Papers

The emergence of movable antenna technology enables flexible antenna placement, allowing transceiver antennas to more effectively exploit spatial degrees of freedom. In this paper, we investigate a monostatic near-field sensing system, aiming to minimize the worst-case squared position error bound (SPEB) over the entire near-field region by jointly optimizing the transmit and receive antenna placements, along with the transmit power allocation. Toward this end, we first derive the closed-form expression of SPEB by calculating the Cramér-Rao bound (CRB) for estimating the target's angle and distance. It is shown that the derived SPEB is governed by the power-weighted moments of the transmit antenna locations and the moments of the receive antenna locations, motivating the development of a moment-based optimization algorithm. Functional analysis of the SPEB's moment structure proves that the optimal antenna distributions can be realized by a centro-symmetric structure, and shows that the worst-case target is located at the array broadside on the Rayleigh boundary. Moreover, by leveraging moment-based analysis and the Richter-Tchakaloff theorem, we derive closed-form antenna distributions with three points supported at the aperture center and two edges. Specifically, the optimal transmit antenna distribution can be exactly realized by activating only three transmit antennas while the optimal receive antenna distribution consists of three clusters. Numerical results show that the proposed closed-form design significantly outperforms conventional antenna placements and power allocation benchmark schemes.

We prove that there is a 1-generic set which computes a uniformly almost everywhere dominating function. The proof has two ingredients: a Baire space version of the Chong-Downey Sigma01-density criterion and the forcing of Cholak-Greenberg-Miller.

We study inference on the drift of a critical Galton–Watson process with immigration, a count time series with a unit root. Climate change motivates such nonstationary models for weather-related disaster counts. The drift is the expected increase in the count per period. Ordinary least squares is inconsistent for the drift, so we study state-weighted least squares, giving greater weight to observations following small counts, where conditional variance is lower. Under strict recurrence, we apply null-recurrent regenerative limit theory to obtain a polynomial convergence rate and a standard normal studentized limit. Our main contribution is the recurrence boundary, where we establish a logarithmic convergence rate and a parameter-free non-Gaussian studentized limit. Estimating the optimal weights yields the same first-order limiting distribution as knowing them. Simulations show lower root mean squared error than time-weighted least squares with weights $1/t$, and improved state-weighted confidence-interval coverage when the unit root is imposed.

Selective state space models (SSMs) have recently emerged as a compelling alternative to transformers, combining competitive performance with substantially improved inference efficiency. At each SSM layer, a sequence of hidden states are propagated by a recurrence, mixing information of different tokens. Despite using a different mechanism, this mixing plays a role analogous to attention in transformers. In fact, recent works have shown that the two architectures may be closer than they first appear, as this recurrence admits a formulation akin to linear attention. In transformers, attention is known to drive the tokens to cluster, i.e., to reach consensus, collapsing in the limit to a single direction. Thus, we ask: does the recurrence at the core of SSMs drive the tokens to consensus, as attention does in transformers? To answer this question, we take a dynamical systems perspective on SSMs, modeling the evolution of tokens across layers as an ordinary differential equation. By exploiting input-to-state stability arguments, we establish local exponential stability of the consensus equilibria and characterize their domain of attraction for time-varying weight matrices, a setting not addressed by previous results. We thereby show that the resemblance between SSMs and transformers does run deeper: the recurrence at the core of SSMs aggregates tokens just as attention does. Numerical experiments on a pretrained Mamba-2 model point to the output gate as the component that regulates the extent of this consensus, preventing the tokens from reaching it in full.

A set B is AEWG if almost every set computes a weakly 1-B-generic. Extending two results of Hirschfeldt, Jockusch and Schupp, we prove that there is an AEWG superhigh r.e. set and every low2 r.e. set is AEWG.

We present a solution of the limiting five-expert prediction problem with geometric stopping, based on stochastic calculus and partial differential equations. If $\delta$ is the stopping probability, the minimax expected regret from tied initial scores is $45\pi^2/(512\sqrt{2\delta})+o(\delta^{-1/2})$ as $\delta\downarrow0$. The adversarial rank control selecting the best and third-best experts maximizes the limiting Hamiltonian at every state. Following the four-expert construction of Bayraktar, Ekren, and Zhang (2020), we represent the value correction as a discounted boundary-local-time expectation for a degenerate obliquely reflected Brownian motion. The hyperbolic systems governing its boundary traces are derived and solved explicitly. To verify the nonlinear Hamilton–Jacobi–Bellman equation, we combine heat-equation minimum principles in equality directions, ODE minimum principles applied to convex combinations of multiple controls, and comparisons across controls. These arguments reduce the 48 regional control inequalities to a few lower-dimensional boundary problems, with the remaining signs established by explicit monotonicity arguments and by factorization of multivariate integral kernels that are positive polynomials in some of their variables. We prove global $C^2$ regularity across both regional interfaces and changes in rank ordering. We also show that the alternating COMB control only attains the limiting Hamiltonian maximum exactly when the two highest scores coincide and the third- and fourth-highest scores coincide.

The rapid expansion of data centers presents both economic opportunities and systemic risks for Caribbean Small Island Developing States~(SIDS). This paper develops a PyPSA-based capacity expansion framework for optimal co-location of data centers with renewable energy infrastructure, incorporating behind-the-meter battery storage, workload flexibility (70%/30% inflexible/flexible split), and multi-bus transmission topology. Two case studies are examined: Trinidad~&~Tobago (single-bus, 92\,MW Brechin Castle solar, 5\,MW data center) and Jamaica (4-bus 138\,kV JPS network, 15\,MW data center at Kingston). Jamaica's optimum deploys 405\,MW solar, 278\,MW wind, 800\,MWh grid-scale BESS, and 84\,MWh behind-the-meter BESS, achieving 42.5% renewable penetration; Trinidad's gas-anchored optimum builds no new RE, and a sensitivity analysis shows solar entering at gas costs above \$65--70/MWh, a carbon price of \$25–50/tCO$_2$, or solar CAPEX of \$800/kW. A post-hoc distributionally robust stress test evaluates hurricane resilience, and a novel 10-factor siting framework addresses the regulatory gap in SIDS that lack data-center-specific zoning. Results show that energy-optimal locations may score poorly on societal criteria—noise, freshwater stress, zoning readiness. This demonstrates the need for integrated planning tools in small island contexts.

We study a quadrature-surface free boundary problem driven by the Riemannian $p$-Laplacian on a smooth compact finite-dimensional Riemannian manifold. We formulate the problem as a shape optimization problem and develop an intrinsic admissible class based on a uniform Riemannian $RC$-$GNP$ condition~\cite{DS3}. We establish compactness of admissible domains and stability of the associated Dirichlet problems under strong $RC$-$GNP$ convergence. In particular, the states converge strongly in $W^{1,p}$ and the associated $p$-torsional energy is continuous. Under additional boundary regularity, we derive the first variation of the shape functional and obtain the free-boundary Euler–Lagrange condition \[ |\nabla_g u_\Omega|_g^p=\sigma H_{\partial\Omega}+k^p. \] We also formulate the contact optimality inequality, prove the Riemannian comparison of second fundamental forms at tangential contact, and obtain a sufficient condition on a reference domain which rules out contact and yields a solution of the free-boundary problem. Explicit radial examples on geodesic balls of the round sphere are included.

For an algebraically closed field $k$ of characteristic $p>0$, we study unramified $\mathbf{Z}_p$-towers $\{X_n\}_{n\ge 0}$ of smooth, projective and connected curves over $k$. Our main result provides an Iwasawa-theoretic description of the Newton polygons of their Jacobians, establishing that the slopes of the new part at level $n$ are eventually affine functions of $p^{-n}$ with rational coefficients.

We introduce a new single-loop algorithmic framework for smooth nonconvex–concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of $O(L^2D_Y\bar\Delta_0\varepsilon^{-3})$, where $L$ is the gradient Lipschitz constant, $D_Y$ bounds the diameter of the dual feasible set, and $\bar\Delta_0$ is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to $O(L^2D_Y\Delta_\phi\varepsilon^{-3})$, up to an additive lower-order cost, where $\Delta_\phi:=\phi(x_0)-\inf_x\phi(x)$. We further establish a lower bound of $\Omega(L^2D_Y\Delta_\phi\varepsilon^{-3})$ for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves $\mathcal{O}\!(L^{3/2}D_Y^{1/2}\Delta_\phi\varepsilon^{-5/2})$ gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves $O\!(\sqrt{\kappa}\,L\Delta_\phi\varepsilon^{-2})$ leading complexity for both stationarity criteria, where $\kappa=L/\mu$ is the dual condition number, up to an additive initialization cost. The $\varepsilon^{-2}$ accuracy dependence is optimal under fixed regularity and initialization bounds.

We investigate structural properties of the ideal-null sequence spaces c_{0,I} and c_{0,I}(X), and of the quotient l_infty/c_{0,I}, emphasizing the interaction between Banach space theory and combinatorial, topological, and measure-theoretic properties of the underlying ideal. We study classical problems related to c_0, including the dual space, compactness criteria, bounded and compact c_{0,I}-valued operators, Sobczyk-type extension phenomena, and complemented copies of c_0. We identify c_{0,I}^* isometrically with a space of bounded finitely additive measures on I and obtain a canonical atomic-singular decomposition, with l_1 as an isometrically complemented summand. A Dini principle for ideal convergence yields characterizations of relatively compact subsets of c_{0,I} and of compact operators with range in c_{0,I}. We prove a Sobczyk theorem for c_{0,I} and show that its separable-injectivity constant is exactly 2 for every proper ideal. We also construct, in ZFC, a class of statistical ideals for which c_{0,I} is not complemented in l_infty, and characterize complemented copies of c_0 in c_{0,I}(X) via the corresponding properties of c_{0,I} and X. Finally, we describe quotients associated with direct and Frolik sums of ideals, characterize Banach-lattice copies of c_0(kappa) in l_infty/c_{0,I} through I-almost disjoint families, identify ad(I) with the cellularity of the associated Stone space, and determine its behavior under Fubini products.

We consider a soft version of the spherical Sherrington-Kirkpatrick model, where the spherical constraint is replaced by a radial confinement term. We prove that, for a real symmetric disorder matrix under suitable spectral assumptions, the free energy exhibits two distinct fluctuation regimes. In the high-temperature regime, the fluctuations are asymptotically Gaussian. In the low-temperature regime, the fluctuations are governed by the largest eigenvalue and converge to the GOE Tracy-Widom distribution. These results show that the soft radial constraint preserves the fluctuation transition of the classical spherical Sherrington-Kirkpatrick model.

In this paper, we investigate the Fourier uncertainty principle through infinite banded matrices by way of a variational approach to minimizing the uncertainty product. Traditional methods for analyzing the Fourier uncertainty principle rely heavily on the Heisenberg–Weyl structure that underlie the Fourier transform and therefore do not generalize well beyond this setting. We provide three distinct approaches to developing the uncertainty principle based on the spectrum of a shifted quantum harmonic oscillator Hamiltonian, $H_{a,b}$. In the Hermite–Gauss basis, $H_{a,b}$ has an infinite, tridiagonal form which is central to the analysis. One approach exactly solves for the spectrum through analytic methods, another decomposes $H_{a,b}$ into an infinite sum of $2\times 2$ matrices, and the third approach is an algebraic approach to establishing the spectrum via a unitary equivalence with the quantum harmonic oscillator. The approaches taken herein provide alternate avenues that are more broadly applicable to other canonically conjugate operators for other integral transforms which maintain a tridiagonal form.

In this paper, we study the upper box dimensions, the packing dimensions and the quasisymmetric minimalities of some homogeneous Moran sets. We obtain the upper box dimension formula and the packing dimension formula for three classes of homogeneous Moran sets under some conditions. And two classes of them with packing dimension 1 are shown to be quasisymmetrically packing-minimal.

We reformulate average-cost (ergodic) control of a Markov jump process as a finite-horizon stochastic optimal transport (SOT) problem that jointly optimizes over the controlled evolution and the marginal law from which it starts and returns to (i.e., a self-transport). The constraint relating the marginal flow to the controlled generator is the basic adjoint relationship (BAR), so we call the resulting problem BAR-SOT. For any time horizon T>0, its optimal value is a constant scaling of the long-run average-cost rate. We give three equivalent formulations (through controlled processes, a Fokker–Planck constraint, and relaxed marginal measures) and show that the optimal dual is a stationary potential plus a term linear in time, with slope equal to the rate. A relative-entropy penalty on the control yields a cost-tilted Schrodinger bridge problem, computable by a Sinkhorn-type iteration when every transition rate is controlled, whose value converges to the unregularized optimum as the penalty vanishes. We develop the theory for finite Markov decision processes and then for general controlled Markov jump processes. A neural parametrization of the dual, trained as a physics-informed neural network (PINN), that encodes Harrison's equivalent-workload formulation (see Harrison 2000) matches the strong reinforcement-learning baseline of Dai and Gluzman (2022); numerically conditioning the fit of a sub-dominant transverse correction then improves on both that baseline and the best priority heuristic. On the input-queued switch, a graph-attention parametrization of the dual improves on the strongest matching heuristic we are aware of, with a parameter count that does not grow with the switch size.

We provide a unified description of a stable variational family of traveling vortex dipoles for the two-dimensional incompressible Euler equations, connecting the classical Lamb dipole to asymptotically radial dipoles in the large-impulse regime. We show that this family undergoes a sharp transition from the Lamb dipole, where the mass constraint is inactive, to non-explicit traveling waves for which the mass constraint becomes active, and determine the critical value exactly. We further prove that the variational solutions are unique up to translation both near the Lamb dipole and in the large-impulse regime. As a consequence, the corresponding individual traveling waves are orbitally stable. Our results provide a unified picture of these two distinct asymptotic regimes within a single variational family of traveling vortex dipoles.

We show that an autoregressive time series on Banach space can be canonically identified with a finite direct product of self-contained and self-determined natural components defined on separate subspaces by the spectral projections for each spectral point of the characteristic linear pencil. We assume the resolvent of the pencil is analytic everywhere except for the spectral points which may be poles or isolated essential singularities. Each natural component is represented as a forward, backward or outward flow fixed by an upstream boundary condition in the far distant past, the far distant future, or at a nominated initial time—depending on whether the corresponding spectral point is outside, inside, or on the unit circle.

We study the structure of random $d$-regular triangle-free graphs and show that a sharp phase transition occurs at $d=\frac{\sqrt 3}{2}\sqrt{n \log n}$. For smaller $d$, asymptotically almost surely the graph is non-bipartite, whereas for greater $d$, asymptotically almost surely the graph is bipartite.

We study decompositions of holomorphic isometries of complex hyperbolic space into holomorphic involutions. We prove that, for every n>=3, the involution length of PU(n,1) is 4. This improves the higher-dimensional upper bound 8 of Paupert–Will, as well as the projective upper bound 5 implied by Bünger's linear decomposition theorem, to the optimal value 4. Combined with the two-dimensional result of Paupert–Will, this shows that the involution length of PU(n,1) is 4 for every n>=2. The core of the proof is a three-involution decomposition theorem for square roots: every g in PU(n,1) has a square root of involution length at most 3, and this uniform bound is optimal. The key construction is carried out first on a two- or three-dimensional indefinite block and then completed in higher dimensions by a necessary and sufficient spectral pairing criterion on the positive-definite orthogonal complement. The four-factor lower bound is provided by complex reflections in a point. As a consequence, every element of PU(n,1) is a single commutator, and the two elements in the commutator representation can be simultaneously reversed by the same nontrivial holomorphic involution. This extends the two-dimensional single-commutator result of Paupert–Will to every n>=2 and thereby verifies Djoković's Conjecture A for the family PU(n,1), n>=2.

The Taylor diagram is used to evaluate and compare predictive models with observed data and has many applications in climate and environmental sciences. Three statistics of interest–the centered root-mean-squared error, standard deviation, and the product-moment correlation coefficient–are related by the law of cosines; Taylor diagrams leverage this fact to allow visualization of all three statistics in a two-dimensional plot without any loss of information. In this work, we present a novel model evaluation tool–the Wasserstein-Taylor diagram–formed from integrating the Wasserstein distance from optimal transport into a Taylor diagram framework. This tool is built upon the fact that the law of cosines still holds in the Taylor diagram if the centered root-mean-squared error and product-moment correlation coefficient are replaced by the centered 2-Wasserstein distance and quantile correlation coefficient, respectively. The advantage of this Wasserstein-Taylor diagram is that one can use the distribution perspective afforded by the Wasserstein distance to compare observed and predicted data sets of different sizes and even to compare statistical models with only observed data. We further show that the quantile correlation coefficient on empirical measures converges to the quantile correlation coefficient on the sampled measures.

Adaptive cross approximation (ACA) constructs low-rank CUR approximations from selected rows and columns of a matrix, making it attractive when individual entries are inexpensive to query but forming or repeatedly multiplying by the full matrix is not. Its practical performance, however, is not well explained by existing worst-case bounds, which can be exponentially large and often fail to predict when greedy largest-entry pivoting performs poorly. By viewing the matrix as a cross-gram matrix, we reinterpret the classical determinantal formula for the CUR residual through the lens of exterior algebra. In this representation, each residual entry is a weighted inner product between two $(k+1)$-blades divided by the normalized volume of the pivot block. The formula separates three effects: the weighted magnitudes of singular-vector rows, their angular geometry, and the conditioning of the selected pivots. From this we recover a classical $\sigma_{k+1}$-type estimate, characterize when one pivot annihilates or nearly annihilates additional rows and columns, obtain a quasi-optimality estimate for positive-semidefinite matrices, and bound the rank-one update through an explicit angle-based formula. We use this framework to explain the failure of greedy ACA on an asymmetric spiral-kernel example and to motivate a geometry-aware weighted-mass pivoting rule for radial-basis Galerkin stiffness matrices. In the reported experiments, weighted-mass pivoting often reduces the Frobenius residual relative to greedy pivoting and randomly pivoted Cholesky while retaining comparable runtime on parallel hardware.

The problem of identifiability in linear parametric models (LPMs) whose nodes follow either an ordered logit model or a regular one-parameter exponential family is evaluated. The results go beyond classical structural equation models as well as results for nodes with observations from a homogeneous family of distributions. The main result establishes that the orientation of every edge joining an ordinal node to an exponential-family node is identifiable from the joint distribution alone at every parameter value, provided the ordinal node has at least three categories and the exponential-family node at least three points of support, with no restriction on the sufficient statistic. Converses show that both requirements are necessary: the three-category requirement is binding only for affine sufficient statistics, and the three-point requirement is binding under the canonical link. The guarantee extends to orienting every such mixed ordinal-exponential family edge of a given $d$-node undirected skeleton. Numerical experiments illustrate the theoretical results by successfully separating orientations within a Markov equivalence class, which are indistinguishable by conditional independence alone.

In this paper, we study Renyi and Tsallis information entropies for a Hamiltonian system with position-dependent mass confined in harmonic oscillator potential. Gegenbauer polynomials are used to obtain the position eigenfunction of such a system, and the modified Bessel function of the second kind is used to determine the equivalent momentum eigenfunction. By means of probability densities of both representations, we analytically and numerically evaluate the Heisenberg-like uncertainty of this system. Because the Renyi and Tsallis information entropies in position representation are described by integral functionals of the Gegenbauer polynomials, these quantities are much more difficult to calculate. To get around this problem, we evaluate these information entropies at the system s asymptotical limit, which corresponds to the behavior of an undeformed harmonic oscillator. Nevertheless, no approximation method is used to obtain the Tsallis and Renyi information entropies in momentum representation. In both representations, we find that these information entropies approach the Shannon entropy when the entropic parameter alpha-> 1 and are closed to the results of similar models of the literature. Finally, we evaluate the related entropic uncertainty relations numerically to validate the latter obersevations.

This paper presents a study of dispersion for the two-dimensional Friedlander model with Neumann and Robin boundary conditions, thereby extending previous works treating only Dirichlet boundary conditions. By considering an Airy-type special function, we obtain a parametrix construction similar to the Dirichlet case but with additional terms requiring careful control. By studying the dispersion of this parametrix, we obtain the same dispersive estimates as in previous results for the Dirichlet case, and therefore the same Strichartz estimates.

Accurate prediction does not establish that a world model has recovered a physical law: distinct dynamics can generate identical records under the same observation protocol. We formulate law recovery on a fixed physical domain under an explicit catalog of experiments, sensor uncertainty, and an acquisition budget. A rate–distortion converse separates the information needed to describe a law from the information the apparatus can reveal. Its constructive counterpart gives a finite response codebook and an explicit decoding budget. On compact world classes, uniform recovery is possible exactly when every pair of different laws is experimentally distinguishable; equivalently, the apparatus can recover all the entropy of every finite law source. An inverse response modulus quantifies stability. For Lipschitz fields on a $d$-dimensional state–action domain, noisy full-state readouts after resets require minimax budget $\Theta(\varepsilon^{-(d+4)/2})$ for squared law error $\varepsilon$, compared with $\Theta(\varepsilon^{-(d+2)/2})$ for direct field observations. Exact crossing-time symmetries establish the lower bound under adaptive experiment selection and arbitrary durations with constant inputs. Reproducible synthetic cases illustrate the separate roles of intervention, calibration, and repeated measurement. Together, the results identify which evidence supports a claim of physical-law recovery and the cost of acquiring it.

The core-diagonal closure of Bothe and Druet identifies multicomponent Darken diffusion as a thermodynamically consistent Maxwell-Stefan model in which the constitutive data are species tracer diffusivities. Building on the Stieltjes-coordinate and De Giorgi regularity framework introduced in our recent preprint for additive Maxwell-Stefan frictions, we prove global maximal-$L^p$ strong well-posedness, without smallness assumptions, for the affine Darken class with composition-dependent scalar coefficients and uniformly positive tracer diffusivities. We show that this is exactly the maximal Darken class preserving the fixed Stieltjes eigenframe underlying the global argument. For general smooth, uniformly positive Darken tracer diffusivities we also derive the direct constrained flux form, exact mixing-entropy symmetrization, Fisher dissipation, composition-uniform normal ellipticity, and local strong well-posedness. Solutions in the affine class remain uniformly separated from the simplex boundary, become classical for positive times, and converge exponentially to equilibrium. Constant multiplicative frictions, including repeated factors, are covered by grouping species with equal factors.

Young's law on a rigid solid and the Neumann balance at a three-fluid junction describe different admissible motions. We connect them through an ordered continuum construction. An affine viscoelastic third phase first acquires permanent deformation memory in the no-relaxation limit at fixed modulus. Independently specified material-surface laws distinguish surface free energy from mechanical surface stress. On the deformable support, displacement of solid material and migration of the wet-dry partition give separate equilibrium conditions. Stiffening an anchored support suppresses displacement at a fixed macroscopic observation distance but leaves reversible migration possible. The surviving variation gives Young's law; the constrained mechanics remains as a reaction carrying normal and, in general, tangential capillary loads. Two complementary results substantiate this reduction. A clamped finite-strain model with circular liquid interfaces yields uniform convergence of reduced energies and recovery of the spatial Young angle for global almost minimisers. A separate quadratic half-space model gives convergence of stationary migration derivatives and a nonzero limiting traction distribution. The results explain how a local Neumann-like ridge can coexist with a Young angle measured against the distant wall, while distinguishing geometric convergence from the control of migration and reaction forces.

For the neutral periodic one-dimensional Riesz gas with pair potential locally $-|x|^a$, $0<a<1$, we prove a particle-displacement variance bound of order $\beta^{-1}$, uniformly in the number of particles. Log-concavity also gives exponential displacement tails. Every stationary periodic thermodynamic limit is simple, has intensity one, and admits a stationary ordered matching to the unit lattice with the same bounds. Each limit satisfies the canonical Gibbs equations for the full-line interaction, with an ordinary symmetric spatial principal value for the exterior potential. The matching implies uniformly bounded interval number variance and a positive limiting second moment of the reciprocal-lattice Fourier average at sufficiently low temperature. The main estimate compares the inverse random Hessian, through deterministic electrical flows, to a transient long-range network.

It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-homotopy characterizes exactly those matrices in the special linear group over rings of continuous or holomorphic functions that are finite products of involutions. We give an upper bound $I(n, d)$, depending only on the size $n$ of the matrices and the dimension $d$ of the parameter space, for the number of involutions needed in the factorization. Our upper bound is not optimal, however we show that for fixed $n$ the optimal upper bound tends to infinity as $d\to \infty$. In some cases we are able to determine the optimal bound.

Let $X$ be a three-dimensional real Banach space and let $f:S_X\to S_Y$ be a surjective isometry. We study the propagation of an affine formula for $f$ on a relatively open part of $S_X$. A finite family of antipodal distance coordinates propagates a linear anchor except at three explicitly described degeneracies: a facet, an open face-star, or a family of chord cones with a common cylindrical kernel. We prove that, if the norm has no fixed-direction cylindrical open cone, every affine open anchor is linear and global. This yields a source-geometric criterion for the Mazur–Ulam property and covers, among other non-strictly-convex examples, the Euclidean double cone. We also give a segment-saturation criterion which proves the property for every prism $Z\oplus_\infty\mathbb R$, where $Z$ is an arbitrary real Banach plane. Finally, in the remaining cylindrical regime, we prove that a chord-saturated same-kernel network cannot be confined to one proper projective quotient arc. We then show that this multi-arc difficulty and the transverse-ruled part of producing an initial affine anchor reach the same final obstruction: upgrading norm calibration on an ambient open cone, together with calibration on a few spherical segments, to pointwise calibration of the sphere map on a two-dimensional open patch.