Papers

The size of minimal systems of generators and these systems themselves for groups $ESL_n \left[ \mathbb{Z} \right]$ and $ESL_n \left( \mathbb{F}_p \right)$ were found. Moreover we obtain triples of involutions with\textit{ two commuting involu\-tions} generating $ESL_5 \left[ \mathbb{Z} \right]$, $ESL_5 \left[ \mathbb{Z} \right]$ relatively, that are Mazurov triples.

Let $R$ be a commutative Noetherian ring. We prove that if the one-variable formal power-series ring $R[[x]]$ is a unique factorization domain, then so is the two-variable formal power-series ring $R[[x,y]]$. This resolves a question raised by Bayart in 1973 for Noetherian coefficient rings. The proof uses the divisor theory of Noetherian normal domains, expressed through finite rank-one reflexive modules.

We introduce an intrinsic spectral sparsity model for nonparametric density estimation on compact connected Riemannian manifolds. Instead of penalizing coefficients in an arbitrarily chosen Laplace–Beltrami eigenbasis, we group each complete eigenspace and measure the Hilbert norm of its spectral component. The resulting block-variation space is basis independent and isometry invariant. We establish its structural, atomic, and nonlinear approximation properties and clarify its relation to Sobolev, Besov, and coefficientwise spectral $\ell^1$ classes. We then construct a coordinate-free block-shrinkage estimator and prove a nonasymptotic signal-dependent $L^2$-oracle inequality that adapts to the unknown set of detectable eigenspaces. Under polynomial spectral growth, the risk theory separates the number of spectral blocks from their multiplicities and exhibits two regimes: one driven by a single high-dimensional eigenspace and the other by cumulative spectral complexity. Under matching spectral-growth and nondegeneracy assumptions, corresponding minimax lower bounds show that this multiplicity dependence is intrinsic, with sharp consequences for spheres and the rotation group $SO(3)$. Finally, we develop a positive, normalized, block-penalized exponential spectral sieve for log-densities and derive likelihood oracle inequalities together with expected Kullback–Leibler, Hellinger, and $L^2$ risk bounds. The resulting framework provides a geometry-respecting theory of sparse density estimation that remains invariant under changes of eigenbasis.

We study the abstract elementary class of acts with pure embeddings. In particular, we show that stability and superstability in this class can be characterized in terms of the monoid $S$ being LO (for every $s,t \in S$, we have that $s \in St$ or $t \in Ss$) and weakly noetherian (every ideal is finitely generated), respectively. Moreover, under mild set-theoretic assumptions, we characterize the stability spectrum via the minimal cardinality of a generating set for every ideal. As an application, we obtain a Baer-like criterion for pure injective acts when $S$ is LO. We use this to provide an alternative proof that the class of acts has enough pure injectives when $S$ is LO.

In this paper, we prove that the Artinian Gorenstein $\mathbb{K}$-algebra $A_{F_s}$ of codimension $n$, socle degree $d$ and Macaulay dual generator $F_s := \ell_1^d + \dots + \ell_s^d \in \mathbb{K}[X_1, \dots, X_n]$ where $\ell _1, \cdots , \ell_s$ are general linear forms satisfies the strong Lefschetz property (SLP). This result allows us to study whether the Waring rank of $F_s$ is exactly $s$. Furthermore, we show that $A_{F_s}$ is the doubling of a suitable 0-dimensional scheme $Z_{F_s}$ in $\mathbb{P}^{n-1}$, the so-called tight annihilating scheme of $A_{F_s}$, and we compute the minimal free resolution of $A_{F_s}$ in terms of the minimal free $R$-resolution of $I(Z_{F_s})$. Finally, we determine the linear general Jordan type of $A_{F_s}$.

The distance matrix of a finite point cloud can be visualized as a heatmap. When the data arise from a time series, the sublevel sets of this image are known as recurrence plots and are widely used in time series analysis. Motivated by this perspective, we establish a relationship between the distance-matrix filtration of the time series and the Čech (or Vietoris–Rips) filtration of its state-space embedding in the form of a degree-one chain map. We study the induced maps in homology, showing that the map from $H_0$ into $H_1$ is essentially surjective and providing an example where the map from $H_1$ into $H_2$ is nontrivial. These chain maps can be applied to simplify image persistence computations arising in the computation of cycling signatures, a topological tool for time series analysis. Moreover, these computations yield finer information that allows the analysis of transitions between different types of cycling motion.

We prove that in a semi-dispersing Minkowski billiard, that is a billiard in the complement of a collection of convex sets in a normed space with a smooth strictly convex norm, any billiard trajectory of finite length has only finitely many reflections off the walls.

Gibbs states are probability distributions defined on Hamiltonian G-manifolds. They are parametrized by an open set of the Lie algebra g and arise as natural equilibrium states with regard to the action of G on the system. In this paper, we focus on nilpotent orbits of SL(n,R). We show that for n>2, those orbits do not admit any Gibbs state.

We construct an explicit real quantum Latin square of order six with cardinality $29$, the last unresolved value in the order-six spectrum. We first construct a punctured $6 \times 6$ array in $\mathbb{R}^5$ whose punctured rows and columns are orthonormal bases, and then adjoin a common diagonal vector in an orthogonal one-dimensional summand. The six diagonal entries lie on one ray. Of the thirty off-diagonal entries, two rays occur twice and the other twenty-six occur once; exact support and coordinate-ratio comparisons establish this count. Together with the known constructions, this closes the cardinality spectrum of quantum Latin squares of order $6$.

For $X\in\{\mathbb{N},\mathbb{Z}\}$, let $\alpha_X(\ell)$ and $\beta_X(\ell)$ denote the supremal upper and lower densities of subsets of $X$ admitting $\omega$-permutations without monotone $\ell$-term arithmetic progressions. We strengthen the published lower bounds for the three-term upper-density parameters by proving \[ \alpha_{\mathbb{N}}(3)\geq\frac23,\qquad \alpha_{\mathbb{Z}}(3)\geq\frac23. \] We also prove $\beta_{\mathbb{Z}}(4)=1$ by constructing four-permutable subsets of the integers whose lower symmetric densities approach one.

We introduce the tame étale fundamental group $\pi_1^t(X/K)$ of a rigid space X over a non-archimedean field K. We show that if X is qcqs and K has topologically finitely generated tame Galois group (e.g. algebraically closed or a local field), then $\pi_1^t(X/K)$ is topologically finitely generated. If X is moreover the rigid generic fibre of a strictly semistable formal scheme such that the smooth locus of its special fibre admits a projective snc compactification, then $\pi_1^t(X/K)$ is topologically finitely presented. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids), in particular on an analogous finiteness statement for the tame log étale fundamental group, and on the 'vertical compactification' of a map of adic spaces.

We study (binary) nearly-perfect covering codes, which are codes that attain the Van Wee bound with equality. They act as the covering counterparts to nearly-perfect error-correcting codes, which attain the Johnson bound with equality. These codes have been completely classified for covering radius $R=1$. We prove that no code with $R\geq 2$ can attain the original Van Wee bound with equality, since it omits the dependence on the minimum distance of the code. We refine the bound to account for the minimum distance and show some nearly-perfect covering codes. By proving some structural properties of such codes, we prove all nearly-perfect covering codes with $R=2,3$ must be equivalent to the codes we showed. We also prove that for any $R\geq 3$, there are at most a finite number of nearly-perfect covering codes.

In this paper, we systematically study a variant of the scaled relative graph (SRG), referred to as the $\theta$-symmetric SRG, and apply it to the stability analysis of cactus networks. Compared with the previous SRG definition, the $\theta$-symmetric SRG enables the characterization of phase lead and lag behaviors, and serves as a more natural multivariable extension of the classical Nyquist plot. We first analyze the gain and phase aspects of $\theta$-symmetric SRG separately and build a connection between $\theta$-segmental phase and a norm minimization problem. This connection makes it possible to compute $\theta$-segmental phase via semidefinite programming. We further derive the submultiplicative and subadditive properties of $\theta$-symmetric SRG. These algebraic properties are crucial to determine the nonsingularity of product-type and sum-type return difference matrices, which topologically correspond to the cyclic and parallel-feedback extreme cases of cactus networks. By taking the cyclic interconnection as the fundamental starting point, we establish necessary and sufficient conditions for its robust stability. Integrating this with the parallel case, we synthesize a unified stability framework for general multi-loop cactus networks. The $\theta$-symmetric SRG framework is less conservative and provides a more intuitive geometric interpretation of system behaviors compared with some existing approaches. Several examples are included to demonstrate the effectiveness of the proposed methods.

To each finite symmetric measure $\sigma$ on the real line, with support a compact subset of $(-2,2)$, we associate a measure $\mu$ on the unit circle by transporting mass at $\pm x$ to $e^{\pm i\theta(x)}$, $\theta(x)=2\arcsin(x/2)$. The linear prediction errors, Verblunsky coefficients, and Toeplitz determinants of $\mu$ are then expressed through the orthogonal polynomial data of $\sigma$ at the single edge point $x=2$. In particular $E_m(\mu)=\tfrac12 t_m\|P_m\|_\sigma^2$ with $t_m=P_{m+1}(2)/P_m(2)$. Under a condition on the first coefficients, decay of the recurrence coefficients of $\sigma$ forces the $t_m$ to increase from $t_1$ onward, a Turán-type monotonicity in the degree placing every prediction margin of $\mu$ past the first above the corresponding coefficient of $\sigma$. Taking $\sigma$ to be the Lommel-polynomial measures, with atoms at rescaled reciprocals of the zeros of the Bessel function $J_\nu$, yields a two-parameter family of purely atomic circle measures with a normalized determinant limit equal to a value of $J_\nu$.

We disprove the conjecture of Gau, Wang and Wu that the numerical range of a finite-dimensional partial isometry, when circular, must be centred at the origin. More precisely, we prove that there is an \(\varepsilon>0\) such that, for every \(a\in(0,\varepsilon)\), one can find a rank-four partial isometry \(V_a\in\M_6(\R)\) satisfying $$ W(V_a)=\{\zeta\in\C:|\zeta-a|\leq 3/4\}. $$

Cardinality estimation - counting the number of distinct elements in a data stream - requires a tradeoff between memory and accuracy. ExaLogLog recently established the state of the art for this tradeoff by combining wide registers with a Fisher-information-optimal maximum likelihood (ML) estimator, achieving the best known memory-variance product (MVP) among HyperLogLog variants. Here we present Arithmetic Variable LogLog (AVLL), which surpasses ExaLogLog at every memory point tested using arithmetic encoding and eliminating uncommon states to consume 64-bit words completely with 11 registers each, yielding a 5.5x register-count advantage. Its four-component blended estimator, HLDLC, exploits this density advantage to surpass ExaLogLog's ML accuracy without iterative solving. At 1 KB, AVLL achieves 1.63% width-weighted mean absolute error compared to ExaLogLog's 1.71% - a 4.7% improvement. The corresponding empirical MVP is 3.4, surpassing ExaLogLog's practical MVP of 3.78 and its theoretical optimum of 3.67. This holds at every tested size from 0.25 to 4 KB. AVLL inherits DynamicLogLog's early exit mechanism, which filters most elements before any register is touched. With thousands of simultaneous sketches per thread, AVLL is 2.7-4.5x faster than ExaLogLog due to the reduced memory bandwidth from early exits. Like DynamicLogLog, AVLL stores relative NLZ values with a shared offset, so its memory scales as O(B + log log C) rather than O(B x log log C) - decoupling maximum representable cardinality from register width. These results hold under both high-complexity (all-unique) and low-complexity (nonuniformly high duplication rate) data distributions, with zero accuracy degradation from duplication. AVLL is implemented as a single self-contained Java class with all correction formulas embedded, available in the BBTools suite at https://bbmap.org.

Let $A$ be a finitely generated commutative algebra over a field $\mathbb K$ of characteristic zero. We prove that every finitely generated Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally finite on $A$ is finite-dimensional. Consequently, for a Lie subalgebra generated by finitely many locally finite derivations, the following are equivalent: it is finite-dimensional, it acts locally finitely on $A$, and all its elements are locally finite. A key ingredient is a second theorem: every Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally nilpotent is solvable; when $A$ is reduced, its derived length is at most $\dim A$, and this bound is sharp. In general, the $(\dim A)$-th term of its derived series is nilpotent, of class bounded in terms of $A$ only. We also prove that every solvable Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ generated by finitely many locally finite derivations is finite-dimensional and acts locally finitely on $A$. For an affine variety $X$ over an algebraically closed field, we deduce that a subgroup of $\operatorname{Aut}(X)$ generated by finitely many connected algebraic subgroups is algebraic if and only if every element of the Lie algebra generated by their tangent algebras is locally finite. For two unipotent one-parameter subgroups, this gives an answer to a 2005 problem of Popov. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proof uses derivations vanishing to second order along a suitable finite subscheme, on which local finiteness forces local nilpotence.

We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.

We construct a 321 point set in R^63 that cannot be partitioned into 64 subsets of smaller diameter, proving b(63)>=65. Starting from the G_2(4) Euclidean representation and the Jenrich Brouwer 320 point core in dimension 63, we add one projected and rescaled point while preserving the relevant clique obstruction. The example and proof were generated entirely by ChatGPT using GPT 5.6 Sol. The author has personally verified the result and assumes responsibility for that verification, but claims no credit for the originality of the construction.

Dadush et al.\ (2024) recently developed a scaling-invariant layered least squares algorithm for linear programming whose complexity depends on the optimal condition measure $\bar{\chi}_A^*$. Their work builds on Vavasis and Ye's (1996) algorithm whose running time depends only on the constraint matrix $A$ through the condition number $\bar{\chi}_A$. Monteiro-Tsuchiya (2003) defined the optimal condition number $\bar{\chi}_A^*$ as the maximum $\bar{\chi}_{AD}$ achievable over all positive diagonal column rescalings $D$. Dadush et al.\ (2024) introduced the optimal circuit imbalance measure $\kappa_W^*$, which serves as a lower bound for $\bar{\chi}^*_A$. Instances with artificially large optimal circuit imbalance measures $\kappa_W^*$ can be easily constructed; however, finding naturally occurring examples where this optimal scaling-invariant measure grows exponentially is of independent interest. In this paper, we show that the Ben-Tal Nemirovski (BN) linear programming approximation of the unit disk provides such an example. By explicitly constructing circuits in the kernel of the BN formulation, we prove that the optimal circuit imbalance measure $\kappa_W^*$ grows exponentially in the number of approximation steps. Since $\kappa_W^*$ lower bounds $\bar{\chi}_A^*$, our result demonstrates that the BN approximation yields an exponentially ill-conditioned family of constraint matrices.

In our previous work, we proposed and experimentally demonstrated a physics-based simulator for combinatorial optimization problems (SSBM), which exploits a distinctive spontaneous symmetry breaking (SSB) phenomenon identified in a dissipative system exhibiting robust causality described by an iterative equation. We further conducted numerical simulations in which initial fluctuations of distinct patterns were applied to large-scale benchmark problems, and reported that the SSBM exhibits a unique behavior absent in conventional Ising machines: under specific conditions, pseudo-spin patterns progressively converge during the solution-search process and ultimately coalesce into a single pattern, with the cut value of the converged solution reaching 99.7% of the best-known cut value. In the present paper, we elucidate the conditions under which the SSBM can be configured to emulate the behavior of a continuous-variable Ising machine. We further demonstrate that by additionally incorporating an effect analogous to physical annealing into these conditions, the SSBM becomes capable of identifying the best-known cut value for the K_2000 benchmark instance.

We associate to a collision $p\ne q$ of a homogeneous Keller map $F=\operatorname{id}+h$ the polarization subalgebra generated by $p$ and $q$. This collision hull is the smallest invariant linear subspace containing the collision; the restricted map remains a noninjective Keller map, and its dimension controls that of the standard symmetric lift. The construction is compatible with scalar extension and equivariant under linear conjugacy, so it gives a canonical linear carrier for a marked failure of injectivity. We compute this carrier exactly in two recent cubic-homogeneous reductions of the three-variable Jacobian counterexample. For Thompson's 24-variable map, the growth $2,4,11,20,20$ recovers MacFarlane's 20-dimensional invariant subspace. For the 19-variable homogenization of Van Rijn's 12-variable degree-three map, the growth $2,4,11,19,19$ fills the whole space. Hence no invariant linear restriction retaining the displayed collision can improve the corresponding 38-variable symmetric lift. We give the resulting 340-monomial homogeneous quartic explicitly: it is Hessian-nilpotent, violates Zhao's Vanishing Conjecture, and its gradient Keller map has an exact collision over $\mathbb{Q}(i)$. The analogous 20-variable application gives the previously studied 40-variable, 350-monomial quartic. All finite calculations are checked by exact companion code.

We find the adjoint of a map on the space of almost holomorphic modular forms, analogous to the generalized Ramanujan-Serre derivative on the space of quasi-modular forms. As applications, we explicitly find special values of certain shifted Dirichlet series associated to quasi-modular forms and prove a rationality result for the shifted Dirichlet series.

We study the simultaneous limits of homogenisation $(\varepsilon\to0)$ and dimension reduction $(h\to0)$ for thin heterogeneous nematic liquid crystal films in the Landau–de Gennes $Q$-tensor framework. The elastic energy density is governed by a tensor $\mathbf{A}(x'/\varepsilon,x_3/h)$ that is periodic in the in-plane fast variable and measurable in the normalised thickness variable. Surface anchoring on the top and bottom faces is modelled by a weak anchoring energy of strength $h^\gamma$, $\gamma\geq0$, in a general set-valued framework covering the principal classical anchoring geometries. The simultaneous limit reveals two principal features. First, the anchoring scaling yields a hierarchy of effective behaviours depending on $\gamma$: a hard constraint $\mathcal{Q}\in H^1(\omega;\mathscr{A})$ for $0\leq\gamma<1$, a finite surface density contribution for $\gamma=1$, and vanishing anchoring for $\gamma>1$. For $0\leq \gamma<1$ with uniaxial anchoring, the homogenised energy reduces on the constrained class to anisotropic Oseen–Frank and Ericksen energies. Second, the effective elastic response depends on the scale ratio $\rho=\lim h/\varepsilon\in[0,\infty]$: each regime has a distinct corrector structure and yields a two-dimensional LdG energy with homogenised tensor $\mathbf{A}^{\mathrm{hom}}_\rho$, characterised by a regime-dependent cell problem. To the best of our knowledge, this is the first rigorous treatment of the simultaneous homogenisation and dimension reduction limit for the Landau–de Gennes energy, and the first in which the anchoring strength enters as a scaling parameter, producing a hierarchy of qualitatively distinct effective models.

Let $S$ be a numerical semigroup with multiplicity $\mult$, conductor $\cc$, embedding dimension $\ee$, type $\typ$ and genus $\gnus$, and let $\nn=\cc-\gnus$. Wilf's conjecture asserts that $\ee\,\nn\ge\cc$; the inequality $\gnus\le\typ\,\nn$ of Fröberg, Gottlieb and Häggkvist settles it when $\typ\le\ee-1$. The Apéry set of $S$ with respect to any $s\in S\setminus\{0\}$ carries a partial order whose maximal elements are the pseudo-Frobenius numbers translated by $s$; for $s=\mult$ its minimal elements are the minimal generators other than $\mult$. Comparing the two extremal statistics bounds the type by $\typ\le\ee-1+\Xii(S)\le\ee-1+\Theta(S)$, where $\Theta(S)$ measures the redundancy of the covering of the gaps of $S$ by the pseudo-Frobenius numbers and $\Xii(S)$ refines it. With an exact decomposition of the Wilf number this yields the genus bound $\gnus\le\ee-1+\typ(\nn-1)$, strictly stronger than $\gnus\le\typ\,\nn$ precisely when $\typ\ge\ee$, and reduces Wilf's conjecture to an inequality free of $\cc$ and $\nn$. We determine the equality case of $\gnus\le\typ\,\nn$, recovering a classification of Singhal; answer a question of Moscariello and Sammartano whenever $\ee\ge\typ+1$; and correct Kaplan's classification of the equality case for $\cc\le2\mult$, from which an infinite family is missing.

Let \(k\geq 3\), \(n\geq 1\), and let \(H_{k,n}=\Sh(C_k,n)\) be the group generated by the standard \(k\) pile perfect shuffle and cyclic pile permutation on a deck of \(kn\) cards. We prove that \(H_{k,n}\) is \(2\)-transitive whenever \(n\) is not a power of \(k\). Residual commutators give translations supported on two pile labels, and a strongly connected digit digraph propagates these translations throughout the deck, a separate argument resolves the antipodal support case. We then combine this result with fixed point ratio bounds for primitive groups and explicit boundary calculations to determine \(H_{k,n}\) for all \(k\) and \(n\). If \(n=k^f\), then \(H_{k,n}\cong C_k\wr C_{f+1}\). If \(k=4\) and \(n=2\cdot4^j\), then \(H_{4,n}\cong\AGL(2j+3,2)\). In every other case, \(H_{k,n}\) is \(\Alt(kn)\) or \(\Sym(kn)\), according to the parity of its generators. This proves Conjecture~1.10 of Amarra, Morgan and Praeger and Conjecture~5.1 of Xia, Zhang, Zhang and Zhu. More generally, we classify \(\Sh(P,n)\) for every pile group \(P\) containing \(C_k\), and obtain the odd \(k\) part of their Conjecture~5.2.

Integrating heat storages in district heating networks (DHNs) supports managing dynamic characteristics such as heat-demand variations and changing energy prices, and the intermittency of renewable sources. A common application are DHNs with combined heat and power (CHP) units, where a storage allows shifting heat extraction to periods with favorable electricity prices. Although the benefits of heat storage in DHNs are well established, determining the optimal storage size remains challenging due to the diversity of DHNs in terms of heat sources, production and consumption patterns, network heat losses, and fuel costs. This paper presents a scalable, automated methodology for optimizing short-term heat storage in DHNs using mathematical optimization. The nonlinear, physics-based approach models the DHN, heat producers, and storages simultaneously to minimize total economic cost through optimal storage sizing, explicitly considering time-varying heat-production costs, heat demands, and heat losses. The methodology is demonstrated on a 3rd-generation CHP-DHN in Belgium with 30 consumers, exceeding the scale of previous physics-based storage-sizing studies. For this system, optimal storage integration reduces the 20-year cost by 730 k EUR (16.5%), from 4.41 M EUR to 3.68 M EUR. The reduction results from a 1.07 M EUR decrease in heat-production cost achieved by shifting heat extraction to periods of low electricity prices, while the storage investment amounts to 323 k EUR. A comparison to a commonly used simplified storage sizing method, which fails to identify the optimal storage size, demonstrates the advantage of the proposed holistic, physics-based optimization approach in ensuring feasible designs, accurate cost assessments, and optimal storage sizing.

In this paper, we consider an orthogonal time frequency space (OTFS) system in time-varying channels with overspread Doppler shifts, typically found in non-terrestrial multi-satellite links. The overspread Doppler shifts with magnitude greater than half of the subcarrier spacing, result in aliased Doppler shifts in the delay-Doppler (DD) domain due to the OTFS modulo operation. This makes channel estimation very challenging and the traditional channel estimation methods become ineffective. To address this challenge, we propose a DD training frame and a two-stage channel estimation method. The training frame comprises a cosine pilot signal and a pilot symbol. In the first stage of the channel estimation, the pilot symbol in the DD domain is utilized to estimate the delays, aliased Doppler shifts, and channel gains of the propagation paths. In the second stage, the received time domain signal is converted into the frequency domain to detect the peaks of all the Doppler shifts using the cosine pilot signal. Then, we present a threshold-based method to pair the estimated actual Doppler shifts with their corresponding delays and channel gains. The complexity of the proposed channel estimation is also discussed. Finally, the performance of the proposed channel estimation is validated in terms of the normalized mean square error (NMSE) and bit error rate (BER) in various scenarios.

Uncertainty in nonlinear dynamical systems is often organized by transport structures that are not apparent from posterior geometry alone. We introduce Dynamical Uncertainty Geometry (DUG), a framework that separates three ingredients that are frequently conflated: posterior uncertainty, future observations, and model-induced transport. DUG indexes posterior credible sets and transport-conditioned subsets by a common enclosed-probability coordinate, allowing physically meaningful classes to be tracked across credibility levels rather than examined at a single threshold. The framework combines a mass-ranked credible filtration, a description of future experiments through their induced probability laws and Fisher geometry, and a labelled transport skeleton representing dynamically distinct outcomes. We establish stability results for the resulting mass-indexed persistence modules under simultaneous perturbations of posterior density and probability measure, including continuum, refinement, and finite atomic formulations. These results provide quantitative control of persistence computed from numerical approximations and weighted grids. Two benchmark problems illustrate the framework. In a short-arc orbit-determination setting, DUG identifies dynamically distinct return classes within a connected uncertainty region and highlights a difference between information-based and local Fisher-based observation-design criteria. In the Earth-Moon planar circular restricted three-body problem, DUG reveals multiple first-hit transport outcomes coexisting within a connected credible region and provides diagnostics for assessing their numerical resolution. Together, these examples show how topological summaries of posterior geometry, when coupled to transport labels and future experiments, yield a richer description of uncertainty than either posterior probabilities or dynamical classifications alone.

Automated-driving rulebooks rank rule violations lexicographically, and model predictive control enforces that ranking either exactly, through $L{+}1$ sequential programs per tick, or approximately, through a weighted sum tuned by the separation heuristic $w_1\gg w_2\gg\cdots\gg w_L$. We show the heuristic answers the wrong question. For a convex priority-ordered program, a weighted sum reproduces the lexicographic optimum precisely when its weight, augmented by a unit performance coefficient, supports the upper image of the achievement map at the lexicographic point; the admissible weights form the unit-performance slice of an outward normal cone. Under hinge penalties this slice is a polyhedron obtained by projecting a scaled-KKT system, and a linear program returns an interior weight with a certified margin; under squared-hinge penalties no finite weight is exact whenever the limiting multiplier is nonzero, with violation along the local minimizer branch decaying as $O(1/w)$. Calibrated on held-out logs, the resulting weights have near-equal tier components in nine of eleven calibration-eligible scenario classes and roughly double legal-tier event precision against a matched heuristic weight in closed-loop nuPlan experiments on a 25-rule rulebook. The certificate is, however, pointwise: no single weight is valid across the sampled ticks of an episode, the median lifetime is one sampling interval (zero subsequent ticks at the native rate), and persistence tracks active-set stability. These findings motivate monitored weighted solves with selective cascade fallback, although the compliance-pattern monitor detects only a subset of measured lapses.