Papers

Replica Symmetry Breaking (RSB) theory for spin glasses, which is exact in infinite dimensions, provides detailed predictions for the behavior of these systems. One of these predictions is that vanishing magnetic fields induce chaotic behavior. Here, vanishing has a precise meaning: we refer to an externally applied magnetic field with intensity scaling as $1/\sqrt{N}$, where $N$ is the number of spins in the system. RSB predicts that this chaotic behavior depends only on the probability density function of the overlap, $P(q)$, as computed at zero field. We proceed to test the RSB prediction in the 3D Edwards-Anderson model, by first computing the $P(q)$ numerically in the absence of a magnetic field. We then use this function as input to analytically compute the statistics of the states in the regime of a vanishing magnetic field, using a representation based on the Bolthausen-Sznitman coalescent process (which is exact for the mean-field Sherrington-Kirkpatrick model). This mixed numerical-analytical approach allows us to determine the prediction of RSB for the overlap distribution function measured in the presence of a small vanishing field, as well as the increase in chaoticity as the field strength grows. The above RSB predictions are quantitatively compared with the outcome of numerical simulations of the three-dimensional Edwards-Anderson in the presence of a small vanishing field, finding very good agreement. This agreement provides strong support for the idea that three-dimensional spin glass at zero magnetic field and low temperature is well described by RSB theory.

Echo State Networks (ESNs) are fundamentally important from the conceptual perspective of AI, because they provide simple and tractable frameworks for understanding temporal information processing, memory, and learning in dynamical systems. They are composed by an interconnected reservoir whose connectivity matrix remains constant over time and only the output weights are updated during the training period. In particular, ESNs employing online training allow the incremental adaptation of the output weights as data is received, making them particularly suitable for applications in which new input data becomes available over time (non-stationary environments). A key objective in the study of such learning structures is to determine which statistical properties of the reservoir enhance computational efficiency and maximize prediction performance. To this end, we investigate the effects of reservoir topology on learning performance, considering weight matrices generated from Erdos-Renyi random networks and Watts & Strogatz small-world networks. As test data we use the Lorenz and Rossler models in the chaotic regime and the FitzHugh-Nagumo model in the limit cycle regime. In the case of random Erdos-Renyi reservoirs, our simulations show that best performances are obtained for sparse reservoirs and especially when the link-density is near and immediately above the critical point of mean-field bond percolation. Watts & Strogatz reservoirs are shown to demonstrate best performance also for small values of link-density, compatible with the mean-field bond percolation when taking into account the specific nonlocal ring structure of the Watts & Strogatz reservoirs. These studies on the influence of the reservoir architecture and hyperparameters on enhancing the accuracy of ESN predictions provide an efficient framework for real-time forecasting in dynamically evolving systems.

We investigate the role of different correlation functions in reconstructing the phase diagram of the spin-1 bilinear-biquadratic chain, parametrized by the angle $\theta$ between the bilinear and biquadratic couplings, using our previously developed label-free machine-learning pipeline \cite{Ming21, Ming23}, in which an autoencoder compresses iDMRG correlation matrices into latent features and K-means clustering with silhouette analysis groups them into phases. Without any prior labels, the scheme reveals four natural phases, the Haldane, dimerized, critical trimerized, and ferromagnetic phases, and accurately locates the Takhtajan-Babujian ($\theta=-\pi/4$), $SU(3)$-symmetric Uimin-Lai-Sutherland ($\theta=\pi/4$), and Ferro-Dimer ($\theta=5\pi/4$) transition points; a shallow supervised refinement further sharpens the phase boundaries. We find that different correlation functions play fundamentally different roles in phase identification. The $SU(3)$-symmetric correlation alone selects a four-cluster partition that can be associated with the four phases, whereas the spin-spin and dimer-dimer correlations must be combined to recover the complete phase diagram. In contrast, the trimer-trimer correlation behaves as a marginal correlation: it is insufficient for phase identification by itself but improves the precision of the predicted phase boundaries when combined with the other correlations. Furthermore, the latent representations and silhouette values reveal signatures that we relate to the $\theta$ dependence of the correlations within each phase and, tentatively, to the underlying renormalization-group structure. These results demonstrate that machine learning can not only identify quantum phases but also uncover the physical information encoded in different correlation functions.

Recovering the interaction structure of complex stochastic dynamical systems from sparse time-series observations is an inherently ill-posed inverse problem. The posterior over network configurations is high-dimensional, highly degenerate, and shaped by the non-equilibrium nature of the available data. To address this, we propose a fully Bayesian, ensemble-based inference framework. Our approach grounds a continuous-time Stochastic Differential Equation (SDE) likelihood formulation—acting mathematically as an Onsager-Machlup quasi-potential for non-equilibrium paths—within a non-parametric Minimum Description Length (MDL) structural prior. This framework rigorously penalizes model complexity and avoids kinetic overfitting by forcing continuous parameters into a finite vocabulary of discrete categories. The joint posterior is optimized using a block-coordinate scheme combining sub-quadratic Markov Chain Monte Carlo (MCMC) topological sampling with deterministic Nelder-Mead parameter optimization. Using highly non-linear macroscopic population dynamics as a benchmark, we demonstrate that inference requires non-stationary transient data to resolve causal information. Furthermore, we prove that propagating structural uncertainty via ensemble-based posterior analysis produces significantly more robust and reliable predictions for downstream perturbation outcomes than traditional single-point (MAP) estimates. The framework is highly generalizable, capturing fundamental topological constraints even under severe generative model mismatches, offering a robust tool for uncertainty-aware inference in complex stochastic systems.

Medium-range order (MRO) is an important feature of amorphous materials, but its nature in terms of atomic arrangement has remained elusive. Here, using large-scale machine-learning-driven atomistic simulations of amorphous silicon, we probe the MRO of this archetypal network glass. By characterizing the structure in three dimensions, we reveal that the symmetry of the atomic arrangement around a central atom shows a crossover as a function of distance. At short distances, the structure is determined by local bonding constraints, which evolve from predominantly tetrahedral environments in low-density amorphous silicon to highly coordinated and strongly disordered local environments in very-high-density amorphous silicon. At larger distances, however, distinct cubic- and octahedral-symmetry patterns emerge in the three-dimensional atomic-density distribution, despite the absence of these symmetries at shorter length scales. We find that this crossover is accompanied by a quasi-periodic spatial organization of the topological rings in the structure. The characteristic spacing between ring centers matches the length scale of the first sharp diffraction peak (FSDP) in the structure factor, establishing a direct connection between this reciprocal-space signature of MRO and its real-space organization. These results demonstrate that the FSDP, the spatial correlation of the rings, and emergent orientational symmetries are not independent manifestations of MRO, but reflect different aspects of a common structural organization. Our findings provide a unified real-space picture of MRO in amorphous silicon and suggest that intermediate-range structural order in network glasses emerges from the interplay between local bonding constraints and extended atomic arrangements driven by efficient packing.

Aging and relaxation of glasses proceed through localized rearrangements that also generate long-ranged mechanical responses in their surroundings. We develop a theory that treats these rearrangements as internal microseismic sources, and thereby separates the direct effects emerging from an event from the effects from the far-field displacements it generates. These far-field disturbances propagate and relax through the viscoelastic medium through which they move, producing a stochastic background of weak but numerous perturbations at any given location. Combining the description of local events from the Random First Order Transition theory with this propagating background, we derive a unified equation for the dynamical structure factor of a relaxing glass. The theory identifies a finite wave-vector and time window in which the growing viscoelastic far-field disturbances can produce compressed exponential relaxation, which has been observed using X-ray photon correlation spectroscopy. At higher wave vectors that resolve local jumps, heterogeneity of the local event dynamics instead produces stretched exponential relaxation. The crossover between these behaviors depends on the event statistics and the scattering phase accumulated across the region reached through viscoelastic propagation from the numerous distant sources.

Mandlebrot's coastline paradox reframes measurement as a joint property of an object and its ruler. Here we ask how observational resolution shapes the apparent structure of neural computation. We introduce a scale-indexed decomposition that separates the predictive information captured by direct dependencies from the residual outside that model family. Exact constructions show that projection can convert higher-order dependence into pairwise structure. Projection can increase the direct fraction of surviving information even as total predictive information decreases. A schematic branch-to-soma cascade grounds these effects in local coincidence detection, saturating integration and somatic mixing. Resolving organized nonlinear subunits improves approximation and raises effective representational rank. In a nested hierarchy, these gains occur without increasing information about the target. This distinction separates information availability from access by a restricted readout. A point-neuron or direct population model therefore establishes an effective description at a specified scale. It does not uniquely identify the finer computation that generated the observed relation. Which resolution is adequate depends on the observable and perturbation of interest.

Non-Abelian anyon chains whose interactions favor a single fusion channel host quantum critical ground states described by minimal-model conformal field theories (CFTs) protected by a topological symmetry. We show that their measurement-only counterpart, where non-commuting fusion charge measurements act locally on the anyons of a chain of $L$ sites, provides a fundamental example of a non-unitary CFT controlling a non-Abelian learning problem. Mapping the monitored dynamics to a random tensor network, we numerically find $1/L$ scaling Lyapunov gaps, logarithmic entanglement scaling with the chord length, and an effective central charge in the Shannon entropy of the measurement record. The conserved topological charge sharpens with dynamical exponent $z= 1$ and is learnable by a decoder on an $O(L)$ time scale. As the topological symmetry is a non-invertible symmetry of the $SU(2)_k$ anyon theories, we establish the monitored golden chain $(k=3)$ as a non-Abelian generalization of the weakly (Kramers-Wannier) self-dual quantum critical point of the toric code $(k=2)$, and the first of a family for $k\geq 3$.

Quantum simulators with disordered spin Hamiltonians offer a natural route to Haar-like unitary ensembles, but face a competition. Disorder enhances randomness, yet strong disorder suppresses dynamics and localizes the system in the computational basis. This suggests an optimal intermediate disorder strength for approximating a unitary $k$-design, which has so far been identified mainly empirically. Here, we develop a perturbative theory for the moment gap $\Delta^{(k)}$, the deviation of the $k$-th moment operator from its Haar value, for the full spectrum of the interacting XXZ chain in a fixed-magnetization sector. We find the non-monotonic asymptotic scalings $\ln \Delta^{(k)}\sim -\sigma^2$ at weak disorder and $\ln \Delta^{(k)}\sim -1/\sigma$ at strong disorder, whose crossover sets an optimal disorder strength $\sigma_c\sim J_{xy}$, where $J_{xy}$ is the energy scale of the non-random interaction. We further show that a multi-quench protocol further suppresses the gap exponentially in the number of quenches. Numerical simulations confirm both scalings and this exponential suppression, and we demonstrate the usefulness of the protocol for classical shadow tomography. This work suggests that strong disorder may block both thermalization and the formation of unitary designs through a common localization mechanism.

The valence-bond-glass state of the 4d^1 double perovskite Ba2YMoO6 combines Mo^5+ singlets with residual paramagnetic "orphan" spins whose coupling is hard to reach with bulk probes. We probe them by electron paramagnetic resonance of a dilute Gd^3+ ion on the Y^3+ site via broadband coplanar-waveguide transmission at 5 K. The isostructural nonmagnetic niobate Ba2YNbO6, doped with the same Gd^3+ probe, is a fixed-lattice control. The Gd^3+ linewidth nearly doubles in the magnetic host. Dipolar coupling to the sparse orphans is nearly an order of magnitude too weak; the excess is instead reproduced by a transferred f-d superexchange of J ~ 0.2 K, so the width is a static measure of orphan-spin coupling. The half-field (Delta m_S = +/-2) satellite collapses in the magnetic host, a dynamical dephasing channel, while the unchanged g-factor marks a zero-mean random internal field. Together the two channels probe the coupling and dynamics of valence-bond-glass orphan spins.

Rare events, such as crystal nucleation in mildly supercooled liquids, are characterized by waiting times that can be much longer than the timescale of the underlying microscopic dynamics. The mean first-passage time (MFPT) formalism [J. Wedekind and D. Reguera, J. Phys. Chem. B 112, 11060 (2008)] provides a route to reconstruct one-dimensional free-energy profiles from the MFPT and steady-state distribution measured along a reaction coordinate, but conventional implementations become inefficient for sufficiently high barriers because spontaneous crossings are rarely observed. We introduce a "top-start" extension that initiates trajectories near the barrier top and reconstructs the free-energy profile from their subsequent evolution away from the barrier toward lower-free-energy regions. We establish the equivalence between the top-start and two-region constructions using a Markov-chain transition-matrix formulation. We validate the method using one-dimensional random-walk simulations across low, medium, and high free-energy barriers, with transition-matrix calculations providing an exact benchmark for the corresponding discretized models. The top-start reconstruction remains accurate in the high-barrier regime where conventional MFPT sampling becomes prohibitively inefficient.

We identify exact altermagnetic ground states of a classical Heisenberg model on a quasicrystal. Considering the Ammann–Beenker tiling as an example, we determine the ground states by exploiting the parity structure of its parent four-dimensional hypercubic lattice. The ground state hosts coplanar altermagnetic order characterized by the preserved spin-group symmetry $[C_{2x}\parallel C_{8z}]$. Without translational symmetry, the momentum-space spin splitting that usually identifies altermagnets is not directly accessible. We instead couple the spins to itinerant electrons and show spin-resolved transport with $g$-wave symmetry, confirming the altermagnetic nature in a nonperiodic system. Our results establish a geometric route to altermagnetic order in quasicrystals, connecting exact magnetic ground states to their spin-resolved transport signatures.

We identify the limit free energy of mean-field multi-species spin glasses with +/-1 spins, allowing for the presence of external fields, and without assuming any convexity property of the covariance function of the Hamiltonian. The limit is given by an "uninverted" variational formula over Brownian martingales. The proof consists in first identifying the limit free energy of configurations with a fixed magnetization. Once the magnetization is fixed, we can show the convergence of the enriched free energy to the solution of a Hamilton-Jacobi equation. The magnetization constraint is also used to show the convexity of the initial condition in this equation, and this yields a variational representation of the solution, as a Hopf formula. Convex duality then turns this formula into an optimization over martingales whose initial values are the prescribed magnetizations.

We study the contributions of half-BPS surface defects to supersymmetric Casimir energy and the universal logarithmic term of supersymmetric Rényi entropy in four- and six-dimensional SCFTs. Using equivariant integration of defect anomaly polynomials and its extension to supersymmetric Rényi backgrounds, we obtain closed expressions for these observables in terms of defect 't Hooft anomalies. For Gukov–Witten defects preserving transverse rotations in four-dimensional $\mathcal{N} = 4$ super Yang–Mills theories, both observables depend on a single R-symmetry anomaly. For $\mathcal{N} = (0, 4)$ defects in six-dimensional $\mathcal{N} = (1, 0)$ theories, additional terms arise from the gravitational anomaly. We provide complementary checks using chiral algebras, five-dimensional localization, and holographic probe branes at large $N$, while localized chiral fermions reproduce the gravitational-anomaly contributions. The framework also recovers Wilson-surface results in six-dimensional $\mathcal{N} = (2, 0)$ theories.

We prove that the simulated annealing algorithm efficiently samples from the Sherrington-Kirkpatrick model at inverse temperature $\beta < 1$ to an arbitrary inverse polynomial accuracy in total variation.

We revisit the statistical mechanics of Transformer attention via modern Hopfield models. We present a free-energy formalism to characterize parameter regimes and critical thresholds for robust attention inference. The resulting free-energy landscape incorporates randomness in query states and attention weights. Attention then emerges as the equation of motion of the corresponding statistical-mechanical system (i.e., modern Hopfield model) through the standard free-energy minimization principle. This allows us to identify first-order phase transitions in attention between glassy and robust-inference phases. We further extend to attentions with stochastic queries and keys. Together, our results provide a rigorous statistical-mechanics framework for characterizing phase structure and inference reliability across a family of Transformer attention mechanisms.

Every 3-regular graph of girth at least $g$ has a cut containing at least a fraction $M_g$ of its edges, and the best rigorous bounds at large girth are $0.912\le \lim_{g\to\infty}M_g \le 0.92386$. Lower bounds on $M_g$ follow from the performance of local algorithms: on a graph of girth $g \ge 2p+2$ a depth-$p$ local algorithm sees the same tree around every edge, so its expected cut fraction on that tree bounds $M_g$ from below. We apply this strategy to the Variational Iterative Rotation Algorithm (VIRAL), a classical local algorithm, and evaluate it on the depth-$p$ tree for every even depth up to $p=32$. The resulting bounds exceed those recently obtained from the Quantum Approximate Optimization Algorithm (QAOA) for every $p\ge 4$, by $0.005$ to $0.011$, and they extend the range of girths covered from $g=36$ to $g=66$. Notably, at $g=66$ we obtain $M_{66} \ge 0.9127$ at $99.99999\%$ confidence level, which raises the asymptotic lower bound from $0.912$ to $0.9127$.

Active gels are intrinsically non-equilibrium materials whose structural and dynamical heterogeneity emerges from the interplay of attractive interactions, thermal fluctuations, and self-propulsion. We investigate a model active gel, simulated using large-scale Brownian dynamics simulations, to explore the effects of temperature $T$, active force strength $f_0$, persistence timescale $\tau_p$, and active particle concentration $\rho_a$. We characterize the structure of the evolving gel network through morphological analysis, while dynamical changes are quantified by the relaxation timescale and dynamical heterogeneity (DH). We show that an optimal choice of activity parameters yields a percolating gel network with faster gelation and larger voids and clusters, enhancing mechanical stability. Conversely, the structural relaxation timescale and DH decrease with increasing activity, indicating enhanced particle rearrangements and reduced spatial correlations of mobile and immobile regions. An effective temperature ($T_\text{eff}$) framework for colloidal gels maps structural quantities and relaxation timescales in an active gel onto those of a passive gel. More interestingly, the growth of DH with increasing swim force can also be described by the same effective-temperature framework, in stark contrast to observations in active glasses. Our findings show that, in colloidal gels, temperature and active force compete, giving rise to similar structural ordering and, consequently, similar DH at the same $T_\text{eff}$. These findings highlight the crucial role of structural ordering in governing DH in active colloidal gels, as compared to active glasses.

The yielding of amorphous solids has been extensively studied in the athermal limit, but the influence of thermal fluctuations is much less understood, in particular for glasses of different stability. Using molecular dynamics simulations of a model glass where the swap Monte Carlo algorithm can be employed, we study the yield stress at finite temperature and shear rate, both in steady flow and at shear start-up, over a broad range of glass stability. Thermal fluctuations soften both the dynamic yield stress in steady state and the start-up yield stress. In all cases, the temperature dependence is compatible with the $T^{2/3}$ law expected for thermally activated yielding near a saddle-node instability, with no systematic dependence of the exponent on stability or shear rate. Extrapolations to zero temperature recover the athermal yield stresses. The amplitude of the softening at start-up, by contrast, depends strongly on stability: highly stable glasses are much less sensitive to temperature than poorly annealed ones. A description of yielding at finite temperature must therefore account explicitly for the stability of the glass.

Uniform orbital detuning can restore spatial extent even as quasiperiodic modulation strengthens. In a trimer ladder with fixed hoppings and orbital modulation signs, increasing detuning and modulation amplitude together drives successive avoided crossings that redistribute orbital weights, suppressing the projected onsite potential in both lower molecular bands while leaving the third strongly modulated. Complete-band participation scaling supports two reentrant high-participation windows, with wave-packet propagation corroborating spatial recovery at representative controls over the studied sizes and times. These diagonal cancellations leave residual interband modulation, which generates a self-energy whose magnitude depends on the molecular level separation. Gaussian quadrature improves convergence of closed-walk determinant estimates used to cross-check energy-resolved attenuation. Comparing the coupled and decoupled ladders under the same finite-size participation criterion shows that restoring the full off-diagonal modulation narrows both sampled windows, with a larger absolute contraction of the second.

The spectral form factor (SFF), which is the Fourier transform of the two-point spectral correlation, is a fundamental diagnostics of quantum chaos. Nevertheless, its experimental detection in quantum many-body systems was less unexplored until recently. Here, we establish a relationship between the SFF with the quantum Fisher information (QFI) by means of quantum sensing protocols. Furthermore, when the eigenvalues and eigenstates of the disordered Hamiltonian are statistically independent, we prove that QFI is typical both in short and long timescales while numerics suggest that it is also typical at intermediate timescales, which ensures self-averaging of QFI. The self-averaging property renders the experimental measurement of QFI with our protocol less noisy in contrast with the direct measurement of SFF which is known to be non-self-averaging. Our results reveal an intriguing connection between quantum noise and the signatures of quantum chaos.

A fruitful approach towards understanding generalization in machine learning involves characterizing how models separate generalizable structure from idiosyncratic or even corrupted features of training data. Fisher information has emerged as a diagnostic tool for this separation, leveraging loss geometry to distinguish generalization from memorization. Here we investigate this connection in vision models trained on CIFAR-10 with controlled noise that can be learned only through memorization. We show that projecting model weights onto the top Fisher eigenvectors decouples general task performance from noisy sample memorization, and we investigate how this decoupling presents differentially across layers. We show that the same phenomena are present in small transformers trained to solve a synthetic in-context learning task, and are reproduced in closed form in a simple, analytically solvable linear model. Our results reveal that Fisher information captures aspects of the geometric structure underlying how neural networks allocate capacity between generalization and memorization.

Quantum localization, the phenomenon by which wave diffusion is halted, here understood as the superposition of propagating waves, can have many different origins, such as onsite random potentials in Anderson localization. However, some systems exhibit complex geometries and could potentially reproduce similar results without external potentials. In this work, we study how localization of single particle states is affected by the underlying topology of a simple but highly predictive complex network model: the geometric soft configuration model $\mathbb S^1$. By tuning the model's temperature and heterogeneity parameters, we are able to generate different structural patterns and focus on the transition from a small-world to a non-small-world regime. Our study suggests that small-world networks tend to have extended states, which spread the whole system's size, in contrast to non-small-world networks which tend to localize, showing how complex connectivity can play the role assigned to random energetic disorder.

We numerically study spectral and wave-function properties, as well as time evolution, in finite-size commensurate approximations of the 3D Aubry-André model. We explicitly show, through scaling analysis, the existence of two critical points. One of the critical points is a localization-delocalization transition in real-space, similar to the critical point in the 3D Anderson model. The other is a localization-delocalization transition in a dual momentum space, which does not have any analog in the 3D Anderson model. We estimate their values from spectral measures, such as the gap ratio and the spectral form factor, as well as from wave-function measures, such as the inverse participation ratio and the participation entropy. Additionally, we demonstrate that both critical points exhibit hallmarks of critical points between quantum ergodicity and localization. First, the eigenstates and the eigenspectrum at the critical point fulfill the mutual universal relation between the eigenspectrum compressibility $\chi$ and the eigenstate fractal dimension $d^{(1)}$ [S. Jiricek et al., Proc. Nat. Acad. Sci. 123, e2518027123 (2026)]. Second, the dynamics of the wave packet at the critical point is governed by the critical dynamical exponent $z$, characterized by the spectral fractal dimension and the dimensionality of the lattice [M. Hopjan and L. Vidmar, Phys. Rev. Lett. 135, 060401 (2025)]. Thus, similarly to the 3D Anderson model, the critical dynamical exponent is sub-diffusive with the value of $z=3$.

The local Hilbert space dimension is a fundamental parameter in quantum many-body systems, but its role in controlling the ergodicity-breaking transition remains less explored. We systematically study this question by introducing toy models of qudits in zero dimensions that exhibit an ergodicity-breaking transition. Using the hybridization condition, we analytically obtain the dependence of the critical value of the interactions on the local Hilbert space dimension. Numerical studies of spectral correlation measures show agreement with the theoretical predictions, hence confirming the existence of the ergodicity-breaking transition in these models. We further introduce a distinct higher spin version of the model and numerically obtain the same critical behavior, demonstrating that the observed critical point is independent of the microscopic structure of interactions considered.

We formulate the Generalized Constraint-Controlled Rheological Model (GCCRM) for entangled linear and long-arm-branched polymer melts under sinusoidal periodic shear deformation (SPSD), spanning the linear and strongly nonlinear regimes. Molecular architecture, conformation, chain stretch, flow-active constraints, convective constraint release, dynamic tube dilation, and activated arm retraction define a common upstream molecular state. This state feeds two time-domain constitutive pathways (a configurational stress and a dissipative internal-variable stress) which are recombined before Fourier analysis. For branched polymers, the architecture is described by an entire backbone of molecular weight $M_{bb}$ and $N_a$ long arms of molecular weight $M_a$, with a compliance-weighted representative-constitutive-cell partition. The linear limit is normalized so that the configurational moduli sum to the entanglement modulus, while dynamic tube dilation and stretch drive distinct contributions to dissipative activation. Under reduced closures, GCCRM generates a smooth nonlinear elastic response with an onset of nonlinearity, an approximately logarithmic growth regime, an architecture-dependent maximum, and distinct high-deformation asymptotes. Fourier harmonics, ELR, VLR, and Lissajous curves are treated as post-processing observables of the total stress. The framework establishes a consistent constitutive architecture whose quantitative validation requires one parameter set per material across SPSD and, prospectively, transient shear histories.

The origin of the anomalous low-frequency vibrational density of states (VDOS) universally observed in amorphous solids has been debated for decades. In this work, we model amorphous solids as a zero-temperature scalar spring network, the simplest model that captures both the amorphous structure and the features of the anomalous VDOS. Using this scalar network, we mathematically demonstrate that disorder can lead to a shift of the VDOS toward low frequencies, thereby accumulating the anomalous part there. Disorder is defined through the topological connectivity of the network. The validity and universality of the theoretical framework are general and valuable for real-world scenarios because it rests only on Weyl's theorem for Hermitian matrices and is free of perturbation and phenomenological assumptions. This work strips away complex factors and reveals, at the level of mathematical principles, a profound and clean mechanism for the origin of the anomalous VDOS in amorphous solids.

Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed $0<\alpha<1$. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error $\epsilon$ across a planar cut of cross-section $A$ scales as $\exp[\tilde{\Theta}((A\log\epsilon^{-1})^{2/3})]$ when $\log\epsilon^{-1}\gtrsim A$. Thus $\alpha=1$ is the sharp threshold for a universal Rényi area law within this adiabatically connected class.

We develop a dynamical mean-field framework for the full finite-time and stationary asymptotic Lyapunov spectra of random recurrent neural networks. At finite observation time, an exact log-determinant representation of tangent-propagator singular values leads to a closed-contour matrix Dyson equation for the smoothed spectral tail. In the stationary setting, a cocycle representation yields an operator equation for the asymptotic cumulative spectrum. Both formulations retain complete local gain histories and the repeated action of quenched connectivity. Direct singular-value calculations and network QR spectra support the predictions in the regimes examined, with sampling uncertainty and numerical cutoffs assessed separately. The stationary equation also recovers the known maximal exponent, and controlled gain-history replay illustrates how temporal correlations redistribute instability. This work was carried out in collaboration with ChatGPT 5.6 Sol. Toward the end of preparation we became aware of David G. Clark's contemporaneous work, Lyapunov spectrum of random neural networks[arXiv:2610.12426], which reports overlapping mean-field results for the full Lyapunov spectrum using a cavity approach.

In the machine learning community, the reinforcement-learning-driven reasoning process has been intensively studied since around 2025. In this article, we present a negative result on the hardware implementation of this reinforcement-learning-driven reasoning process in a physical neural network model in the presence of a reward external potential energy for a given reward. In this hardware implementation, the existence of a chain of thought in dynamics contradicts the Lyapunov stability of the system.