Papers

Core stability gives approval-based committee elections a strong form of coalitional proportionality, but deterministic existence is not known for an arbitrary number of seats. We prove that every finite election with nine seats has a deterministic Hare-quota core committee. Starting from an eight-seat committee that is both a global Proportional Approval Voting (PAV) maximizer and core stable, we suppose that all one-candidate extensions are blocked and extract an inclusion-minimal closed response system. Singleton responses are eliminated by exact equality rigidity. For the remaining responses, rational Farkas certificates bound every nonpositive PAV add-marginal drift below by $-n/189$ and force every successor drift above $n/126$. The uniform response chain is irreducible, so stationarity makes these bounds incompatible. The computer-assisted component comprises an exact pointwise check over 7,356 voter types, 36 one-blocker cells, and a symmetry-complete family of 19 two-blocker motifs. Exact certificate archives and standard-library verifiers support these checks. The theorem settles the nine-seat case but does not decide deterministic core existence for arbitrary committee size.

Chung, Graham, and Spiro introduced slow Fibonacci walks and used them to partition the integers $n\ge2$ into two sequences, the down-integers and the up-integers. They studied the local spacing of these two sequences and conjectured that their $\ell$-step gap sets agree for every $\ell\ge1$. We show that the conjecture fails at $\ell=4$ by proving \[ 9\in U_4\setminus D_4 . \]

We present an exposition of Bourgain's celebrated 1986 proof of the circular maximal theorem.

Clinical decision support frequently involves heterogeneous measurements, expert assessments, and guideline-based knowledge whose relationships to clinical decisions are gradual rather than purely binary. This paper develops a clinical decision-support framework based on bipolar fuzzy relational optimization, in which favorable and unfavorable clinical relationships are represented jointly through positive and negative fuzzy relational matrices together with a fuzzy requirement vector. The simultaneous participation of each decision variable and its complement in the relational constraints enables bipolar clinical information to be incorporated within a unified optimization model. The minimum \(t\)-norm is adopted to provide a noncompensatory, bottleneck-type interpretation of the interaction between fuzzy relational grades and decision levels. We characterize the feasible recommendation set by deriving clinical admissibility intervals and effective evidence activation sets, which provide the basis for feasibility analysis and systematic reduction of the relational system. The complete feasible set is then represented as a finite union of clinical recommendation regions associated with admissible Clinical Evidence-Assignment Functions. A region-wise optimal candidate is constructed for each admissible evidence assignment, and comparison of the resulting finite collection of candidates yields a globally optimal recommendation. An algorithm integrating feasibility analysis, system reduction, region-wise optimization, and global selection is developed and illustrated through a numerical CDSS example. The results show that bipolar fuzzy relational optimization provides a structured and mathematically rigorous mechanism for transforming fuzzy clinical relational information into feasible and optimized decision-support recommendations.

For finite-dimensional Banach spaces $E$ and $F$, set \[ \rho(E,F):=\sup_{0\neq z\in E\otimes F}\frac{\pi(z)}{\varepsilon(z)}. \] The square growth of $\rho(\ell_p^d,\ell_q^d)$ was determined by Bonet, Defant, Peris and Ramanujan. We study the rectangular problem with the two dimensions varying independently. If $r=\min\{n,m\}$ and $\eta(t)=\min\{1/t,1/t'\}$, then, whenever $p$ and $q$ lie on the same side of $2$, \[ \rho(\ell_p^n,\ell_q^m)\asymp_{p,q} r^{1/2}\min\{n^{\eta(p)},m^{\eta(q)}\}. \] Thus the classical square exponent splits into two dimensional scales. In the mixed range $1\le p\le2\le q\le\infty$, writing $a=1/p$ and $b=1/q$, we obtain the lower estimate \[ \rho_{p,q}(n,m)\gtrsim_{p,q} \max\!\left\{r^{\min\{a+b,2-a-b\}}, m^{b-1/2}r^{1/2}\min\{n^{1-a},m^{1/2}\}\right\}, \] and the upper estimate \[ \rho_{p,q}(n,m)\lesssim_{p,q} \min\!\left\{n^{1-a}r^{1-b},m^b r^a,r\right\}. \] Moreover, if $(n-m)(a+b-1)\le0$, then \[ \rho_{p,q}(n,m)\asymp_{p,q} r^{\min\{a+b,\,2-a-b\}}. \] In the interior mixed range $1<p<2<q<p'$, if \[ \alpha_{p,q}:= \frac{(a+b-1)(\frac12-b)}{a-\frac12}, \] then \[ \rho_{p,q}(n,m)\lesssim_{p,q}m^{1-\alpha_{p,q}}, \qquad \sup_{n\ge1}\rho_{p,q}(n,m)\asymp_{p,q}m^{1-\alpha_{p,q}}. \] The corresponding statements in the reversed mixed range follow by duality and symmetry.

We study an asynchronous consensus dynamics on $N$ agents: at each step a uniformly chosen agent replaces its state by $f(Y_1,\dots,Y_r)$, where $f$ is a fixed monotone aggregation rule and $Y_1,\dots,Y_r$ are the states of $r$ agents sampled uniformly with replacement. Let $T$ be the first time at which all agents agree. Let $f:\{0,1\}^r\to\{0,1\}$ be monotone and non-constant. The expected consensus time is governed by the two endpoint degrees $D_0(f)=\#\{i:f(e_i)=1\}$ and $D_1(f)=\#\{i:f(\mathbf{1}-e_i)=0\}$, where $e_i$ is the $i$th standard basis vector and $\mathbf{1}$ the all-ones vector. If $D_0(f)\ne 1$ and $D_1(f)\ne 1$, then $E[T]=\Theta\bigl(N(1+\log d_f)\bigr)$, where $d_f$ counts the agents that must change state before the nearest attracting consensus is reached, so that $E[T]=O(N\log N)$ uniformly over initial states. If $f$ is a dictator, $E[T]$ is given by a voter-model formula and equals $\Theta\bigl(N^2 Ent(p_0)\bigr)$ with absolute constants, uniformly over the initial state, where $Ent$ is the binary entropy and $p_0$ the initial fraction of agents in state $1$. Otherwise exactly one of $D_0(f),D_1(f)$ equals $1$ and the other equals $0$. The worst-case expected consensus time is then $\Theta(N^{2-1/m})$, where $m\ge 2$ is the least number of coordinates that force the value $1$ in the residual rule of $f$, defined in the paper (of its dual, when $D_1(f)=1$).

Distributed systems often operate under uncertainty, where minimizing expected loss may overlook rare but severe events. This paper studies a distributed risk-averse convex optimization problem in which agents cooperatively minimize the average of local conditional value-at-risk (CVaR) objectives over a time-varying network. Each agent has access only to noisy evaluations of its local loss function, rather than to its CVaR objective or gradient. We therefore develop a zeroth-order algorithm that uses sampled losses to construct empirical CVaR estimates and their gradient estimates. At each iteration, agents combine neighboring decisions and perform a local update. Under convexity and Lipschitz continuity assumptions, we prove that the agents reach exact asymptotic consensus. We also establish a finite-time expected suboptimality bound for the weighted ergodic iterate. With diminishing step sizes and fixed sample sizes, the local last iterates converge almost surely to a common optimum, and their limiting expected CVaR gap is bounded in terms of the smoothing and finite-sample errors. This distributed bound matches the parameter dependence of the centralized benchmark provided in this paper. Finally, simulations on a distributed sensor network estimation problem illustrate the efficacy of the method.

Let $A\subseteq\mathbb{R}$ support a probability measure whose Fourier–Stieltjes transform tends to zero at infinity. We prove that, for every $\varepsilon\in(0,1)$, there is a closed, $1$-periodic, nowhere dense set $E\subseteq\mathbb{R}$ such that \[ m(E\cap I)\ge1-\varepsilon \] for every interval $I$ of length $1$, while $E$ contains no affine copy of $A$. Thus every set supporting a Rajchman measure satisfies the Erdős similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Ivašev-Musatov, showing that for every Hausdorff gauge $h$ there is an $h$-null compact Rajchman support $K$ satisfying \[ \overline{\dim}_{\mathrm B}^{\log} K=\dim_{\mathrm P}^{\log} K=1. \] The value $1$ is sharp for both dimensions.

In this paper, we provide an alternative proof of a result due to Gross (1960s), extending it to the Fréchet space setting. Our argument adapts the proof technique developed by Bogachev in the Banach case, as presented in the monograph Gaussian Measures, to the more general framework of separable Fréchet spaces, simplifying several steps along the way.

We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space $\Pi(X)$ of homotopy classes of paths (relative endpoints) to the path space $X^I$ that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if $X$ has the dpp, then its universal cover is contractible and, in particular, $X$ is aspherical. The space $\Pi(X)$ is a fiber bundle over $X$ whose fiber is the universal cover of $X$, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from $\Pi(X)$ to $X^I$ and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.

In this manuscript, scalar norm-type accelerators are used for first time in order to increase the order of convergence of known iterative methods for solving the matrix equation $X^{-1}-A=0$, while decreasing the number of matrix-matrix products. Specifically, our proposed scheme upgrades the order of convergence in a $50\%$, achieving the sixth-order of convergence with the computational cost of fourth-order Hyperpower method, in terms of matrix-matrix multiplications. In the main results, not only the convergence and its order is proven, but also its efficiency and stability. This new technique provides good numerical results, even compared with existing fourth-, sixth- and eighth-order schemes in large matrices of up to $10^6$ entries, but also promises the opening of a new kind of procedures with good performance and scalability. Finally, an application to digital image restoration is made to show the behavior of the method, compared with existing ones.

We prove a structure theorem for the dimension towers~$\{d_k(D)\}_{k\geq0}$ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~$K=\Q(\qD)$. If~$\eps$ denotes the first totally positive power of a fundamental unit of~$K$ and $t_k = \eps^k + \eps^{-k}$ the trace of its $k$-th power, then the SIC dimension tower~$\{d_k=1+t_k\}_{k\geq0}$ is the level~$m=3$ row of an infinite two-dimensional cyclotomic array~$\{\Psi_m(t_k)\}_{m\geq1,k\geq0}$ attached to~$K$, while the auxiliary factors~$(d_k+1)$ and $(d_k-3)$ are its ramified levels $m=2$ and $m=1$. Here~$\Psi_m$ denotes the minimal polynomial of~ $\zeta_m+\zeta_m^{-1} = 2\cos{2\pi/m}$. This construction arose initially from an attempt to formulate relations among SIC dimensions in $q$-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family $c_{m,k}=1-\zeta_m\eps^k$ over the cyclotomic field tower $\{K(\mu_m)\}_{m\geq1}$. This framework sheds new light on the mod-$p$ analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~$2mD$, the valuations $v_p(\Psi_m(t_k))$ at every fixed level~$m$ are described exactly in terms of a single local unit valuation, which is then related, through the~$p$-adic class number formula, to the corresponding $p$-adic $L$-value.

This paper establishes the algebraic and categorical foundations of spectral duality structures (SDS). We show that the class of all SDS admits the structure of a graded symmetric monoidal category, where the degree $K$ is the fundamental invariant indexing the strata. We define operations of Cartesian product and disjoint union, morphisms preserving involution and weights, and prove that $C^\ast=1/(1+\sqrt K)$ is a functor constant on each stratum. We classify objects up to isomorphism by the combinatorial type $(k,f)$, the degree $K$, and a multiset of inversion classes $[r]=\{r,r^{-1}\}$. In each degree, skeletons are exactly the finite $C_2$-sets over the additive group of the reals with $C_2$ acting by $t\mapsto -t$, which explains the monoidal structure and identifies the classification with the decomposition into orbits. Furthermore, we prove that the response rank is the rank of an explicit matrix built from the amplitudes and the weights, and hence exactly the number of non-trivial pairs $k$ under the full-rank condition, and we connect the categorical structure with the Fisher–Rao geometry of Paper V. The SDS, originally discovered in stochastic resetting problems, is thus an autonomous mathematical object with a rich algebraic structure and a natural geometric realisation.

This paper introduces and studies a Principal-Agent competition model in which two risk-neutral Principals compete to hire a single Agent. The Agent chooses to work exclusively for one Principal, making the Agent's reservation utility endogenous, as it is determined by the competing offers. We characterize the Nash equilibria of the resulting game by relating them to the individual Principal-Agent contracting problems and analyzing the Agent's selection bias. We further introduce two numerical algorithms to approximate Nash equilibria by locating sign changes of the value functions of the Principals' individual problems. The first is a Newton-type method that exploits differentiability of the value functions, while the second is a bisection method that does not require differentiability. The proposed procedures are illustrated through several examples involving exponential, logarithmic, and constant relative risk aversion (CRRA) utility functions within the polynomial framework of Renner and Schmedders (2015). We also provide technical results regarding the regularity of the value function of a classical Principal-Agent problem, including differentiability under suitable conditions.

We propose a framework for multiple zeta values in $\lambda$-rings that unifies both classical multiple zeta values and multiple $q$-zeta values. As in the classical case, we show that the multiple zeta function defines an algebra homomorphism from the quasi-shuffle algebra to the $\lambda$-ring. We also prove a formula for multiple polylogarithms in $\lambda$-rings, similar to the iterated integral expression in the classical case. As an application, we explicitly describe the algebra of regular multiple $q$-zeta values, compute the corresponding Poincaré series, and exhibit a small spanning set.

Let a finite population of n labelled examples carry a class-weighted loss, with pi*n in a rare positive class weighted by N0/N1. We study estimation of total risk from a subsample K << n under designs allocating K0 and K1 draws to the two strata. We derive the exact finite-population variance of the weighted risk estimator under class-conditional sampling without replacement and solve for the optimal allocation. The class multiplier inflates positive-stratum dispersion by the imbalance ratio, causing that ratio to cancel from the optimal allocation and making equal, rather than proportional, allocation the natural default. Simple random sampling is dominated by an explicit between-stratum term; an exact bias identity shows that cluster-representative selection has no general unbiasedness guarantee; and a Serfling bound transfers the allocation result to selection error over a finite candidate set. Under the implemented truncation, the realised allocation ratio is gamma=min(2*pi/f,1), independent of n, yielding the parameter-free efficiency prediction A(pi,f)=gamma/[pi(1-pi)(1+gamma)^2]. A separate measurability result bounds the record occupied by a labelled example, determining the required train-test separation and controlling departure from block independence under absolute regularity. We test these predictions on forecasting the onset of statistically explosive price regimes, dated ex post by the Phillips-Shi-Yu procedure, using 350 U.S. equities from 2004-2011 with under 1% positive rows and five purged forward blocks. The predicted ordering of the four designs holds, and at the 10-day horizon the five design points are ordered exactly as predicted by A (Spearman rho=1, exact p=0.0167). The predicted dependence on pi across horizons does not hold; we identify the channels lying outside the design-based argument.

In this paper, we study the autocovariance matrix estimation and inference problems under heavy-tailedness, high-dimensionality, general nonlinear temporal dependence, and potentially nonstationarity of time series. We consider two types of tail-robust autocovariance matrix estimation methods: the element-wise Huber's $M$-estimator and a computationally more efficient element-wise truncated estimator. Both estimators are designed to achieve sharp error bounds in matrix max-norm. The nonasymptotic properties of these estimators are proved based on new variants of Bernstein-type inequalities under functional dependence for the potentially nonstationary processes which may be of independent interest. Moreover, we prove a high-dimensional Gaussian approximation result, as a limiting distribution, for our element-wise truncated autocovariance estimator. A Gaussian multiplier bootstrap result is also given to facilitate the practicality. Our theoretical results are nonasymptotic, which gives explicit error bounds in terms of the sample size, dimensionality, moments, and the strength of temporal dependence. Numerical evidence is provided to support our theoretical results. Finally, we illustrate the benefits of the proposed methodology for detecting change points in monthly macroeconomic data.

Let $X\sim Bin(N,p)$ and \(Y=|X-Np|\). We derive an exact reduction for every odd absolute moment of \(Y\), expressing it through finitely many central masses and central tail probabilities. The tail coefficients satisfy a sum rule and are divisible by \(q-p\), so that in the symmetric case the odd ladder collapses to masses alone. We then obtain the complete central-tail expansion at a bounded lattice displacement, with Bernoulli-polynomial coefficients in integer powers of the large parameter. As an application, the first two tail coefficients yield a second-order expansion for the median of the beta distribution, whose formal one-parameter degeneration reproduces the first two terms of Choi's expansion for the gamma median.

The goal of this paper is to generalize recent results about the topological classification of the dynamics of a real planar polynomial vector field near infinity. Given a polynomial vector field $X$, using Newton polyhedra one can define its generalized upper principal part $X_{\Gamma}^{U}$. By dropping some monomials of $X_{\Gamma}^{U}$, we define its minimal generalized upper principal part $X_{G}^{U}$. We prove that there exist an open and dense set $\widetilde{\mathfrak{U}}_{1}$ in the set of polynomial vector fields with Newton degenerate upper principal part satisfying the following property: if $X\in\widetilde{\mathfrak{U}}_{1}$, then $X$ and $X_{G}^{U}$ are topologically equivalent near infinity, provided that $X_{G}^{U}$ satisfies some additional non-degeneracy assumptions. Our techniques rely on toric compactification and the Normal Form Theorem.

We study uniform observability for Laplace eigenfunctions with mixed boundary conditions whose reflection signs do not define a scalar character. For the DDN/NND sectors of the equilateral rhombus, the resulting $\mathbb Z_2$ monodromy is resolved by a degree-two branched arithmetic translation surface of genus two. On an explicit class of admissible open sets, every exact mixed-sector eigenfunction satisfies a local $L^2$ lower bound uniform in the eigenvalue, multiplicity, and choice within the eigenspace. The proof combines exact unfolding, semiclassical defect measures, and the Veech dichotomy to exclude concentration near the saddle/conic network and inside periodic cylinders. We also obtain a reflection-overlap observability result for the full rhombus and isolate the remaining saddle-network obstruction for arbitrary open observation sets.

Classical extremum seeking (ES) is commonly interpreted as approximating gradient descent, but this interpretation is less clear for nonsmooth objectives in the continuous-time multivariable setting. We propose a minimal modification of the classical multivariable perturbation–demodulation architecture: rationally independent perturbation frequencies and matched demodulation signals. For any locally Lipschitz static objective, the Kronecker–Weyl theorem shows that, at every fixed perturbation amplitude, the long-time averaged dynamics are exactly the negative gradient of a kernel-smoothed objective. Because rationally independent frequencies render the perturbation and demodulation signals nonperiodic, we employ general averaging theory rather than periodic averaging theory. If the gradient flow of the smoothed objective is globally uniformly asymptotically stable, then the ES dynamics are practically globally uniformly asymptotically stable. We also derive a general matching condition relating the perturbation occupation density, demodulation signal, and smoothing kernel, yielding a family of alternative designs. Numerical examples include a nonsmooth objective function, which may be interpreted as the penalty function of a nonlinear program, and the Rastrigin function, for which smoothing eliminates all undesired local minima.

We investigate Chern's conjecture for minimal submanifolds of higher codimension in the unit sphere. We establish two integral inequalities for closed minimal submanifolds. Then we obtain a new rigidity theorem under a pinching condition on the squared norm of the second fundamental form.

This paper considers the energy landscape of the t-SNE algorithm. While this algorithm has enjoyed broad adoption, the non-convexity of the associated energy has made it difficult to rigorously understand what the algorithm captures in many settings. In particular, a number of well-known numerical examples, several of which are reproduced in this article, suggest a complicated energy landscape with many local minimizers that do not respect the topology or clustering structure of the underlying data. This work seeks to provide first steps towards a rigorous explanation of these phenomena. Specifically, for a general family of energies, which include both the original t-SNE algorithm and recently identified large data limits, and for densities in feature space which obey a continuous symmetry, we construct infinite families of distinct critical points. These critical points are based upon identifying pairs of discrete symmetries, one in the original feature space and the other in the target embedding space, which are preserved under gradient dynamics. These critical configurations exhibit many characteristics, such as topology breaking and spurious clustering, which are often observed empirically. Finally, numerical and analytical examples are given throughout as a means of illustrating the approach.

Fukui and Kawakami showed that the trigonometric Haldane–Shastry (HS) spin chain can be deformed by introducing twisted boundary conditions. In this work we revisit the resulting twisted HS chains, which we call Fukui–Kawakami (FK) chains. We analyse their spectra in dependence of the twist parameter, utilising a direct connection with the (untwisted) HS chain. This allows us to explain why part of the spectrum can be described by Fukui and Kawakami's twisted Bethe equations, and can be constructed from Yangian highest weight states. These states can be labelled by combinatorial data called `motifs', as in the HS chain, which cover some of the (deformed) descendants. We furthermore show that there are other descendants which do not follow from these Bethe equations, but whose energy can be described as a sum of two single-particle energies, suggesting additional hidden symmetries. We then focus on the special case of antiperiodic boundary conditions, and show that this chain coincides with the `minimally polarised' long-range model recently introduced by Basu-Mallick, Finkel, and González-López. Remarkably, this connection implies that the antiperiodic chain has a hidden $\mathfrak{gl}(1|1)$-symmetry, which we use to relate the HS motifs to their `supersymmetric' $\mathfrak{gl}(1|1)$ counterparts.

We determine the Boltzmann–Grad limit of the Ehrenfest “wind-tree” model with a periodic configuration of polyhedral scatterers. The central result is a loss-of-memory effect in dimension $d\geq 3$ for a generic choice of polyhedron. This yields a Markovian limiting transport process on an extended state space and a generalised linear Boltzmann equation for the time evolution of the particle density.

The integral Chow motive of an Enriques surface $S$ is determined by a torsion summand $\mathfrak{t}(S)$, which contributes the torsion in the cohomology of $S$. We study isomorphisms between and endomorphisms of motives of the form $\mathfrak{t}(S)$. This is closely related to the integral Hodge conjecture for products of two Enriques surfaces.

We study edge-state transport in a two-dimensional topological insulator with a periodically deformed edge subjected to a uniform magnetic field. Zeeman coupling breaks time-reversal symmetry and enables elastic backscattering, producing oscillations of the forbidden-band widths. In the strong-field regime, the gaps can close completely at discrete field values. In the weak-field regime, we identify an important class of periodic deformations for which the dominant semiclassical scattering is controlled by complex infinity rather than by the nearest turning points. We develop a semiclassical treatment of this process and establish its agreement with perturbation theory and direct numerical calculations. The gap modulation should produce observable oscillations of the edge conductance. Unlike conventional magnetic quantum oscillations, which are periodic in inverse field, the predicted oscillations are periodic in the magnetic field itself, with a period determined by the Fermi velocity and effective g-factor

We prove that for any cocompact Fuchsian group the class of Medvedev degrees attained by subshifts of finite type is the class of $\Pi_1^0$ degrees. This result relies on the construction of a rigid hierarchical structure in a graph model of the hyperbolic plane, and on the fact that every such group admits a representation in $\operatorname{PSL}_2(\mathbb{R})$ such that the matrices in the image have coefficients that are computable real numbers.

The enumeration of pattern-avoiding permutations has been a popular area of study over the past several decades, but comparatively little attention has been given to the topic of pattern-avoiding involutions. In this paper, we derive the algebraic generating functions of two Wilf-equivalence classes of involutions avoiding a single pattern of length $4$, $\operatorname{Av^I}(2431)$ and $\operatorname{Av^I}(3421)$. We then adapt the Mosaic method, a fast counting algorithm for permutations, to count involutions and apply it to substantially extend the known initial terms of the counting sequences for the remaining two Wilf-equivalence classes avoiding a pattern of length $4$, $\operatorname{Av^I}(1324)$ and $\operatorname{Av^I}(4231)$. Based on these extended sequences, we empirically analyze the asymptotic behavior of the counting sequences of these two classes.

Let $$X_n=\sum_{i=1}^s t_i A_i^{\otimes n},$$ where $A_1,\ldots,A_s\in M_d(\mathbb C)$ and $t_1,\ldots,t_s\in\mathbb C$ are fixed, while $n$ grows. Direct computation of determinants or Schatten norms of $X_n$ is exponential in $n$. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms. We give an exact representation-theoretic method which, for fixed $d$ and $s$, computes $\|X_n\|_p$, $0<p<\infty$, and determinants in polynomial time in $n$. Schur–Weyl duality yields a simultaneous block decomposition, while Jacobi–Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers. For $d=3$, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term $3\times3$ trace-norm problem with $n=18$ is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately $2.4\times10^{18}$ bytes. Direct and reduced computations agree to relative error below $3.4\times10^{-15}$ throughout their common range $n\leq9$.