Regularity

We study the Bayes-optimal spherical linear model as the ambient dimension and sample size grow proportionally, under a quantitative Marchenko--Pastur spectral-regularity condition on the design. This condition is satisfied by normalized i.i.d. designs with standardized entries of finite fourth moment, but does not require entrywise independence or impose conditions on the singular vectors. Under this condition, we prove a quantitative all-temperature TAP approximation and characterize the posterior geometry. For the natural finite-aspect-ratio TAP functional, the normalized spherical free energy and the TAP optimum differ by $O_P(p^{-1})$. Each is within $O_P(p^{-1/2})$ of its explicit deterministic equivalent, and this fluctuation scale is sharp. Uniformly over all global TAP maximizers, the normalized squared Euclidean distance to the spherical posterior mean is $O_P(p^{-1})$. We also prove that the posterior mass outside a data-dependent band determined by the ridge estimator has sharp exponential order. More precisely, uniformly over sufficiently small band widths $\varepsilon$, the logarithm of this mass is at most $-cp\varepsilon^2+O_P(1)$. For every fixed geometrically admissible width, a spherical-cap construction gives a matching exponential-order lower bound on this mass. For every deterministic sequence of widths $\varepsilon_p\gg p^{-1/2}$, the corresponding bands capture asymptotically all posterior mass.

Numerical simulations reveal how vortices stretch and transfer energy, but establishing smooth evolution requires bounds that remain valid beyond the simulated resolution. Here I develop a computer-assisted framework that establishes global regularity for continuous families of three-dimensional periodic Navier-Stokes flows. Its central construction combines finite reference trajectories with a common error bound that covers an interval of centre fields and infinitely many smooth perturbation modes. The method retains the complete nonlinear residual before spectral truncation and controls the evolution until viscous decay guarantees regularity for all subsequent times. Applications to cyclic-shear, Arnold-Beltrami-Childress and three-component Taylor-Green fields yield explicit perturbation radii and include initial conditions outside the direct Fourier-Wiener smallness criterion. A parameter-uniform extension covers a connected family of non-Beltrami Taylor-Green centres without repeating the proof for individual parameter values. An ensemble of 4,096 configurations, supplemented by 1,600 refinement trajectories and public turbulence data, connects the mathematical observables to spectral transfer and vortex geometry. Matched neural-operator experiments show that physics-informed training improves physical prediction, while also revealing that these gains do not necessarily improve the discovery of proof-limiting initial conditions. Together, these results provide a reusable method for establishing regularity across prescribed flow families and a quantitative setting for evaluating how learned predictions can assist rigorous computation.

Non-Latin handwritten character recognition (HCR) remains understudied. Dominant methods consider it as generic image classification, which uses model scale to implicitly learn stroke structure. Structural-prior efficiency---the principle that explicitly encoding script-geometric regularities as architectural inductive biases can be both more accurate and require fewer parameters. We introduce GraphemeNet, a unified multi-script architecture, governed by two orthogonal binary axes. Axis 1 operationalises stroke-level geometric regularity via Persistent Scaffold Injection (PSI): a script-specific asymmetric convolution injects a stroke scaffold as a weighted residual at every encoder stage, continuously anchoring learned features to script geometry---distinct from skip connections, auxiliary losses, or attention reweighting. Axis 2 selects between global average pooling with gated fusion and cross-scale attention with a Stroke Topology Module (STM), depending on whether glyph discrimination requires spatial relational reasoning. A Linear Capsule Routing (LCR) with $O(n)$ routing is shared universally. On fourteen benchmarks across eight writing systems, the architecture generalises with only scaffold and decoder topology varying per script, consistently challenging, outperforming published baselines, and establishing structural-prior efficiency as a broadly applicable principle for multi-script HCR.

We propose a geometric parametrization of the filters in a single convolutional layer: the parameter is no longer an ordered family of filter vectors, but a fixed-dimensional subspace of the filter space. For one-dimensional finite-stride convolution, the filter-to-convolution-operator correspondence gives an injective linear map $\mathcal{C}:\mathcal{K}\to H$. This map sends filter subspaces in $\mathrm{Gr}(q,\mathcal{K})$ to operator subspaces in $\mathrm{Gr}(q,H)$; composing it with the Plücker embedding yields a projective parametrization $Φ:\mathrm{Gr}(q,\mathcal{K})\to\mathbb{P}(\bigwedge^q H)$. Using $T_U\mathrm{Gr}(q,\mathcal{K})\cong\mathrm{Hom}(U,\mathcal{K}/U)$, we compute the differential of the induced Grassmannian map and show that the differential of $Φ$ is injective at every point. We then use the vanishing equations for Plücker coordinates and standard affine coordinates on a Grassmannian to prove that $\mathrm{Gr}(q,\mathcal{C}(\mathcal{K}))\hookrightarrow\mathrm{Gr}(q,H)$ is a closed embedding, and hence that $Φ$ is a closed embedding. Consequently, the parameter space is isomorphic to its projective image, the parametrization is finite and birational onto its image, every fiber is a singleton, and the resulting projective neural variety is smooth. For $k=4$ and $q=2$, we also use Singular to recover the image ideal and check its dimension, degree, chart rank, and smoothness. This computation illustrates, rather than replaces, the general proof. Finally, we discuss possible connections with filter redundancy and low-rank convolution, while distinguishing the proved geometric results from application proposals requiring numerical validation.

A long-standing open problem in robot manipulator control is whether global regulation can be achieved by classical PID control. This paper provides an answer to this question for classical PID controllers with triple parameters (k_p,k_i,k_d) in R^3. We find and prove that for one-degree-of-freedom manipulators, the classical PID control guarantees global stability and asymptotic regulation under standard structural assumptions, and further derive explicit quantitative design conditions for the PID gains. However, for multi-degree-of-freedom cases, we can construct a robot manipulator satisfying the same structural assumptions for which no choice of PID gains (k_p,k_i,k_d) can achieve global asymptotic regulation. These results provide a fundamental understanding of the abovementioned open problem, revealing both the fundamental capability and intrinsic limitation of the classical PID control for robot manipulator dynamics.

We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $α,β$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+α+β<1$; in the fully spin-polarized class it reduces to $s+α<1$. In particular, if $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, $$\left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}).$$ For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$.

Systematic trading rests on one article of faith: that regularities found in the past persist. This paper does three things. First, it states that faith as five axioms, each a commonplace practitioners already accept: (A1) a decision may use only what was known when it was made; (A2) what looks like the market changing its rules is the market changing its unobserved state, the machinery being the same in every era; (A3) the future may replay stretches of the past, though not in history's proportions; (A4) states persist for a while; (A5) whatever predictability exists is slight, even for a rule that knows the state. What turns these into axioms is quantification, and the quantities are declared rather than estimated: an invariance defect $\varepsilon_0$, a recurrence bound $Λ$ at a block scale $b$ (one declaration in two parts), coherence times $\ell_i$, a signal ceiling $ρ$ and an invariance ratio $κ$. These five declarations are the whole of the premises' empirical content. Second, it proves that the axioms force a five-stage canonical form for a quantitative investment system -- a declared representation, a predictor within a capacity ceiling, contiguous purged block evaluation aggregated by $\mathrm{CVaR}_{1/Λ}$, a budgeted and deflated search, robust sizing at a fraction of the estimate that the budget bounds -- each stage necessary: a procedure omitting it does strictly worse under a law the axioms admit. Third, it tests the axioms where they are falsifiable, each only at its declared constants, on real market series: no axiom is so far overturned.

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon--Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert--Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space, we first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates and the method of continuity to incorporate the lower-order terms. A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup with sufficient regularity at the initial time, thereby enabling analysis of the forward-in-time evolution via the global space-time Fourier structure of anisotropic Barron norms. We also show that a corresponding uniform-in-time estimate of the spectral Barron regularity generally fails. Finally, we derive $n^{-1/2}$ two-layer approximation bounds in mixed Sobolev norms for non-constant periodic activations and, under additional anisotropic Barron regularity, for non-periodic activations satisfying a polynomial-decay condition.

We study adaptive gradient descent for continuously differentiable, possibly nonconvex objectives under one-sided Hölder regularity. Unlike classical Hölder- or Lipschitz-gradient assumptions, which control the full gradient variation, our condition bounds only the directional term appearing in the descent inequality. This can allow less conservative step sizes when large gradient changes are orthogonal to, or favorable along, the update direction. We propose an adaptive scalar-step method based on an estimate of positive one-sided Hölder curvature, combined with a simple sufficient-decrease safeguard. For nonconvex objectives on a convex region containing the accepted update segments, we prove an explicit best-iterate stationarity bound with a rate determined by the Hölder exponent. Unlike predetermined diminishing step-size schemes, the method adapts to the local descent geometry. We evaluate the approach on two full-batch benchmarks designed to separate directional curvature from full gradient variation. On a binary classification problem, the method achieves the lowest final cross-entropy, objective value, and gradient norm, together with the largest classification margin among the compared scalar gradient methods. On a nonconvex Hölder regression problem, it attains the lowest final objective gap and gradient norm. These results indicate that one-sided Hölder curvature is an effective adaptive step-size signal when full-gradient variation is inflated by directions that do not hinder descent.

The goal of this paper is to propose a unifying model for Nerode-style characterizations of regularity across functions with different output domains. Building on Hauser's work in communication complexity, we generalize the setting by relaxing the computability assumptions and allowing non-Boolean output domains. We consider functions of type $Σ^* \to \domain$, where $Σ$ is a finite alphabet and $\domain$ is an arbitrary domain. For several domains, we show that the model coincides with known models of computation. We further conjecture that an analogous correspondence holds for other domains that currently lack a Nerode-style characterization of regularity, and we provide ample supporting evidence. In the model, an input string $w$ is split as $w = w_1 w_2$ and distributed between two cooperating parties, Alice and Bob, who exchange a constant number of messages to compute the value of the function. Each message is either an element of the output domain or a signal drawn from a finite set of signals, and the parties must produce the correct output for every admissible split $w = w_1 w_2$. We further extend the framework to infinite alphabets in the setting of nominal sets, and investigate its expressiveness on languages of words with atoms.

Musical scales are near-universal in human music, and most readers will feel they already know what a scale is. On closer inspection, however, the literature lacks a consensus definition: which conditions are necessary and sufficient shifts across disciplines and traditions, and the term turns out to cover several distinct objects. I argue this is less a failure of rigour than a sign that ``scale'' names several related objects: prescriptive abstractions, instrument tunings, statistical regularities in performed pitch, perceptual categories, social conventions. I adopt an empirical definition -- a scale as a statistical regularity in pitch organisation relative to a tonic -- that is portable across traditions and computable from recordings, and situate it alongside the other senses of the term. Even this empirical core is not purely observational, as convention enters in deciding which pitches belong to a scale. And a further step of grouping scales into named categories is a separate convention, which I approach through prototype theory and illustrate with examples from Irish music. Separating these layers provides a basis from which scales can be re-examined empirically and cross-culturally.

We study nonasymptotic convergence of primal-dual methods for a class of nonconvex constrained optimization problems with a convex-composite structure. In this class, both the objective and the functional inequality constraints are given by convex Lipschitz outer functions composed with smooth nonlinear inner mappings. The analysis is complicated by constraint violation in a nonconvex functional inequality system and by the lack of an a priori bound on the multipliers. To address these issues, we restrict the dual variable to an auxiliary compact set and analyze a smoothed prox-linear augmented Lagrangian method through a nonsmooth nonconvex-concave minimax reformulation. The main contribution is a finite-time mechanism for converting stationarity of the truncated minimax problem into a KKT certificate for the original constrained problem. We show that, for a sufficiently large penalty parameter, all but a controlled number of iterates enter a near-feasible region. On this region, a local conic regularity condition uniformly bounds the associated prox-linear multipliers and thereby makes the artificial dual truncation inactive at the selected iterates. Building on this mechanism, we establish explicit convergence rates for the proposed method in terms of the KKT residual. With dual regularization, a global dual error bound together with a bias-balancing argument gives an $O(K^{-1/3})$ rate. In the unregularized case, under additional local structural assumptions including piecewise linearity of the outer functions, a local dual error bound yields the sharper $O(K^{-1/2})$ rate.

We study the small-noise asymptotics of Euclidean heat regularizations of probability measures supported on manifolds with corners. Near a boundary or corner stratum, the relevant regime is a conical boundary layer in which the observation point approaches the stratum at the same scale as the Gaussian smoothing parameter. After rescaling this layer, the support is replaced to leading order by its inward tangent cone. We prove a two-term expansion for the heat-regularized density in this regime. The leading coefficient is the Gaussian mass of the linearized cone, weighted by the density on the support and by the adapted corner Jacobian; the first correction records the variation of the density, the Jacobian, and the quadratic geometry of the embedding. A localization argument then yields the corresponding expansion for the full heat regularization, with the nonlocal contribution exponentially small. From this density expansion we derive logarithmic asymptotics and uniform expansions for the score, the log-Hessian, and the scale derivative of the score. These formulas show how lower-dimensional support, boundary faces, corners, and curvature are encoded in the singular differential structure of small-noise Gaussian regularizations.

This paper introduces range regularization for federated learning with linear systematic components to enhance statistical accuracy and induce cross-client regularity conducive to quantization, coding, and resource efficiency. Our approach identifies features with shared weights across different clients and adaptively clusters the weights of personalized features at extreme values, a process we refer to as polar clustering. Theoretical analysis of the associated estimators poses significant challenges due to the seminorm nature and non-decomposability of the regularizer. We develop new proof techniques for the nonasymptotic analysis of statistical accuracy and faithful pattern recovery. Moreover, a fast optimization algorithm that leverages varying degrees of local strong convexity is proposed to reduce iteration complexity. Experiments support the efficacy and efficiency of the proposed approach.

A central difficulty in training Joint-Embedding Predictive Architectures (JEPAs) is preventing representation collapse. LeJEPA addresses this by enforcing an isotropic Gaussian target on the embeddings via Sketched Isotropic Gaussian Regularization (SIGReg). This target is in tension with the manifold hypothesis, which expects embeddings to concentrate on a low-dimensional subset of the ambient space. We propose \emph{UR-JEPA}, which targets a uniformly $n$-rectifiable measure of local tangent dimension $n$ at small scales, realized through a Gaussian-kernel smoothed Carleson-type square function $\mathcal{L}^{\text{CGLT}}$, with a complementary Jones $β$-number formulation. On Inet10, UR-JEPA($\mathcal{L}^{\text{CGLT}}$) attains $0.9141 \pm 0.0014$ for a $+0.83$,pp gain over LeJEPA($\mathcal{L}^{\text{SIGReg}}$) with $\sim 30\%$ lower seed standard deviation; on matched-recipe Galaxy10~SDSS, a single-seed ImageNet-$100$ run, and a $3$-seed EuroSAT remote-sensing run, the two methods lie in the same peak-accuracy band at convergence, with UR-JEPA retaining its lower-seed-variance signature. On EuroSAT the in-domain pair is competitive at $96.0$ to $96.1\%$ with large remote-sensing foundation-model transfer at a $25\times$ smaller backbone. The distinction is geometric: direct visualization of the projector output distribution shows that on all four datasets UR--JEPA($\mathcal{L}^{\text{CGLT}}$) produces a global PCA spectrum with a $4$ to $5$ order-of-magnitude drop at index $\sim 20$ to $25$ out of $D = 32$, while LeJEPA's spectrum is near-flat (top-to-bottom ratio at most $3.6$). Per-dimension marginals are simultaneously near-Gaussian for both methods (mean Shapiro-Wilk $W \in [0.992, 0.996]$) as a Diaconis-Freedman consequence. At matched accuracy the two regularizers therefore yield structurally distinct projected representations.

Generalization in robotics requires prior knowledge about how the world is structured, yet this structure changes from one situation to the next. This paper investigates the proposition that generalization arises from adaptively composing regularities -- predictable relationships within the robot-environment system -- into situation-appropriate structures for behavior generation. We examine this proposition by analyzing the mechanism in AICON (Active InterCONnect), a framework representing regularities as interacting processes in a differentiable network, where sensory feedback realizes composition and gradient descent generates behavior. To isolate adaptive composition as the key mechanism, we study a simple simulated problem in which all relevant regularities can be identified. We expose the resulting model to a wide range of novel conditions not considered during design, and we find that it generates context-appropriate behavior in all but one case, where encoded regularities are provably insufficient. Ablations reveal that the network automatically modulates which regularities influence behavior based on their informativeness. These results suggest that adaptive composition of regularities constitutes a powerful inductive bias for building generalization into behavior generation.

Large language models (LLMs) have emerged as powerful representation learners whose internal features increasingly align with human cognition. We study whether modern LLMs can serve as a lens for understanding neural representations in the human brain, focusing on emotional valence in EEG. We first build a one-dimensional valence direction, the V-axis, from modern LLMs using only nine emotion-evocative sentences. We validate it through zero-shot transfer to sentiment benchmarks and cross-model consistency across fourteen LLMs. We then show that this LLM-derived direction maps onto human neural activity. On a public EEG cohort of 123 subjects watching affective videos, a single linear projection on EEG features tracks the V-axis position of each stimulus. Moreover, 36 EEG emotion classifiers trained without exposure to the V-axis spontaneously rediscover the same direction in their internal representations, suggesting that the same valence structure emerges in both language models and human electrophysiology. Yet this convergence does not provide an effective training signal. We test twenty-five alignment strategies, including knowledge distillation, representational similarity, contrastive, and topographic losses; none improve decoding, and sixteen significantly reduce accuracy. We formalize this result as the saturation regularity: once task labels alone drive a brain-decoding network onto the target direction, additional supervision mainly distorts an already-saturated basin, while the load-bearing within-class residual receives little useful gradient. This regularity also indicates where improvement should come from: the residual subspace unreachable by supervision. Motivated by this insight, we ensemble across residual diversity rather than supervising the basin, improving balanced accuracy by 10.5% over the prior best on FACED, with the same effect replicated on SEED-V.

Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the regularity of transport maps and the generalization properties of one-step Wasserstein-guided generative models for PDE-induced probability measures. We consider normalized target densities associated with linear elliptic and parabolic equations on bounded domains, as well as diffusion and Fokker--Planck equations on the torus. Under standard structural assumptions, we prove that these target measures satisfy doubling conditions. By combining this fact with regularity theory for optimal transport between doubling measures, we show that the optimal transport map from a uniform source measure to the target measure is Hölder continuous. This regularity yields an approximation-theoretic justification for one-step generative models that learn PDE-induced distributions via a single pushforward map. As a representative instance, we study DeepParticle and derive excess-risk bounds characterizing the discrepancy between the learned map and the population-optimal map. We also establish a robustness estimate under target shift and illustrate the theory with experiments which support the derived rates.

We characterize the fundamental limits of high-dimensional mean testing under arbitrary truncation, where samples are drawn from the conditional distribution $P(\cdot \mid S)$ for an unknown truncation set $S$ that may hide up to an $\varepsilon$-fraction of the probability mass. For distributions with $p$-th directional moments of magnitude at most $ν_{P,p}$, truncation induces a bias of order $O(ν_{P,p}\varepsilon^{1-1/p})$. This bias creates a sharp information-theoretic detectability floor: when the signal $α$ falls below this threshold, the null and alternative hypotheses are indistinguishable even with infinite data. Above this floor, we prove that a simple second-order test achieving near-optimal sample complexity $n = O\!\left(\frac{\|Σ_P\|}{(α-4ν_{P,p}\varepsilon^{1-1/p})^2}\sqrt{d}\right)$. We further identify a structural escape from this finite-moment bias barrier. Under a directional median regularity assumption, truncation bias improves to linear order $O(\varepsilon)$. This reveals an intermediate regime in which estimation requires $Θ(d)$ samples for uniform recovery, while testing recovers the classical $Θ(\sqrt d)$ rate once truncation bias is eliminated. Together, our results provide a unified framework for mean testing under truncation, connecting finite-moment, sub-Gaussian, and median-regular structural regimes.

Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can be arbitrarily conservative and fail to capture the geometry actually encountered by optimization trajectories. In this paper, we develop a trajectory-restricted framework for linear convergence based on localized geometric regularity. We introduce restricted variants of the Polyak--Łojasiewicz inequality, error bound, and quadratic growth conditions that are required to hold only on subsets of the domain. We show that classical convergence guarantees extend under these localized conditions, and in key cases, we develop new arguments that yield explicit relationships between the corresponding constants. The resulting rates are governed by geometric quantities associated with the regions traversed by the algorithm. For polyhedral composite problems, we prove that convergence is controlled by restricted Hoffman constants corresponding to the active polyhedral faces visited along the trajectory. Once the iterates enter a well-conditioned face, the effective condition number improves accordingly. Our work provides a geometric quantification for fast local convergence after active-set or manifold identification and more broadly suggests that linear convergence is fundamentally governed by the geometry of the subsets explored by the algorithm, rather than by worst-case global conditioning.

Point-supervised Temporal Action Localization (PTAL) adopts a lightly frame-annotated paradigm (\textit{i.e.}, labeling only a single frame per action instance) to train a model to effectively locate action instances within untrimmed videos. Most existing approaches design the task head of models with only a point-supervised snippet-level classification, without explicit modeling of understanding temporal relationships among frames of an action. However, understanding the temporal relationships of frames is crucial because it can help a model understand how an action is defined and therefore benefits localizing the full frames of an action. To this end, in this paper, we design a multi-task learning framework that fully utilizes point supervision to boost the model's temporal understanding capability for action localization. Specifically, we design three self-supervised temporal understanding tasks: (i) Action Completion, (ii) Action Order Understanding, and (iii) Action Regularity Understanding. These tasks help a model understand the temporal consistency of actions across videos. To the best of our knowledge, this is the first attempt to explicitly explore temporal consistency for point supervision action localization. Extensive experimental results on four benchmark datasets demonstrate the effectiveness of the proposed method compared to several state-of-the-art approaches.

Selective State-Space Models (SSMs) such as Mamba have become central to long-sequence modeling. Still, their stability is poorly understood: their state-space coefficients are modulated online by a token-dependent gating signal, making the recurrence neither linear time-invariant nor classically nonlinear. We study continuous-time selective SSMs through passivity, dissipativity, and Input-to-State Stability (ISS), explicitly separating the selection signal $x(\cdot)$ from the driving input $u(\cdot)$. We obtain four results: exponential forgetting under strict dissipativity; a canonical $\mathrm{AUC}_{\mathrm{loc}}$ quadratic storage for the frozen-selection subsystem that accommodates discontinuous gating; a parametric LMI together with universal kernel constraints and "irreversible forgetting" under universal quadratic storage; and sufficient conditions for global ISS uniformly over admissible selection schedules. We then bridge to practice by deriving a sampled block LMI for the Mamba selective-scan core, which is used as a differentiable training-time regularizer. Across seven standard time-series datasets and four prediction horizons, the regularizer reduces sampled Mamba-core LMI violations by roughly $92\%$ in $28/28$ pairs at a clean-MSE cost of less than $0.018\%$. It improves internal Mamba passivity and state-norm diagnostics under injected perturbations. Our results turn classical control-theoretic tools into verifiable structural and training criteria for selective SSMs, while honestly scoping which guarantees transfer to a deep selective-scan architecture.

Despite their widespread use, the principles governing the organisation of syntactic dependency trees remain poorly understood. I analyse dependency trees from 124 typologically, genetically, and geographically diverse languages. Their topology departs systematically from randomness. Relative to uniformly sampled random trees, dependency trees exhibit greater structural robustness and lower branching heterogeneity. I propose that these universal regularities emerge naturally from incremental grammatical encoding. I model this process using sublinear preferential attachment. The model accurately reproduces the observed topology. More generally, the results demonstrate how a universal statistical property of syntax can emerge from a simple, cognitively motivated generative process. They further illustrate a broader principle of efficiency by construction: communicatively efficient syntactic structures can emerge without direct optimisation for communication.