Papers

Under a fairly general condition on the underlying jump rates, we prove that the Exclusion Process with fixed particle density exhibits cutoff at time $t_{{mix}}\sim \frac{1}{2}{t}_{{rel}} \log N$, where $N$ is the volume and ${t}_{{rel}}$ the relaxation time of the single-particle dynamics. Our result covers, in particular, the standard setting of uniform nearest-neighbor jumps on large discrete tori in any fixed dimension and, more generally, on any sequence of vertex-transitive graphs with bounded degree and polynomially diverging diameter. Our (short and entirely human) proof combines Wilson's method, the Octopus Inequality, Fourier analysis on Hamming slices, and a sharp comparison with the Dirichlet form of a natural accelerated variant of the dynamics.

Consider an open Jackson network with Poisson arrivals and exponential services, in which no node is overloaded and exactly~$k_0$ nodes have load equal to~$1$. We prove that the queue-length process is recurrent if and only if $k_0\leq 2$. For that, we show that the network satisfies a sector condition, so its capacities are comparable to those of its symmetrisation; the latter is a reversible network whose recurrence/transience can be dealt with by standard means. We also discuss what the Lyapunov-function method gives in this setting; in particular, we show that every balanced queue is empty at arbitrarily large times, also in the transient case.

We propose a novel domain decomposition framework based on energy minimization, applicable to a wide class of linear and nonlinear partial differential equations. This energy-minimizing domain decomposition (EMDD) method addresses problems whose solutions are minimizers of an energy functional, either unconstrained or constrained to a manifold, and thus covers source and eigenvalue problems in a single formulation. Besides introducing the general framework, we derive the algorithmic realization of the method for four classes of model problems, namely linear and nonlinear source and eigenvalue problems, and derive the computational complexity of the method. Numerical experiments demonstrate the robustness and efficiency of the method across all four model problems and show that it compares favourably against existing model-specific domain decomposition methods.

Shift-invariant permutations of $\mathbb F_2^n$ are particularly attractive in symmetric cryptography because of their regularity and implementation efficiency. Constructing such permutations with an explicit algebraic structure remains a challenging problem. In this paper, we study a family of shift-invariant transformations of $\mathbb F_2^n$ constructed from a recursively defined sequence of mappings $\{\gamma_j\}_{j\ge 0}$ associated with landscapes. First, we prove that, under an explicit dimension condition relating $n$ to the support of the landscape, the sequence $\{\gamma_j\}_{j\ge 0}$ has the polynomial composition property if and only if the corresponding landscape is quasi-conserved on $\mathbb F_2^n$ with a nonzero special index. Then, we determine the eventual zero or periodic behavior of $\{\gamma_j\}_{j\ge0}$, including the exact index at which the sequence either first vanishes or becomes periodic, together with the least eventual period, and establish linear independence until the first linear relation occurs. When the polynomial composition property holds, we establish an explicit isomorphism between the group of permutation elements in the monoid generated by these mappings and the unit group of a quotient ring $\mathbb F_2[z]/\langle p(z)\rangle$, where $p(z)$ is a monomial or a binomial determined by the landscape. This framework reduces the tasks of testing bijectivity, computing iterates, and determining compositional inverses and orders of permutation elements to polynomial computations. Finally, we apply the results to mappings represented by binomials and trinomials in the quotient ring, obtaining explicit criteria and formulas, and tabulate more than one hundred representative constructions of shift-invariant permutations, recovering and unifying several previously studied families.

We consider composition operators in $L^2(\mu)$, where $\mu$ is a discrete measure. Our main result shows that, for every integer $p\ge2$, there exists a non-hyponormal composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of a circuit of length $p$ and one infinite branch. In contrast, if the symbol induces a directed graph consisting of one loop and one infinite branch, then the generation of Stieltjes moment sequences does imply subnormality. These two results completely solve the “moment property implies subnormality” problem left open in [Adv. Math. 310 (2017), 484-556] for the entire family of one-branch directed graphs with a circuit of arbitrary length. Combining them with the examples in that paper shows that, for directed graphs consisting of one loop and a finite number of infinite branches, $2$ is the smallest number of branches for which the moment property does not imply subnormality. Finally, we construct a noninjective composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of one loop, one infinite branch, and one leaf. This operator gives a negative answer to the injectivity problem posed in the aforementioned paper. Both constructions use the Krein and Friedrichs measures of an arbitrary normalized S-indeterminate Stieltjes moment sequence.

We classify common invariant probabilities for two beta-transformations with a rational affine relation between their bases. Let $\beta>1$ be irrational and $\gamma=a\beta+c>1$, where $a,c\in\mathbb Q$ and $a+c\ne1$. Every finite nonnegative measure that is singular with respect to Lebesgue measure and invariant under both maps is a multiple of the point mass at zero. No entropy, ergodicity or multiplicative independence assumption is needed. The proof constructs a finite signed measure invariant under an irrational rotation and relates its mass to the first moment of the original measure. Combining this result with the known classification of coincident absolutely continuous invariant probabilities gives all common invariant probabilities in this family. The only additional measures are mixtures with a common absolutely continuous probability for adjacent bases in a known quadratic family. We also determine all common fixed points on the excluded line $a+c=1$.

Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.

We study a quasi-linear Schrödinger-type equation arising in the description of particle dynamics within atomic nuclei. Under suitable assumptions on the model parameters, we establish the existence, uniqueness, and non-degeneracy of positive radial solutions. Moreover, we analyze how these solutions depend on the parameters.

To design systems that are protected against events much rarer than the observational record, extreme-value methods are needed to extrapolate distribution tails. Tail-index estimators such as the Hill estimator are central to this extrapolation, but overestimating the tail exponent can lead to substantial underestimation of rare-event probabilities. Motivated by this, we derive large- and moderate-deviation asymptotics for the Hill estimator and use them to construct estimators whose probability of exceeding the true tail index decays at a controlled exponential rate (the decay rate). In the large-deviations regime, we show that a simple rescaled version of the Hill estimator achieves an optimal balance between bias and decay rate among scale-invariant estimators based on the same top $k$ order statistics. Under a second-order condition, we quantify the effect of the Hill bias, analyze a bias-corrected estimator, and identify sufficient conditions for moderate deviations in the boundary case where the second-order parameter $\rho$ equals zero.

We present a systematic approach to characterize Devaney chaos in the solution of homogeneous one dimensional constant-coefficients PDEs on Herzog spaces. We show that for equations where the highest-order time derivative lacks mixed spatial derivatives, there always exists a weighted space in which the solution is chaotic. This analysis is extended to include mixed partial derivatives in the highest-order temporal term. Moreover, we link the chaotic behavior in Herzog spaces to the semidiscretization of the equations by identifying isometric isomorphisms between the two contexts, providing also a characterization for chaos on these semidiscretizations. The work concludes with two applications that provide new insights on the dynamics of the fourth-order Moore-Gibson-Thompson equation and the viscous van Wijngaarden-Eringen equation.

Stochastic processes underpin dynamics across biology, physics, epidemiology, and finance, yet accurately simulating them remains a major challenge. Classical approaches such as the Gillespie algorithm are exact for Markovian, time-independent systems, where propensities depend only on the current state and agents of a given type are statistically identical. While efficient, this framework misses a defining feature of many real systems: heterogeneity and memory at the level of individual agents. Cells may divide or differentiate on distinct intrinsic timescales, individuals may preferentially interact with specific partners, and inter-event-time distributions can deviate strongly from the exponential. We introduce MOSAIC (Modeling of Stochastic Agents with Individual Complexity), a general and scalable framework that embeds agent-specific properties directly into the dynamics. MOSAIC unifies heterogeneous rates, dynamic interaction preferences, and both Markovian and non-Markovian waiting-time distributions within a single stochastic formalism, while retaining Gillespie-like computational cost.

By distilling and combining earlier arguments, we give a self-contained and entirely elementary proof of the strong and real linear polarization inequalities established by Martínez and Ortega-Moreno. Specifically, for any unit vectors $u_1,\ldots,u_n\in\mathbb{R}^d$, there exists a unit vector $v\in\mathbb{R}^d$ such that $$\sum_{i=1}^n \frac{1}{\langle u_i,v\rangle^2}\le n^2. $$ As an immediate consequence, we recover the real linear polarization inequality $$\prod_{i=1}^n |\langle u_i,v\rangle|\ge n^{-n/2}. $$

We study Bayesian optimization (BO) through the lens of information geometry. Pulling back the Fisher information metric through the surrogate posterior map yields a local sensitivity tensor on the input space, which leads to an upper bound on the gradient of reparameterizable acquisition functions. This view explains vanishing-gradient behavior in high-dimensional BO and provides a common interpretation of heuristics such as RAASP and dimension-scaled lengthscales. Building on this analysis, we propose FITR, a trust-region-based BO method that replaces lengthscale-based scaling by local pullback-Fisher weights. FITR is not restricted to GP kernels with explicit lengthscales. On GP benchmarks with an SE kernel, experiments show competitive performance using FITR. The proposed method also easily generalizes to non-isotropic surrogates, although the gains are more task-dependent in that setting.

Let $C_n^m$ be the $m$-th power of the cycle on $n$ vertices, and let $\Gnm$ be its complement, where $m\ge1$ and $n\ge3m+1$. We prove that every induced subcomplex of the clique complex of $C_n^m$ is homotopy equivalent either to a disjoint union of contractible complexes or to a circle, and we establish this dichotomy for a larger class of circular interval graphs. As a consequence, we obtain a closed formula for the linear strand of the Betti table of $R/I(\Gnm)$. We also prove that $R/I(\Gnm)$ is a Buchsbaum ring of depth $2$ and that the independence complex of $\Gnm$ is a triangulated $m$-manifold with boundary for $m\ge2$, namely a Möbius band or an annulus when $m=2$. For the complement $\overline H$ of every graph $H$ in the above class, we show that $\reg(I(\overline H)^k)=2k+\indm(\overline H)-1$ for all $k\ge2$, where $\indm$ denotes the induced matching number. In particular, some power of $I(\Gnm)$ has a linear resolution if and only if $n\ge4m+1$. In this range, we further prove that the colon ideals $(I^{k+1}:M)$, where $I=I(\Gnm)$ and $M$ is a minimal monomial generator of $I^k$, have regularity $2$.

Andrews and Newman \cite{AN19, AN20} defined the minimal excludant or mex of a partition. The mex of a partition is the smallest positive integer which is absent from the partition. Baruah, Bhoria, Eyyunni and Maji \cite{BBEM} studied the sum of mex split according to parity, together with its $\ell$-th moments. Ballantine and Merca \cite{BM20} generalized mex to the least $s$-gap, which is the smallest natural number that does not occur minimum $s$ times in the partition. Baruah and Talukdar \cite{BT} conjectured a corresponding asymptotic equivalence between the sums of odd and even least $s$-gaps for every $s>1$, generalizing a theorem of Barman and Singh \cite{BS} for the classical mex. In this article, we derive exact formulas for the $\ell$-th moments of $\smex(\pi)$ for every fixed $\ell\ge 1$, in terms of partition functions. We give a complete proof of the conjecture of Baruah and Talukdar \cite{BT} for every natural number $s>1$. We also generalize an identity of Hopkins, Sellers and Stanton \cite{HSS} to the least $s$-gap setting.

We study a class of non-negative spatio-temporal random fields that exhibit self-exciting clustering, which we refer to as affine pure-jump Volterra fields. They are defined via stochastic integration of a Volterra kernel against a thinned Poisson random measure and the thinning is such that the field admits a representation as a stochastic integral of the same kernel, but against a random measure whose compensator has a density that is pointwise affine in the field itself. This representation leads to affine transform formulas characterizing the Laplace transform of functionals of the field up to the solution of a deterministic non-linear Volterra integral equation. Affine pure-jump Volterra random fields extend a non-negative finite-first-moment subclass of pure-jump affine Volterra processes to the random-field setting. The framework includes non-negative finite-variation ambit fields, marked Hawkes systems, and branching-type models as special cases, and we make these connections explicit through examples.

We prove that finite-dimensional real normed spaces with isometric unit spheres are linearly isometric, where each sphere carries the distance induced by the ambient norm. This answers the object-level question of Kadets and Martín in finite dimensions, without asserting linear extension of a prescribed sphere isometry. We associate with each norm a family of Plücker contraction bodies generated by volume-normalized maximal minors of linear contractions into finite-dimensional $\ell_\infty$ spaces. Null-Lagrangian identities show that the sphere metric determines these bodies. A finite exposed-face construction recovers almost-norming contractions from the convexified data, and compactness together with an exact volume identity yields a linear isometry.

Quantum algebras ($q$-algebras) have been used in the literature to deform spin-chain models. In particular, in previous works by the present authors, the Kittel–Shore Hamiltonian was deformed using $\mathfrak{su}_q(2)$ algebra. The deformed Hamiltonian was presented for a general spin, and the thermodynamic properties were studied for spin-1/2 particles. In this paper, we focus on the $q$-deformation of the Kittel–Shore model for spin-1 systems. We analyse the impact of this deformation on the thermodynamic properties and phase transitions of the system as a function of the deformation parameter $q$. Specifically, we study the specific heat, magnetic susceptibility, and magnetisation, including a detailed analysis of the Curie temperature characterising the ferromagnetic phase. Furthermore, a finite-size scaling analysis is conducted for both the ferromagnetic and antiferromagnetic cases. Results are systematically compared with those of the undeformed model to reveal the physical effects of the $q$-deformation.

For linear-quadratic $N$-dimensional Nash systems with mean-field-like scaling and strongly displacement monotone data, we prove uniform-in-time a priori derivative estimates exhibiting the expected scaling as $N \to \infty$ in order to pass the system to the limit. As a consequence, in a heterogeneous (i.e., non-symmetric) mean-field setting, we obtain both uniform-in-time convergence of the Nash system on any arbitrarily long horizon $[0,T]$ (as $N \to \infty$), and convergence of both the $N$-dimensional Nash system and the corresponding master equation to their respective ergodic counterparts (as $T \to \infty$).

We study fundamental information-theoretic limits of robust stochastic optimization when the distribution is known only through its exact moment sequence. We develop a unified framework that produces families of distinct distributions sharing all moments yet inducing radically different optimal decisions, thereby establishing strong impossibility results for a range of decision problems under moment ambiguity. Our binary and $N$-way moment-matching constructions separate quantiles, threshold regions, and online selection opportunities while preserving every moment. For the newsvendor problem, moment equivalence causes quantile ambiguity, inducing any fixed or randomized order quantity to fail arbitrarily badly. For revenue maximization, no deterministic or randomized posted pricing scheme can secure a nontrivial approximation relative to the full-information benchmark. For the secretary problem with cardinal observations, the optimal robust success probability under exact moment disclosure is the classical finite-horizon value $s_m$, even for i.i.d. observations. This value converges to $1/e$, compared with the full-information limit of approximately $0.5801$ due to Gilbert and Mosteller (J. Am. Stat. Assoc., 1966). We also recover and expand upon the recent impossibility result of Correa et al. (STOC, 2026) for prophet inequalities with moment knowledge. For every fixed $r\ge1$, exact moment knowledge can yield at best a $\Theta(1/\log m)$ competitive ratio, even when competing against relaxed benchmarks based on expected $r$-th order statistics or when the algorithm is allowed to select $r$ items.

We prove the (uniform) Yau-Tian-Donaldson Conjecture for transcendental Kähler classes in the case of trivial automorphisms. Indeed, we show the existence of Special Kähler Fujita Approximations of big cohomology classes associated to big test configurations, establishing the equivalence between the uniform $K$-stability and the strengthened version for models. More generally, we show that any big cohomology class on a compact Kähler manifold admits such Special Kähler Fujita Approximations: the volume and the analogue of the Riemann-Roch coefficient of the big class are both approximated by those of Kähler classes on higher compactifications.

Pinching-antenna systems (PASS) create reconfigurable wireless channels by moving dielectric antennas along a waveguide. Mechanical position errors translate into phase errors that degrade performance, yet almost all prior PASS studies assume ideal placement. This paper models this position error, derives the resulting phase error, and analyzes its impact on time-division multiple access (TDMA) and non-orthogonal multiple access (NOMA), obtaining their ergodic-rate bounds and closed-form outage probabilities. Analyses and simulations reveal that single-antenna TDMA is immune to position error, whereas multi-antenna TDMA loses array gain and NOMA suffers an interference floor that cannot be overcome by increasing transmit power.

This paper studies constrained optimization problems through the lens of feedback control. Building on the interpretation of Lagrange multipliers as feedback controllers, we propose the saddle-point PID (SPPID) dynamics: a unified proportional–integral–derivative (PID) framework for continuous-time saddle-point dynamics. The proposed dynamics employ PID control for equality constraints and anti-windup PI control for inequality constraints. We show that SPPID is equivalent to a preconditioned primal–dual gradient flow of the augmented Lagrangian. This equivalence reveals the distinct role of each feedback component: integral action enforces constraint satisfaction, proportional action induces the augmented Lagrangian structure, and derivative action modifies the geometry of the primal dynamics via a state-dependent Riemannian metric. For convex problems, we establish convergence to the KKT set together with conditions for global asymptotic stability. For equality-constrained problems, we establish local exponential convergence for nonlinear equality constraints under standard assumptions, and show that projected gradient flow emerges as the infinite derivative-gain limit. For strongly convex problems with affine equality or inequality constraints, we establish global exponential convergence by leveraging contraction theory. Finally, we provide various numerical examples to illustrate the utility of SPPID.

Most studies of stochastic search processes with resetting focus on spontaneous resetting events that occur at a random sequence of times typically generated by a Poisson process. One of the major consequences of spontaneous resetting is that it yields a finite mean first passage time (MFPT) in unbounded domains, resulting in an optimal resetting rate that minimises the MFPT. An optimal resetting rate can also occur in bounded domains provided that the reset point is not too far from the target. Recently, there has been growing interest in event-based rather than spontaneous resetting, where resetting is triggered when a specific threshold is reached. Identifying the threshold with a physical non-target boundary then leads to so-called boundary-induced resetting. In this paper, we explore the effects of boundary-induced resetting on single-particle diffusive search in a two-dimensional (2D) bounded domain $\Omega$ containing one or more small interior targets. The interior target boundaries are totally absorbing, whereas the exterior boundary $\partial \Omega$ is sticky. That is, whenever the particle reaches a point on $\partial \Omega$ it remains attached for a random waiting time $\tau$ after which it immediately resets to a fixed interior point $\x_0\in \Omega$. Using renewal theory, matched asymptotic analysis and Green's functions we calculate the unconditional MFPT for absorption by any of the targets and compare this with the corresponding MFPT of a search process on the same domain with a reflecting surface $\partial \Omega$ and no resetting. We show that boundary-induced resetting only reduces the MFPT relative to the no resetting case if the searcher's reset point $\x_0$ is sufficiently close to one of the targets.

This study derives the asymptotic conditions under which the massless stretched string formulation is valid for the equivalent dynamics of the inertial string carrying a moving point mass. Consequently, the governing equation under such conditions can be written as a singularly perturbed partial differential equation when the inertia of the string is included. When the inertia of the string is ignored, the governing equation reduces to a non-homogeneous hypergeometric ordinary differential equation governing the transverse motion of the moving mass. This hypergeometric equation has a discontinuous solution at the terminating boundary known as the trajectory paradox. However, numerically solving the actual equation yields a continuous solution with a rapid variation near the boundary. The complete solution resembles the inner solution as well, whereas the hypergeometric ordinary differential equation leads to an outer solution, revealing that the trajectory paradox is a boundary layer.

Among the various "no-go theorems" in the foundations of quantum mechanics, Bell inequalities are among the few, which do not implicitly rely on the mathematical structure of quantum mechanics. Instead, Bell inequalities are commonly understood to rest on Kolmogorov's theory of probability and a "locality" assumption. Physical evidence supporting their violation is hence taken as evidence against at least one of these two assumptions. This work provides a comprehensive analysis of the physical debate on Bell inequalities from the point of view of Kolmogorov's theory of probability, focusing on the Bell-Clauser-Horn-Shimony-Holt inequality (BCHSH inequality). The central contribution is the motivation and mathematical study of the notion of contextuality in Bell experiments: in so-called contextual Bell models the detectors themselves behave randomly and the distribution of the random detector parameters depends on the chosen detector settings. It is shown how even "classical" Bell experiments motivate such a description. The need to account for contextuality was pointed out before by de la Peña, Cetto, Brody, Lochak, Bohm, Hiley and others. A broad class of contextual models is proven to satisfy the BCHSH inequality, thereby generalizing a previous result by Gill and Lambare. Nonetheless, even under strong locality requirements, contextual Bell models may violate the BCHSH inequality: based on work by Popescu and Rohrlich, we consider a "proof of concept" example exhibiting the maximum possible violation. Contrary to the much debated "loopholes" in empirical tests of Bell inequalities, contextuality thus reveals a conceptual misunderstanding in the physical conclusions commonly drawn from Bell inequality violations.

We present a direct simulation Monte Carlo (DSMC) method for the kinetic theory of ordered fluids recently introduced by the authors. The microscopic constituents of these fluids carry an order parameter belonging to a manifold. The method applies Strang splitting to couple a Monte Carlo treatment of the binary collisions with a symplectic integration of the mean-field Vlasov transport on the order-parameter manifold. We use the Nanbu-Babovsky approach for Maxwellian collision kernels, and Bird's acceptance-rejection method for general ones. The mean-field Vlasov force is reconstructed from the particle ensemble at each time step. We also include a Bhatnagar-Gross-Krook (BGK) / Andersen thermostat to control the system temperature. We formulate the method for a generic order-parameter manifold, and specialize it to two-dimensional calamitic (rod-like) molecules. Numerical tests validate the method. We confirm the relaxation to the Maxwellian distribution predicted by the H-theorem, and simulate the mean-field alignment driven by quadratic, Kuramoto, and Onsager interaction potentials. With the Onsager potential we successfully capture the isotropic-nematic phase transition at the correct critical temperature.

In this work we construct a new semi-implicit (SI) hyperbolic and thermodynamically compatible (HTC) scheme for the compressible Euler equations of gasdynamics. The challenge therein lies in solving the entropy inequality as the primary evolution equation instead of the total energy conservation law, which in our framework is an additional conservation law that needs to be fulfilled by the numerical scheme as the consequence of a compatible discretization. The proposed method has a predictor-corrector character, where we first solve the isentropic problem without considering the entropy production. Therein, the entropy and the nonlinear convective terms in the momentum flux are discretized explicitly, while the mass flux and the pressure terms are taken implicitly. The discrete momentum equation is then inserted into the discrete mass conservation equation, leading to a mildly nonlinear system for the pressure, which can be efficiently solved with a nested Newton-type algorithm. Based on this prediction, a non-negative production term in the entropy is added, ensuring a cell entropy inequality for the numerical scheme. To guarantee global conservation of total energy at the discrete level, a new modified global Abgrall-type flux correction is introduced. The presented numerical method allows time steps that are restricted only by the material wave speed and not by the sound speed. This makes the scheme particularly useful for flows in the low-Mach-number regime. Numerical results are shown in one and two space dimensions to assess the theoretical properties of the new scheme.

We introduce Boolean trusses, that is, trusses in which every element is idempotent. Every Boolean truss admits a decomposition into a truss arising from a Boolean ring and a rectangular truss. Furthermore, we classify all Boolean trusses up to isomorphism.

We develop a balanced truncation method for linear delay differential systems based on a finite-rank alternating-direction implicit (ADI) iteration for the associated operator Lyapunov equations. The method acts directly on the infinite-dimensional history-state realization and does not discretize the delay interval during Gramian approximation or balancing. The finite-rank reachability and observability factors are represented exactly in Takenaka–Malmquist coordinates, reducing the required computations to finite-dimensional linear algebra. These factors yield a finite-dimensional, delay-free reduced model. For real system data, the implementation remains in real arithmetic. Under suitable conditions on the ADI shifts and a gap in the Hankel singular values, we prove convergence in $\mathcal H_\infty$ to the reduced transfer function obtained by exact balanced truncation of the infinite-dimensional system. The construction extends to finitely many discrete delays. Numerical experiments validate the method against independent reference computations. The complete MATLAB implementation used in the experiments is provided as supplementary material.