We investigate the thermodynamics, photon sphere, and dynamical stability of an AdS Ayón--Beato--García (ABG) massive black hole with graviton mass and magnetic charge. The Gibbs free energy exhibits distinct limiting behaviors: it reduces to that of an AdS massive black hole when magnetic charge vanishes, to that of an AdS ABG black hole when graviton mass is zero, and smoothly interpolates to the AdS massive Reissner-Nordström case in the asymptotic regime. Furthermore, the photon sphere and shadow analysis indicate that increasing the graviton mass expands their radii, while increasing the magnetic charge causes contraction, in agreement with earlier studies of black hole spacetimes. Quasinormal mode (QNM) calculations further confirm dynamical stability, as the imaginary part remains negative, ensuring decay of perturbations. Additionally, the real part of the frequency decreases with graviton mass, while the imaginary part initially grows before saturating at higher values. Together, these results provide meaningful insights into the interplay between graviton mass, magnetic charge, and stability, thereby enriching the understanding of black holes in modified gravity theories.

Utility-scale quantum computing requires the integration and operation of a large-scale qubit register. Semiconductor spin qubits are a primary candidate for this, due to the prospects of building integrated hybrid quantum-classical architectures. However, scaling spin-qubit systems while preserving performance and control has remained a challenge. Here, we demonstrate the operation of an 18-qubit array in germanium based on an extendable 2xN architecture. We achieve simultaneous initialization, control, and readout across the entire array, enabled by parallel operation of modular unit cells. Across the array, we achieve average and median single-qubit gate fidelities of 99.8% and 99.9%, respectively. Finally, we characterize the nearest-neighbor exchange couplings throughout the device and implement high-quality controlled-Z gates to generate a three-qubit Greenberger-Horne-Zeilinger (GHZ) state. These results demonstrate that spin-qubit arrays can be scaled while maintaining high-fidelity operation and establish a modular, extendable architecture for planar semiconductor quantum processors.

We study edge-isoperimetric inequalities in chamber graphs of affine hyperplane arrangements. Our approach is topological: to a set of chambers we associate its thickening in Euclidean space and estimate its edge boundary through the induced stratification by intersections of arrangement hyperplanes. This yields general lower bounds for a broad class of sets. We show that a convex set of chambers of size $\sum_{i=0}^d \binom{k}{i}$, with $k\ge d-1$, has edge boundary at least $\sum_{i=0}^{d-1}\binom{k}{i}$, and we conjecture that convex sets minimize the edge boundary among all chamber sets of a fixed size. We verify this conjecture in dimension $2$. Our main result is a three-dimensional asymptotic inequality for arbitrary subsets of chambers: for arrangements in general position, every set $S$ occupying at most a fixed proportion of the chambers satisfies $|\partial S|=Ω(|S|^{2/3})$. As a consequence, for an arrangement of $n$ hyperplanes in general position in $\mathbb R^3$, the lazy simple random walk on the chamber graph has $\varepsilon$-mixing time $O(n^2\log(n/\varepsilon))$.

Future 6G networks will host massive numbers of embodied intelligent agents, which require real-time channel awareness over continuous-space for autonomous decision-making. By pre-obtaining location-specific channel state information (CSI), channel map can be served as a foundational world model for embodied intelligence to achieve wireless channel perception. However, acquiring CSI via measurements is costly, so in practice only sparse observations are available, leaving agents blind to channel conditions at unvisited locations. Meanwhile, purely model-driven channel maps can provide dense CSI but often yields unsatisfactory accuracy and robustness, while purely data-driven interpolation from sparse measurements is computationally prohibitive for real-time updates. To address these challenges, this paper proposes a data-model co-driven (DMcD) framework that performs a two-stage interpolation toward a space-time continuous channel map, First, a hybrid ray tracing and geometry-based channel model (H-RT/GBSM) is developed to capture dynamic scatterers, providing dense, time-variant channel properties that match measurement statistics as a physically consistent prior. Then, an inductive edge-conditioned graph neural network (InductE-GNN) fuses the prior with sparse measurements to perform real-time spatial interpolation, enabling rapid online adaptation without retraining, ensuring the synchronization with the dynamic physical reality. Evaluations with measured datasets show that the proposed DMcD framework significantly outperforms data-only and model-only baselines, providing accurate and queryable channel information for embodied intelligent agents.

We present Tsim, an open-source high-throughput simulator for universal noisy quantum circuits targeting quantum error correction. Tsim represents quantum circuits as ZX diagrams, where Pauli channels are modeled as parameterized vertices. Diagrams are simplified via parameterized ZX rules, and then compiled for vectorized sampling with GPU acceleration. After the one-time compilation, one can sample detector or measurement shots in linear time in the number of Clifford gates and exponentially only in the number of non-Clifford gates. Tsim implements the Stim API and fully supports the Stim circuit format, extending it with T and arbitrary single-qubit rotation instructions. For low-magic circuits, Tsim throughput can match the sampling performance of Stim.

We propose an active Cahn-Hilliard theory for the dynamics of a new type of phase transition where the driving force is not the direct interactions between the two separating components, but their active sorting by a third polar species. This third species can transport the other two along its polarity in opposite directions, thus separating them. Inspired by recent experiments where molecular motors that walk in opposite directions along microtubules are sorted into separated domains, our theoretical description of this process introduces a new mechanism for active phase separation and could serve as a model for the organization of biological material in space inside cells. We predict the formation of motor domains, and further show that they can either coarsen to form macroscopic phases or reach a finite micro- or mesoscopic steady state size, these latter due to an arrest of coarsening through activity.

In this paper, we develop a kernel-based policy iteration functional learning framework for computing team-optimal strategies in traffic coordination problems. We consider a multi-agent discrete-time linear system with a cost function that combines quadratic regulation terms and nonlinear safety penalties. Building on the Hilbert space formulation of offline receding-horizon policy iteration, we seek approximate solutions within a reproducing kernel Hilbert space, where the policy improvement step is implemented via a discrete Fréchet derivative. We further study the model-free receding-horizon scenario, where the system dynamics are estimated using recursive least squares, followed by updating the policy using rolling online data. The proposed method is tested in signal-free intersection scenarios via both model-based and model-free simulations and validated in SUMO.

Abelian flavor charges on right-handed fermions produce left-handed anarchy: we prove that all abelian discrete Froggatt-Nielsen models with uncharged left-handed doublets yield Haar-random PMNS and CKM matrices, regardless of $\mathbb{Z}_N$ group order, charge assignment, or Majorana mass structure. Scanning $\mathbb{Z}_3$ through $\mathbb{Z}_7$ with 12 charge assignments and $10^5$ Monte Carlo samples each, we demonstrate that the mass spectrum failure previously identified for $\mathbb{Z}_3$ -- the seesaw over-suppression mechanism that pushes $Δm^2_{21}/Δm^2_{31}$ to $\sim 10^{-11}$ -- is specific to $\mathbb{Z}_3$ and avoidable for $N \geq 4$. The mixing angle failure, however, is universal and irreducible. The PMNS angles from every abelian model are statistically consistent with Haar-random unitary matrices, with median $\sin^2θ_{12} \approx \sin^2θ_{23} \approx 0.50$ and $\sin^2θ_{13} \approx 0.31$ across all models tested. The same applies to the CKM: the joint probability of achieving CKM-like mixing from generic $O(1)$ coefficients is $< 2 \times 10^{-6}$. We identify the algebraic origin of this obstruction: abelian groups have only one-dimensional representations, so each generation transforms as an independent singlet with 18 free parameters for three Dirac mass matrices -- far exceeding the 10 physical observables. The transition to non-abelian flavor symmetries such as $A_4$, whose triplet representation reduces free parameters to 4 at leading order, is required specifically for mixing structure. This obstruction applies to the well-motivated subclass of models where left-handed fields are uncharged; models that assign abelian charges to both left- and right-handed fields can evade it.

Nanopore sequencing technology remains highly error-prone, making efficient error correction essential in DNA-based data storage. Prior work addressed high error rates using convolutional codes with their decoder coupled with the basecaller, but such approaches only accommodate a limited number of code classes and incur significant decoding complexity. To overcome these limitations, we propose two algorithms: PrimerSeeker, which efficiently detects primer sequences in raw nanopore sequencing reads, and SynDe, a decoder that operates on the same raw reads and supports any linear error correction code with a low-complexity graphical representation. PrimerSeeker provides primer location estimates close to those of existing approaches while being better suited for real-time primer detection during sequencing. SynDe performs well with convolutional codes augmented with periodic markers, often approaching or exceeding the performance of existing algorithms with a lower time complexity. Remarkably, the confidence scores produced by SynDe reliably identify which of its outputs should be discarded.

Secure network function computation is a critical research direction in network coding, which aims to ensure that the target function is correctly computed at the sink node while preventing the wiretapper from obtaining any information about the security function. In this paper, we focus on the general secure network function computation model, where the target function f and the security function ζ are arbitrary, and the wiretapper can eavesdrop on any subset of edges with size at most a given security level. Using information-theoretic techniques, we establish a nontrivial upper bound on the secure computing capacity, which is applicable to arbitrary networks, arbitrary target and security functions, and arbitrary security levels. This upper bound is shown to degenerate to the existing bounds in the literature when the target and security functions are specific forms. Furthermore, we consider two specific models: one where the target function is vector-linear and the security function is the identity function, and another where both functions are vector-linear. For the former, we derive a simplified form of the upper bound on the secure computing capacity via order-theoretic methods and propose an efficient algorithm to compute this bound with linear time complexity in the number of network edges. For the latter, we characterize the equivalent conditions for the computability and security of linear secure network codes, develop two constructive schemes for such codes, and derive an upper bound on the minimal finite field size required for the constructions, thereby obtaining a nontrivial lower bound on the secure computing capacity.

We establish the linear instability of the semiclassical Einstein-Klein-Gordon system linearised about the Minkowski vacuum spacetime. The proof relies on formulating a forcing problem for both metric and state perturbations within the space of past-compact sections. This geometric framework admits a unique tensor decomposition which, in conjunction with the quantum Møller operator, enables the decoupling of the linearised system into two distinct Cauchy problems. Consequently, the metric perturbations are shown to be governed by a higher-order, nonlocal hyperbolic partial differential equation. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this forcing problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalisation constants. We prove that the system exhibits a late-time linear instability: the metric perturbations grow exponentially, bounded strictly by a universal scale H, thereby indicating a quantum backreaction-driven transition toward a de Sitter cosmological spacetime. Provided the parameters governing the system are restricted to a physically relevant regime, this universal scale is compatible with the measured expansion of our universe.

We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.

It is well known that for $m\geq 2$ there are at most two non-equivalent $m$-knots with diffeomorphic exterior. Such pair of knots will be called $\textit{ non-reflexive knot pair}$. A classical problem in topology is to determine all dimensions where such knot pairs exist. In 1976 Cappell and Shaneson gave a method of constructing non-reflexive knot pairs. In the present paper we construct an infinite family of new examples of Cappell-Shaneson knot pairs, and give examples of Cappell-Shaneson knot pairs that have the same Alexander polynomial but are inequivalent.

Spiking neural networks (SNNs) support energy-efficient machine intelligence because event-driven computation and sparse activity map naturally to low-power digital hardware. In practical implementations, however, membrane states, synaptic weights, and thresholds are represented with finite-precision integer arithmetic. Quantization, clipping, and overflow can therefore alter network dynamics, not just approximate a higher-precision model. This paper adopts an integer-state dynamical perspective, modeling a hardware-oriented SNN as a deterministic map on a bounded integer lattice. Under this view, recurrence, periodic orbits, and regime changes become intrinsic properties of the system. We introduce a lightweight update rule with integer-valued states and shift-based leakage, and demonstrate the approach through exploratory simulations with network sizes N = 30-130, connection densities 0.1-0.9, and bit widths 4/8/16 over T = 1000 steps. The results show bounded and recurrent temporal structure with strong quantization sensitivity. The observed regimes depend heavily on representation semantics and scaling choices. These findings suggest that numerical precision acts as a dynamical design variable and highlight integer-state analysis as a useful framework for hardware-aware SNN co-design, motivating future work on attractor analysis, precision-aware training, and FPGA/ASIC validation.

We study the complexity of inverse cellular automata on configurations of bounded size. Deciding injectivity in this setting is co-NP-complete by a theorem of Durand. We give a simpler proof of this theorem by a direct reduction from UNSAT to this problem, avoiding more complicated intermediate constructions. We also show that one direction of the reduction can be formalized in the weak theory of bounded arithmetic $V^0$. Durand's coNP-completeness result allows one to view inverse cellular automata acting on bounded size configurations as propositional proofs, cf. Cavagnetto, and we prove lower bounds on their size. The proof uses known lower bounds for bounded-depth Frege systems together with the Paris--Wilkie translation of arithmetic proofs into propositional proofs, which allows us to transfer proof complexity lower bounds to our setting.

We identify universal signatures in the bispectrum arising from a transient tachyonic instability of entropic fluctuations during inflation, a phenomenon that naturally arises in hyperbolic field-space geometries. We perform exact numerical calculations directly at the level of fluctuations, without relying on a specific background model, and distinguish two cases. In the light case, with masses around the Hubble scale, our results provide the first-ever computation of the bispectrum due to such tachyonic instabilities. We find a universal magnification of the folded configuration, together with the known non-analytic scaling in the squeezed limit. As an illustrative example, we compute and analyse the bispectrum in angular inflation, demonstrating compatibility with current limits. In the heavy case, with masses well above the Hubble scale, the bispectrum exhibits a distinctive correlation between enhanced folded configurations and a `tachyonic resonance' in mildly squeezed limits, with the resonance scale set by the strength of the instability. While the main qualitative features are reproduced, we show that there exists no UV matching for which a single-field effective description, obtained by integrating out the entropic modes, accurately captures the bispectrum for all kinematic configurations. To facilitate observational applications, we introduce simple bispectrum shape templates suitable for current and forthcoming cosmological surveys. Our model-independent results allow for constraining non-standard inflationary attractors characterised by strongly non-geodesic motion.

We formulate and implement a microscopic framework to derive an optical potential from the solution to an effective Hamiltonian and use it to calculate neutron scattering cross sections for the deformed nuclei $^{24}$Mg, $^{48}$Cr and $^{50}$Cr. This approach is based on a symmetry-restored multi-excitation generator coordinate method (GCM), enabling the consistent treatment of both nuclear structure and reaction observables. Through this method, non-local optical potentials corresponding to a Hamiltonian can potentially be constructed for any nucleus in the whole nuclide chart. We use this to perform reaction calculations employing quadrupole deformed triaxial configurations, obtaining results for $A\approx 50$ chromium isotopes, and study the properties of the calculated non-local optical potentials. This work further advances the unified treatment of structure and reaction, within a framework that exploits the intrinsic symmetries of nuclei.

The coronal magnetic field plays a fundamental role in governing coronal activities, driving space-weather events, and shaping the heliosphere. Due to a lack of direct observations, extrapolation models such as the Potential Field Source Surface (PFSS) model become the primary method to obtain the three-dimensional magnetic field distribution in the corona. However, the PFSS model cannot solve the long-standing open-flux problem, in which the extrapolated open magnetic flux is significantly lower than that inferred from in-situ measurements. To address this issue, we develop an innovative Non-Spherical Potential Field (NSPF) model. The model introduces a Non-Spherical Source Surface (NSSS) defined as an isosurface of the total magnetic field. The NSSS naturally forms concave structures beneath external current sheets, enabling the model to generate substantially more open magnetic flux while yielding a physically plausible distribution of open field regions. As a result, the NSPF model successfully reproduces complex coronal magnetic topologies, interplanetary magnetic field properties, and solar wind source mappings. Our refined coronal magnetic model provides a proper foundation for future research on solar and heliospheric magnetic coupling.

The recent discovery of high-temperature superconductivity in both bulk and thin-film bilayer nickelates La$_3$Ni$_2$O$_7$ has garnered significant attention. However, the corresponding pairing symmetry remains debated in both experiments and theoretical studies due to conflicting experimental evidence from bulk and thin-film materials. In this work, we examine the electronic Raman response across different channels for various pairing symmetries within a two-orbital bilayer model. By comparing Raman susceptibilities obtained from multiorbital and band-additive approaches, we demonstrate that Raman response can distinguish between different pairing symmetries and identify pocket-dependent gap amplitudes for both fully gapped and nodal superconducting states. Specifically, the nodal $d_{x^2-y^2}/d_{xy}$-wave pairing exhibits robust low-energy power-law behavior, distinct from a fully gapped pairing. Additionally, for the $s_{\pm}$-wave pairing, the detailed gap anisotropy on the $β$ pocket can be determined. Possible experimental implications are also discussed. Our results highlight the crucial role of multiorbital effects in shaping the Raman spectra and establish electronic Raman scattering as a powerful and symmetry-resolved probe for determining the superconducting gap in unconventional superconductors.

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes are available that achieve high accuracy while preserving key qualitative properties of the underlying dynamical system, and are successfully used across a broad range of fields. In this work, we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. Two complementary types of error estimates are derived. The first is expressed in terms of operator norms, while the second is formulated using norms of commutators and can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved. The theoretical results are illustrated by deriving explicit error bounds for some representative schemes.

To achieve net-zero emissions, cities must transition away from reliance on private vehicles. However, car-centric urban growth has transformed the automobile from a convenience tool into a necessity for accessing essential services, creating significant "car dependency". This study introduces a novel Car Dependency Index (CDI) that quantifies the accessibility gap between private and public transport across 18 cities in Europe and North America. Utilising high-resolution geospatial data and numerical simulations, we reveal pronounced spatial inequalities, showing that car dependency remains a primary driver of car ownership even when accounting for income. A ``what-if" simulation of the planned metro expansion in Rome predicts a reduction of approximately 60,000 commuting vehicles, yet highlights that isolated interventions have localised impacts. We conclude that systemic, network-level transit expansions are essential to dismantle car-based systems and foster equitable, sustainable urban mobility. Our framework provides policymakers with an objective, scalable tool to identify viable areas for car-free zones and target infrastructure investments effectively.

Boolean networks are dynamical models of disease development in which the activation levels of genes are represented by binary variables. Given a Boolean network, controls represent mutations or medical treatments that fix the activation levels of selected genes so that all states in every attractor (i.e., long-term recurrent states) satisfy a desired phenotype. Our goal is to enumerate all minimal controls, identifying critical gene subsets in disease development and therapy. This problem has an inherent bilevel integer programming structure and is computationally challenging. We propose an infeasibility-based Benders decomposition, a logic-based Benders framework for bilevel integer programs with multiple subproblems. In our application, each subproblem finds a forbidden attractor of a given length and yields a problem-specific feasibility cut. We also propose an auxiliary IP called subspace separation that finds a Boolean subspace that includes multiple forbidden attractors and thereby strengthens the cut. Numerical experiments show that the resulting algorithms are much more scalable than state-of-the-art methods and that subspace separation substantially improves performance.

We construct toral Chern-Simons theory with gauge group $\mathbb T=\mathfrak t/Λ\cong U(1)^n$ from an even, integral, nondegenerate symmetric bilinear form $K:Λ\timesΛ\to\mathbb Z$ by geometric quantization via real polarization. We obtain a unitary extended $(2+1)$-dimensional TQFT by constructing the boundary state spaces and canonical operators and proving that they satisfy the cylinder and gluing axioms. The finite discriminant group $G_K=Λ^*/KΛ$ arises naturally in the theory and controls the genus-$g$ state spaces. At genus one, the theory recovers the finite quadratic data underlying bosonic Abelian topological order.

We report the experimental observation of magneto-chiral anisotropy in the longitudinal and transverse ultrasound propagation in $α$-quartz. To perform such measurement, we have built an ultrasound spectrometer with unprecedented experimental resolution of the order of $Δv/v \sim 10^{-8}$. We present a simple macroscopic Becquerel-like analytical model that accounts for the magnitude of the observed effect and its frequency dependence.

We prove that any algorithm computing the sum-exclude-self of an unsigned $d$-bit integer array of length $n$ under sublinear space must perform two linear passes over the input. More precisely, the algorithm must read at least $n-1$ input elements before any output cell receives its final value, and at least $n - \lfloor t/d \rfloor$ additional elements thereafter, where $t = o(nd)$ bits is the working memory size. This gives a total of $2n - 1 - \lfloor t/d \rfloor$ element reads. A trivial modification of the standard two-pass algorithm achieves this bound exactly for all practical input sizes. The proof uses this toy problem as a worked example to demonstrate the choke-point technique for proving sublinear-space lower bounds.

We present a precise and systematic analysis of $D \to V V$ decays within the factorization-assisted topological-amplitude (FAT) approach, where $D$ denotes the set $\{D^0, \, D^+,\, D^+_s\}$ and $V$ represents the vector mesons $ρ, K^*, ω$, and $ϕ$. Given the limited current experimental data, the FAT approach serves as a available phenomenological framework for predicting charmed meson decays to both vector mesons. In this framework, incorporating flavor SU(3) symmetry breaking effects, we can express nonfactorizable contributions of different modes as a minimal set of universal parameters globally fitted to experimental data. Utilizing 36 experimental data points for $D \to VV$ decays, we precisely extract 10 nonfactorizable parameters associated with the $C$ and $E$ topological diagrams with $χ^2/\mathrm{d.o.f.}=8.43$. We find that a large strong phase in the longitude $E$ amplitude cause strong destructive interference with the $C$ longitudinal component, yielding $f_\parallel >f_L $, contrary to the naive factorization predictions. Additionally, for modes processing exclusively by the $E$ diagram, the amplitude hierarchy $|S|<|D|$ leads to a $D$-wave branching fraction larger than that of the $S$-wave. This explains recent observations that contradict $S$-wave dominance predictions. The predicted branching fractions and polarizations for 28 decay modes are consistent with existing experimental data. Unobserved modes, especially those with branching fractions of order $10^{-3}\sim10^{-2}$, the $D$-wave dominated modes, and modes exhibiting $f_\parallel >f_L $, await measurement by BESIII, STCF, Belle II and LHCb.

We present a new algorithm for finding an $ε$-approximate fixed point of an $\ell_\infty$-contracting function $f : [0, 1]^d \rightarrow [0, 1]^d$. Our algorithm is based on the query-efficient algorithm by Chen, Li, and Yannakakis (STOC 2024), but comes with an improved upper bound of $(\log \frac{1}ε)^{\mathcal{O}(d \log d)}$ on the overall runtime (while still being query-efficient). By combining this with a recent decomposition theorem for $\ell_\infty$-contracting functions, we then describe a second algorithm that finds an $ε$-approximate fixed point in $(\log \frac{1}ε)^{\mathcal{O}(\sqrt{d} \log d)}$ queries and time. The key observation here is that decomposition theorems such as the one for $\ell_\infty$-contracting maps often allow a trade-off: If an algorithm's runtime is worse than its query complexity in terms of the dependency on the dimension $d$, then we can improve the runtime at the expense of weakening the query upper bound. By well-known reductions, our results imply a faster algorithm for $ε$-approximately solving Shapley stochastic games.

We study excited solutions in a Skyrme--Chern-Simons theory in $2+1$ dimensions. In particular, we emphasize the necessity of using a Lagrange multiplier method to obtain excited solutions, due to the appearance of a discontinuity when using a constraint compliant parametrization. These solutions are characterized by an integer number $p$, excited solutions corresponding to $p\neq 0$. The dependence of the global charges on the parameters is analyzed, showing non-standard behaviors. We also find that the presence of the Skyrme--Chern-Simons term does not alter significantly the pattern of energy levels, so $p=0$ solutions (fundamental solutions) have always the minimal energy.

We apply Colombeau-type regularization to the electromagnetic field of a point-charge and show how the Liénard-Wiechert potential can be derived from a generalized function based on the geometry of Minkowski space. Furthermore, for a charged particle in its rest frame, we discuss the electric monopole, magnetic dipole, electron singularity, and self-energy.

Recently, purpose-built analog hardware that can efficiently minimize the Ising energy and thereby solve a variety of combinatorial optimization problems has been receiving widespread attention. In this work, we show how multidimensional, vectorial degrees of freedom, that are either naturally present or can be artificially created in such hardware, could strengthen the capability to find optimal solutions to optimization problems. In order to achieve this, we introduce a simple model of soft vector spins that should be implementable on a variety of analog hardware platforms as well as three different dimensional annealing methods which harness the enlarged phase space of the vectorial degrees of freedom to minimize the Ising energy. We perform simulations on different benchmark problems and show that for all dimensional annealing methods we tested, vectorial degrees of freedom improve upon one-dimensional degrees of freedom when it comes to finding the ground state of the Ising model. In particular, we find that this advantage becomes most pronounced for $d \gtrsim 3$ dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.