion for two such observables to be complementary and, second, we relate maximal sets of complementary observables with informational pure state complementarity equalities, further studied in two companion papers. Equivalently, these results can be formulated in terms of maximal complementary sets of (not necessarily maximal) Abelian subgroups of the Pauli group linking with (possibly coarse-grained) mutually unbiased bases. Finally, we prove that these complementarity sets entail strong entropic uncertainty relations.

We present a family of quadratics in Pauli expectation values, and prove that they constitute state-independent invariants for all n-qubit pure states. This family generalises the two-qubit `pentagon identities', discovered in the reconstruction programme of [P. A. Höhn, Quantum 1, 38 (2017), P. A. Höhn and C. S. P. Wever, Phys. Rev. A 95, 012102 (2017)], where they characterise the space of pure states, as well as the unitary group, and are interpreted as complementarity equalities in the Brukner-Zeilinger information measure. The generalisation to arbitrarily many qubits is nontrivial as it requires new tools which in turn reveal novel structural properties that are absent in the two-qubit case. A thorough analysis of these properties, and their relation with mutual unbiasedness and complementarity in the n-qubit Pauli group, can be found in two companion papers.

In this article, we establish the minimal model program for big gklt Kähler pairs, and in particular we prove Tosatti's transcendental base-point-free conjecture.

Generative models are increasingly adopted in distributionally robust optimization (DRO), but existing approaches trade off model compatibility and adversarial structure: methods that accept arbitrary samplers do not restrict worst-case laws to a generator family, while generator-parameterized adversaries rely on model-specific access such as likelihoods, scores, or training data. We propose Generative Distributionally Robust Optimization (GDRO), a principled framework that accepts any sampleable conditional generator as the nominal model and restricts worst-case laws to a chosen conditional generator family. The key is the sampler-Sinkhorn pairing: samplers represent the conditional laws exactly, while Sinkhorn divergence compares their induced distributions without likelihood access and can be estimated from samples alone. The resulting population problem admits a direct finite-sample approximation and differentiable primal-dual implementation at the active decision context. For Lipschitz losses, the population Sinkhorn radius bounds downstream degradation. Across explicit and implicit generators, our method reduces rare-context inventory regret by 60% and SocialGAN navigation collisions by 50% relative to nominal decisions.

We define a family of finite directed graphs $G(k,c,w)$ on vertex set ${\mathbb{Z}/k\mathbb{Z}}\times {\mathbb{Z}/k\mathbb{Z}}$, where $w\in{\mathbb{Z}/k\mathbb{Z}}^\times$ has multiplicative order $c$, which divides $k$. We show that each $G(k,c,w)$ is a central digraph (its adjacency matrix squares to the all-ones matrix). We deduce explicit descriptions of the graph automorphism groups for graphs of type $G(k,1,1)$; $G(k,2,-1)$ when $k$ is even; and $G(p^2, p, 1+p)$, when $p$ is prime.

Differentiable simulation can accelerate contact-rich trajectory optimisation by exposing local sensitivities of task outcomes to controls. Existing approaches either use finite differences, which are expensive and step-size sensitive; differentiate iterative contact solvers by unrolling automatic differentiation (AD), which stores a growing computation trace; or require intricate, solver-specific KKT sensitivity derivations. We introduce an AD-assisted implicit derivative for regularised smooth contacts and apply it to Mujoco MJX, based on the Implicit Function Theorem (IFT). The method differentiates the stationarity residual at the tolerance-converged solution, avoiding both solver unrolling and hand-assembled KKT systems. IFT keeps compiled temporary memory nearly constant with solver effort, changing by less than 4$\%$ from one to ten iterations versus 10.6$\times$ growth for unrolled AD. IFT memory grows slower with active contacts and model dimension, using 20$\times$ less memory at 256 contacts and 6$\times$ less at 16 contacts and 96 DoF. We further introduce optimiser distillation for residual MPC, amortising batched full-horizon iLQR into a policy that guides short-horizon residual iLQR. Across Finger, Franka, and Unitree, this raises six-step success by 28-98 percentage points over standard iLQR.

We study a two-species chemotaxis system that exhibits two qualitatively different regimes: subcritical dynamics, in which solutions remain smooth, and supercritical dynamics, in which strong aggregation may lead to finite-time blow-up. In the subcritical regime, we use a standard continuous-time physics-informed neural network (PINN) with alternating training for the coupled fields and show that it provides accurate and efficient approximations. In the supercritical regime, however, this formulation is not sufficiently robust to resolve the highly localized structures and steep gradients associated with blow-up. To address this limitation, we introduce a discrete-time PINN based on backward Euler time stepping, combined with a logarithmic transformation to stabilize large solution values and a residual-adaptive collocation strategy that concentrates training points near regions of high activity. This hybrid framework allows us to use a simple and computationally inexpensive PINN when the solution is smooth, while switching to a more robust discrete formulation when singular dynamics emerge. Numerical experiments confirm that the continuous-time formulation accurately resolves subcritical dynamics, while the discrete-time formulation reliably captures singular aggregation and finite-time blow-up in the supercritical regime.

We define a notion of Hochschild homology of a weak assembler and use it to construct a Dennis trace map from the group completion $K$-theory of a weak assembler to its Hochschild homology. We compute the Hochschild homology for some examples, in particular for various cases of restricted polytopes. We also give an explicit description of the Dennis trace map at level $0$, i.e. from the zeroth $K$-theory group to the zeroth Hochschild homology group. Further, we show that the Dennis trace is a refinement of the trace/regulator map defined by Bohmann et al thus showing that group homology is a trace invariant.

We show that the Hochschild cohomology of higher Auslander algebras of type $\mathbb{A}$ vanishes in all positive degrees and is one-dimensional in degree zero. As a consequence, these algebras are rigid and intrinsically formal.

We study resonant solutions of the Heun equation and its confluent limits that arise in the Nekrasov-Shatashvili (NS) limit of four-dimensional $\mathcal{N}=2$ $\mathrm{SU}(2)$ gauge theories with fundamental hypermultiplets. At the resonant loci $2a/\hbar\in\mathbb{Z}$ in the Coulomb branch parameter $a$, the Floquet multipliers coalesce and the instanton expansions of the bulk and surface defect NS functions develop poles of increasing order. We derive blow-up equations involving exclusively NS functions and use them to resum these singular expansions. The resulting resummed bulk and surface defect NS functions reveal the analytic structure of the gauge-theoretic solutions near the resonant loci, including the branch structure of the accessory parameter and of the Floquet solutions that is obscured by the term-by-term instanton expansion. At resonance, the resummed accessory parameters and suitably normalized defect wavefunctions admit finite limits that describe periodic or antiperiodic solutions at the edges of spectral gaps and allow us to construct their logarithmic companions. We then identify distinct nested mass loci governing gap closure and semisimple resonant monodromy. On the larger locus the band-edge accessory parameters coalesce, while on the smaller locus two independent resonant (anti)periodic Floquet solutions survive. We develop the general resummation procedure for $N_f=(n_0,n_1)$ theories with $n_i\leq 2$ $(i=0,1)$, and demonstrate it explicitly for the $N_f=(1,1)$ theory.

Consistency with global flavor anomalies requires the presence of Chern-Simons terms in holographic models of QCD. Such terms are analyzed in a setup arising in the holographic V-QCD model, where chiral symmetry breaking is implemented through the condensation of a complex scalar field, the tachyon. Using the superconnection formalism, the Tachyon-Chern-Simons terms are constructed explicitly in the general case, where the tachyon is any complex matrix in flavor space. This general case covers, among other things, backgrounds where different quark flavors have different masses. These new results are used to analyze the structure of the baryon solutions in the presence of nonzero quark masses. Expressions for the baryon number current and the total baryon number are found, and the baryon number is shown to be equal to the topological instanton number of the baryon solution. The effective four-dimensional pion action is analyzed and is shown to reproduce the chiral Lagrangian, including the Skyrme and Wess-Zumino-Witten terms.

Belief-propagation (BP) decoding is attractive for quantum low-density parity-check (QLDPC) codes because it uses local message passing on sparse Tanner graphs. However, conventional flooding BP often stalls due to stabilizer degeneracy and short cycles. Reinforcement-learning-based sequential variable-node scheduling (RL-S), which learns the update order offline, has shown that adaptive scheduling can improve BP convergence. In this paper, we extend this idea with a second-order local update decoder, RL-S2LU. The proposed decoder preserves BP locality and low complexity, while numerical results show significant error-correction gains over conventional BP and the considered BP-OSD-10 baseline.

Bahadur's Lectures on the Theory of Estimation develop the classical theory of point estimation inside the geometry of Hilbert space, and they record with unusual honesty where the theory strains: the locally best unbiased estimate depends on the parameter, a two-point parameter space yields an estimate Bahadur calls absurd, the odds ratio in binomial sampling has no unbiased estimate, and the virtues of maximum likelihood enter as heuristics and remain heuristics. We present a subset of the lectures, in Bahadur's notation and development, and at each strain make one small modification: for each value in the sample space, an estimate $\tau$ becomes a function on the parameter space rather than a point in it, the continuum of null hypotheses that Fisher described in 1955. Bahadur's own definition of an estimate, square-integrable at every distribution in the family, already supplies the domain. The payoffs are tracked lecture by lecture: estimators that exist at boundary samples where point estimates do not; an elementary lemma showing that no pointwise criterion admits a uniformly optimal estimator, which explains why admissibility, minimaxity, Bayes averaging, and unbiasedness arose as responses; assessment by information, $\Lambda(\tau)$, with the score attaining the Fisher information bound uniformly by a three-line argument; Cramér–Rao attainment and sufficiency recovered as equality cases of that bound under two maps from point estimators to generalized estimators; and the maximum likelihood heuristics converted into exact statements about the score. Nothing classical is overturned; the classical apparatus is explained using Fisher's characterization of estimation as a continuum of significance tests.

We study the functional properties of Mahler functions of several variables associated with a large class of Mahler transformations. We prove that these functions are always analytic on a neighborhood of zero, and meromorphic on the open unit ball in C^n , as well as in C_n^p for all primes p. This enables us to prove a rational-transcendental dichotomy for these functions. As a consequence, we strengthen the lifting theorem recently obtained by Adamczewski and Faverjon. This theorem allows us to systematically compute the algebraic relations between values of multivariate Mahler functions related by a Mahler system, given the algebraic relations between the functions themselves. Finally we prove a multivariate descent theorem in the case of regular points.

Generative semantic communication receivers deliver high perceptual quality but suffer from prohibitive decoding latency. This bottleneck arises because diffusion receivers rely on stochastic iterative decoding, while existing flow matching receivers employ independent endpoint coupling that ignores the physical source–channel link, yielding unnecessarily long and curved sampling trajectories. In this paper, we reformulate receiver-side recovery as a realization-coupled bridge flow matching problem under explicit bandwidth and power constraints. Specifically, we propose Realization-Coupled Bridge Flow Matching (RC-BFM), where the decoder initializes from a channel-induced semantic state rather than isotropic noise. Crucially, training pairs are linked via a realization-coupled entropic optimal transport (RC-OT) plan that preserves the physical channel realization of each transmission while maintaining robustness to stochastic fading. Furthermore, we identify independent coupling as the fundamental source of a conditional train–test distribution shift in conditional flow matching-based receivers, and derive an end-to-end distortion bound whose discretization error decays as \(O(K^{-2})\). Experiments on CIFAR-10 and FFHQ-64\(\times\)64 over AWGN and Rayleigh fading channels demonstrate that RC-BFM achieves a superior fidelity–perception trade-off, reducing decoding latency by over 10\(\times\) compared to diffusion-based receivers.

Self-supervised representation learning from unlabeled channel state information (CSI) can reduce labeling and adaptation overhead in learning-based multiple-input multiple-output (MIMO) systems. Existing CSI pretraining methods typically use reconstruction objectives or contrastive pairs from generic augmentations, which do not explicitly reflect transmission-relevant channel similarity. This paper proposes beam-response contrastive learning (BRCL), a self-supervised CSI pretraining framework based on the transmit-side Gram matrix. For a channel matrix $\mathbf{H}$, $\mathbf{R}=\mathbf{H}^{\mathrm{H}}\mathbf{H}$ determines the received power of any unit-norm transmit beam $\mathbf{w}$ as $|\mathbf{H}\mathbf{w}|_2^2=\mathbf{w}^{\mathrm{H}}\mathbf{R}\mathbf{w}$. BRCL maps each CSI sample to a beam-response profile and uses the induced soft similarity as a label-free relational target for contrastive pretraining. Combined with reconstruction learning, BRCL enforces both sample-level CSI recovery and beam-response-level consistency, yielding transferable CSI representations without task-specific labels or manual positive pairs. Experiments on diverse MIMO channel datasets show that BRCL improves label efficiency and outperforms autoencoder- and channel-charting-based pretraining across beam selection, user selection, and future beam selection tasks.

Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a $G$-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension $E$ of $G$ by $N$: an $E$-graded fusion category induces a $G$-SET input by passing to the quotient symmetry $G$, and the resulting input naturally carries a compatible $N$-grading that implements the further condensation inside the SET phase while preserving $G$. We illustrate the construction using three quantum-double examples: the trivial extension $\mathbb{Z}_2\times\mathbb{Z}_2$, the non-Abelian semidirect product $S_3$, and the nontrivial central extension $\mathbb{Z}_4$. The $\mathbb{Z}_4$ example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.

An antidirected path is an oriented path in which every vertex sees either just incoming or just outgoing edges. We prove that every oriented graph with minimum semidegree at least $k$ contains an antidirected path of length $2 k -1$. This confirms a conjecture of Stein.

We prove that area-minimizing submanifolds in mod $2$ homology are not generically smooth, except in the case of geodesics, minimal surfaces and minimal hypersurfaces. This answers questions of White and Yau that ask about the generic smoothness of area-minimizing submanifolds. We furthermore establish a lower bound on the Hausdorff dimension of the singular sets of area-minimizing submanifolds with respect to open sets of Riemannian metrics. The lower bound is ${(d-3)}$ where $d$ denotes the dimension of the submanifold. As a crucial step, we prove that the cone over the Veronese minimal embedding of $\mathbb{RP}^2$ is mod $2$ area-minimizing, settling another long-standing open problem.

The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. In this paper, we consider a class of semilinear PDEs with nonlinearities in the solution and its first derivative. For this class of PDEs, we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution as the network width and training time $\rightarrow \infty$.

The cycle double cover conjecture of Szekeres and Seymour, the proof of which was recently announced by OpenAI, states that every bridgeless graph has a collection of cycles covering every edge exactly twice. We study the counting version of this statement for cubic graphs, where we count circuit double covers — collections of circuits (connected 2-regular subgraphs) covering every edge twice. We show that every 2-edge-connected 3-edge-colorable cubic graph on $n$ vertices has at least $2^{n/2-1}$ circuit double covers, matching our previously conjectured general lower bound. For every 3-edge-connected cubic graph with girth at least 16 we show a weaker exponential lower bound on circuit double covers. For both of these results we use the same system of linear equations used by OpenAI in their proof, however, we provide additional combinatorial interpretation. We characterize planarity of a cubic graph by solvability of this system of equations for arbitrary nowhere-zero $\mathbb Z_2^k$-flow. We give a condition on the flow that is equivalent to existence of a 5-cycle double cover.

We extend the theory of the first chiral homology group of vertex algebras, developed by van Ekeren and Heluani for elliptic curves, to compact Riemann surfaces of arbitrary genus. Our approach realizes a genus $g$ surface by iterated self-sewing of $g$ handles onto the Riemann sphere, each governed by a sewing parameter $\rho_i$ in a punctured disc. We construct an explicit complex computing the chiral homology groups $H^{\mathrm{ch}}_0$ and $H^{\mathrm{ch}}_1$ of a vertex algebra $V$ on a genus $g$ surface with $n$ marked points, and equip it with a projectively flat connection, with an explicit central-charge anomaly, over the $g$-dimensional space of sewing parameters. We prove a genus $g$ Fourier-space Borcherds identity for the associated modified vertex operators. We show that two finiteness assumptions 1) finite dimensionality of the first Poisson homology $\HP_1(R_V)$ of the Zhu $C_2$-algebra, and 2) finite generation of a certain Koszul homology of the associated graded algebra, imply finite dimensionality of $H^{\mathrm{ch}}_1(X,V)$ for every genus $g$ and every vertex algebra $V$. Using the degeneration $\rho_i \to 0$ together with the factorization theorem of Damiolini-Gibney-Tarasca, we relate the totally degenerate limit of $H_1^{\mathrm{ch}}$ to the Hochschild homology of an iterated construction on the Zhu algebra, and deduce vanishing of the first chiral homology group in every genus for rational vertex algebras.

In this paper, we introduce the notion of a controlled theory, originally developed in the author's thesis, as a structural tool for the study of higher categorical algebra. We define a notion of deformation for pros and controlled theories in a cartesian closed category. Furthermore, we show that deformations of controlled theories naturally produce Lawvere theories enriched over the same base category. We construct functorial one-dimensional categorifications and homotopifications of controlled theories, yielding Lawvere $2$-theories and Lawvere theories enriched in simplicial sets, respectively. As an application, we obtain a new model for $\infty$-groups and construct a model of coherent group-like $E_\infty$-spaces, which we will show in future work models infinite loop spaces.

Quantum resource testing is a fundamental primitive of quantum information processing, profoundly connected to resource manipulation. Its goal is to discriminate $n$ copies of a given resourceful state $\rho$ from all free (i.e., resourceless) states; key instances for applications are entanglement testing and quantum magic testing. The asymptotic characterisation relies on the recently proven Generalised Quantum Stein's Lemma (GQSL), which establishes the rate of decay of the false negative error probability for a fixed false positive error probability. This result, however, is intrinsically asymptotic and thus can provide no finite-resource guarantees, which makes its practical implications unclear. Here, we establish the first rigorous finite-$n$ bounds on quantum resource testing and hence quantum resource manipulation, thus strengthening the GQSL and providing explicit estimates on the number of copies needed to achieve a prescribed performance. As notable consequences, we obtain (a) the convergence of the regularised Rényi relative entropies of a resource, which settles the important open problem from [Fang/Hayashi, arXiv:2508.12901, IEEE ToIT 72:6, 2026]; and (b) the first sample-complexity bound for asymmetric resource testing: for any fixed false positive error probability, a false negative error probability of at most $\delta$ can be achieved with $n=O\left(\frac{\log(1/\delta)}{D^\infty(\rho\|\mathcal F)}\right)$ copies of $\rho$, in the limit where $\delta \to 0$.

We previously constructed closed embeddings of Kac-Moody affine Grassmannian slices using fundamental monopole operators. These spaces are defined via the Braverman-Finkelberg-Nakajima construction of Coulomb branches for quiver gauge theories, and the embeddings do not quantize in general. However, there is variant of the BFN construction that produces zastava spaces, and we show that the closed embeddings do quantize in that case. This allows us to take a limit and construct the limit quantized zastava $\mathcal{A}$ for an arbitrary quiver. This algebra plays the role of the Borel Yangian. We also construct a positive part $\mathcal{A}^+$, which plays the role of the unipotent Yangian. By taking the limit of the monopole formula, we show that both $\mathcal{A}$ and $\mathcal{A}^+$ have Hilbert series given by a version for Hua's formula for Kac polynomials. We also show that $\mathcal{A}^+$ is isomorphic to a certain shuffle algebra. Finally, using these results we obtain a proof of Negut's conjecture on the spherical generation of localized shuffle algebras via the second author's work on generators of Coulomb branches.

We establish sharp kinetic trace estimates and counterexamples across several velocity models. For half-space position domains, without any common bound on velocity support, we prove the natural trace estimate for both Lebesgue and standard Gaussian velocity measures. Density yields natural trace operators and Green's formula on the corresponding kinetic energy spaces. For bounded spatial domains in $d\ge2$, in the bounded-support Euclidean velocity model and in the spherical velocity model, we identify the sharp boundary regularity threshold for the trace weights $\min\{|v \cdot n|,|v \cdot n|^p\}$, $1\le p<\infty$. Writing $\alpha_p=1/(p+1)$, the estimate holds on every bounded $\mathrm{C}^{1,\alpha}$ domain with $\alpha\ge\alpha_p$, and it fails for every $0<\alpha<\alpha_p$ on some strictly convex bounded domain of exact regularity $\mathrm{C}^{1,\alpha}$. In particular, the natural trace ($p=1$) has the regularity threshold $\mathrm{C}^{1,1/2}$. On bounded $\mathrm{C}^{1,1/2}$ domains in $d\ge2$, density yields natural trace operators and Green's formula in the bounded-support Euclidean velocity model with either Lebesgue or standard Gaussian measure, and in the spherical velocity model. Norm-preserving velocity translation rules out the unrestricted Lebesgue trace estimate on every bounded $\mathrm{C}^1$ domain. In the unrestricted Gaussian model, for each $1\le p<2$, we construct counterexamples on every bounded $\mathrm{C}^{1,1}$ domain in dimension $d\ge2$, answering Question 1.8 of Albritton, Armstrong, Mourrat, and Novack (2024) negatively. For $2\le p<\infty$, the Gaussian $\omega_2$ estimate and density instead yield $\omega_p$-trace operators.

A derangetropy operator reweighs a probability density by a fixed profile of its own cumulative distribution function, acting through ranks alone. We prove that these operators are precisely the transformations of absolutely continuous laws equivariant under monotone changes of variable, and that they compose through interval maps, making their dynamics exactly solvable: iteration condenses every law onto its median with a universal Koenigs limit law, the continuous flow is solvable in closed form with the Cauchy family as invariant hyperbolic manifold, and balanced against diffusion the distribution function obeys an overdamped sine-Gordon equation whose unique steady law, the hyperbolic secant, is a globally stable kink. A variational principle selects the canonical kernel, the squared ground state of the Dirichlet Laplacian on the unit interval, whose update is a Bayesian posterior costing exactly one bit for every law, and a unitary lift equips each law with an isospectral Sturm-Liouville system and a quantum carpet: a strong law of large numbers for the critical Sobolev mass, with sharp constant given by Wiener's jump statistic, proves the fractality of the Schrödinger density for arbitrary real bounded-variation data with a jump, previously known only for rational step data. A lift to the dual of the Virasoro algebra, at central charge one, a normalization, identifies laws with disconjugate Hill potentials, tails with conformal weights, and the Cauchy family with the exceptional orbit. Randomized phases yield an exact sampler and a multiplicative chaos, log-correlated in the weakly tempered regime; in several dimensions, dependence stratifies into a conserved interaction, a torsion driven by maximal correlation, and a flat Sinkhorn transport.

We show that finiteness of diagonal dimension, in the sense of Li–Liao–Winter, passes to intermediate sub-C*-algebras. More specifically, we obtain an upper bound for the diagonal dimension of the pair of an intermediate sub-C*-algebra in a C*-diagonal together with the diagonal, which we estimate in terms of the diagonal dimension of the ambient pair and the covering dimension of the diagonal's spectrum. This generalizes a theorem of Archbold and Kumjian stating that intermediate sub-C*-algebras of AF-diagonals are AF C*-algebras.