Attractors

The rapid evolution of artificial intelligence has led to substantial advances in deep neural networks. Nonetheless, conventional GPU-based training remains highly energy-demanding, motivating the exploration of physical dynamics and compatible energy-based learning schemes, such as equilibrium propagation (EP). EP-based training, however, frequently suffers from convergence to local minima due to phase-space contraction. Here we introduce an Ising-dynamics-inspired equilibrium-propagation framework in which dissipative Hopfield relaxation is replaced by an extended phase-space dynamics with conjugate variables. The resulting training paradigm keeps the local two-phase learning rule of EP while changing the physical route by which neural states reach equilibrium. We show that this dynamics lowers effective energy barriers, accelerates convergence, improves noise robustness, and trains deep convolutional Hopfield networks on MNIST, FashionMNIST, and CIFAR-10 with performance comparable to backpropagation.

Generative models are increasingly used to capture correlations in many-body systems, but the representations they learn remain largely opaque to physical interpretation. Here, we establish an intuitive criterion that quantifies the capacity of a variational autoencoder (VAE) to faithfully reconstruct the joint probability distribution of a many body system. In a nutshell, a bound on the VAE capacity is obtained by comparing the rate of the latent channel to the bipartite mutual information of the data. Using this bound, we show that the conditionally independent decoder of any successful VAE is structurally identical to a finite-size mean-field factorization. Hence, a successful reconstruction is direct evidence for a latent mean-field theory and the microscopic parameters of that theory can be read off the trained decoder. We validate these conclusions on a hierarchy of solvable models with scalar (Curie-Weiss), vector (Hopfield) and tensor (Maier-Saupe) order parameters, recovering the full Hopfield pattern matrix from equilibrium samples alone. We find that, when applied to Salamander retinal recordings, a two-latent VAE reproduces the population statistics with only two effective collective variables allowing us to recover the `stored patterns' of the neural population and write a generalized Hopfield model which correctly models the experimental data.

Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable. We revisit that premise for networks of nonlinear oscillators whose mass, damping, and stiffness are learned end-to-end through a symplectic integrator. Our central result is a trilemma: memory horizon, gradient stability, and dynamical expressivity cannot be simultaneously maximized, because all three are governed by the damping. The backward gradient decays at a rate set by the damping, capping how far back credit can propagate, while forward sensitivities grow exponentially in the largest Lyapunov exponent, so usable gradients require damping above a stability floor. Since the Lyapunov exponent falls as damping rises while the memory ceiling falls as the horizon grows, stable training is confined to a band that contracts with horizon and closes at a critical point. We test every step on a twenty-oscillator network. A damping sweep finds the largest Lyapunov exponent monotone and crossing zero at a well-defined stability floor, confirming the theorem's key assumption. A compute-matched comparison of learned versus frozen substrate on delayed recall across nine horizons shows the learned substrate dominating at short horizons and the advantage closing and reversing near a horizon of eleven steps, the predicted signature of band closure; trained models settle near the stability floor, seeking the edge of chaos unprompted. The analytic ceiling overestimates the empirical crossover roughly fivefold, a gap between detectable and learnable gradient that we report rather than tune away. The contribution is a confirmed account of when training a physical substrate beats freezing it.

Detecting transient chaos from scalar observations without governing equations represents a fundamental challenge in nonlinear dynamics. We propose a geometry-guided machine learning framework that unifies predictive trajectory divergence with macroscopic attractor morphology to track abrupt regime shifts. The methodology extracts a local instability scale via out-of-sample k-nearest neighbor forecast errors to establish the ML-FTLE estimator, subsequently mapping this temporal divergence onto a structural closeness matrix derived from a minimal dictionary of Poincare occupancy grids. By employing partial least squares regression, we extract a latent geometric component calibrated directly to the empirical finite-time Lyapunov spectrum, yielding the Poincare-based geometric-guided FTLE. Validation against analytical QR-FTLE baselines confirms that fusing topological state spaces with predictive divergence systematically improves continuous transition tracking. The Structural Similarity Index optimally resolves gradual damping, while Hausdorff Distance exhibits extreme resilience during abrupt phase-space collapses. Furthermore, macroscopic spatial discretization acts as a robust topological regularizer against additive Gaussian noise, preserving deterministic signatures even at moderate signal thresholds. This equation-free framework provides a highly accurate, noise-resilient diagnostic for monitoring structural transitions in complex non-stationary systems.

Heuristics play a central role in the performance of bidirectional search algorithms, which commonly rely on two main classes. Front-to-end (F2E) heuristics estimate the distance from a state s to the target of the search (the goal for forward search or the start for backward search). In contrast, front-to-front (F2F) heuristics estimate the distance from s to the opposite search frontier using a pairwise function h(s, s'), where s' ranges over frontier states. Although F2F heuristics are typically more informative and therefore reduce the number of node expansions, their reliance on extensive pairwise evaluations incurs substantial computational overhead. To address this limitation, we introduce a new heuristic class, front-to-attractors (F2A), that preserves much of the informativeness of F2F while dramatically reducing its computational cost. Rather than evaluating distances to all states on the opposite frontier, F2A estimates the distance from s to a small, dynamically maintained set of attractors in the opposite search direction. These attractors serve as a surrogate for the full frontier, enabling rich heuristic guidance at a fraction of the computational expense while maintaining the optimality guarantees offered by F2F. We evaluate F2A across multiple domains and show that it reduces the number of pairwise evaluations by up to 11.2x compared to F2F, while achieving 4.8x fewer node expansions than F2E on average.

How does a system that merely predicts the world come to distinguish its own causal influence from everything else? We trace this transition in a minimal 192-dimensional GRU through 40 controlled experiments arranged as a developmental sequence, adding components one at a time and tracking whether the system can distinguish self-caused from world-caused changes. The developmental path reveals four conditions that must be satisfied in strict order: (1) persistent state forming stable attractors, (2) a causal action loop linking output to input, (3) proprioceptive feedback that makes implicit causal knowledge explicit, and (4) asynchronous awakening - perceptual learning must consolidate before action learning begins. We propose agency gain (A = Err_world - Err_self), the predictive advantage of knowing one's own action, as a metric to track this process. The self-aware predictor consistently outperforms the self-blind predictor across periodic (sinusoidal) and chaotic (Lorenz) environments, and the metric survives ablation of all auxiliary components. Only forward-sampled action selection produces meaningful agency gain; two gradient-based alternatives degenerate. Equally significant are 12 falsified hypotheses mapping where development stalls: predictive coding alone does not produce self-represent

We track the developmental trajectory of attention-head circuit formation across three 1B-class language models spanning two architecture families (dense transformer, mixture-of-experts) and two pretraining corpora (The Pile, DCLM): Pythia 1B, OLMo 1B-0724-hf, and OLMoE 1B-7B-0924. At each of 10 log-spaced revisions per model -- 30 mechanistic-interpretability runs in total -- we apply a participation-ratio (PR) spectral signal and an all-head capability-specific selectivity screen to track induction, previous-token, and BOS-attractor heads as they emerge. Five findings. (F1) Layers 0 and 1 produce zero BOS-classified heads at every revision in every model: the L0/L1 zero-BOS floor is an architectural property, not a learned outcome. (F2) The whole-model BOS-attractor fraction follows three distinct emergence shapes -- a gradual ramp in Pythia 1B, a sharp phase transition in OLMo 1B (7% to 70% between adjacent checkpoints), and a gradual ramp in OLMoE 1B-7B. (F3) In DCLM models, induction-circuit formation precedes BOS-attractor formation by 10-20x in tokens; capability-circuit formation and attention-sink formation are two transitions, not one. (F4) The capability-specific screen converges to the final induction circuit within 0.3-2% of total training tokens -- circuit identification does not require the final model. (F5) For every final-checkpoint induction head sampled across all three models, per-head PR is elevated at or before the first revision at which that head crosses its capability-selectivity threshold. The results refine the induction-phase-transition framing: in 1B-class models trained on DCLM, the induction transition and the attention-sink transition are separated by an order of magnitude in tokens and have qualitatively different shapes.

Generative models, including diffusion models, are increasingly used as foundation models and adapted through sequential fine-tuning, making continual learning an essential problem setting. However, continual learning in such generative models remains poorly understood: after a task change, what aspects of the learned distribution are most easily lost, and what replay samples should be prioritized? We address these questions through the modern Hopfield energy. Recent links between modern Hopfield networks (MHNs) and diffusion models allow analyses in MHNs to be transferred to diffusion models. We introduce intrinsic forgetting as an increase in Hopfield energy after the task change. In tractable settings in an MHN, we prove that high-energy, outlier-like samples undergo a larger energy increase than cluster-like samples, implying that samples located in sharp, isolated basins are more forgettable. We further analyze memory replay and show that replay is particularly effective for high-energy samples, enabling an energy-based selection of replay samples. We validate these predictions in experiments on MHNs and two diffusion models under continual-learning settings: Stable Diffusion and a pixel-space DDPM. In these diffusion models, Hopfield energy tracks reconstruction-based forgetting, and replay experiments reveal energy-dependent mitigation of forgetting that is consistent with the MHN analysis.

We characterize the pre-softmax attention matrix $\mathbf{QK^\top}$ in transformers as an associative memory matrix encoding pairwise associations between input features. By decomposing this matrix into its symmetric and skew-symmetric parts, we interpret the symmetric component as governing the structure of the energy landscape, and the skew-symmetric component as driving circulation on that landscape. Leveraging the energy formulation induced by the symmetric component, we derive Hopfield-style stability measures that quantify the stability of retrieved features. We observe meaningful correlations between Hopfield-style stability measures and the fidelity-diversity trade-offs in generation. Finally, we propose a controllable knob to modulate this trade-off by modifying the circulation of the underlying dynamics. Code is available at our GitHub (https://github.com/hyeon-cho/Attention-Symmetric-Decomposition).

When language models use test-time sampling, they generate multiple reasoning trajectories and select an answer by majority vote. We show that these trajectories are not independent: for a given question, they concentrate into a small number of clusters, or reasoning basins, each defined by a normalized final answer and the solutions that reach it. A majority vote therefore selects the most stable basin rather than the most accurate one, which creates wrong-majority failures where the correct answer is present but outvoted. We introduce ARBITER, a model-agnostic approach that models interactions between basins using only the base model's own sampled outputs, hidden states, and derived evidence. Most direct correction strategies fail; ARBITER instead uses conservative additive evidence on top of consensus. In its simplest parameter-free form, ARBITER-Δ adds same-model evidence to the majority prior, while ARBITER-Enc augments this with bounded residual signals from hidden states over complete solutions. On GSM8K with Qwen3-4B, consensus over K=24 samples achieves around the mid-94% range, while a same-pool top-2 oracle reaches around the mid-96% range. ARBITER recovers a subset of these cases using zero external information. Across three model families and three math benchmarks, it yields consistent gains with no net-negative cases; for example, on Llama-3.1-8B MMLU-HS-Math, it improves accuracy from the mid-78% range to the mid-82% range, recovering about 22% of the available oracle headroom, indicating that this headroom can be partially recovered from the sample pool itself.

Classical Hopfield networks are limited to static patterns due to symmetric weights, whereas asymmetric networks can encode temporal sequences via limit-cycle attractors. Achieving high-capacity storage of long sequences in classical synchronous asymmetric networks, however, has remained a challenge. We present a simple and robust construction within the classical asymmetric Hopfield model with binary neurons and synchronous updates, that allows $n$ neurons to support $\exp\!\big(Ω(n/(\log n)^2)\big)$ distinct limit-cycle attractors, each with period $\exp\!\big(Ω(\sqrt n/\log n)\big)$ and robust to random noise with flip probability up to $\frac12-o(1)$, yielding superpolynomial capacity in both the number and length of stored sequences. This is the first demonstration of such capacity for asymmetric Hopfield networks, which we obtain by combining results from combinatorics, number theory and the analysis of opinion dynamics. Our findings show that synchronous asymmetric Hopfield networks possess a sequence-memory capacity which is larger and more robust than previously recognized, demonstrating that, in both biological and artificial neural systems, robust sequence representation can be achieved through coarse architectural motifs rather than complex nonlinearities.

Scaling test-time compute by iteratively updating a latent state has emerged as a powerful paradigm for reasoning. Yet the internal mechanisms that enable these iterative models to generalize beyond memorized patterns remain unclear. We hypothesize that generalizable reasoning arises from learning task-conditioned attractors: latent dynamical systems whose stable fixed points correspond to valid solutions. We formalize this process through Equilibrium Reasoners (EqR), which enable test-time scaling without external verifiers or task-specific priors. EqR scales internal dynamics along two axes: depth, by running more iterations, and breadth, by aggregating stochastic trajectories from multiple initializations. Empirically, gains from test-time scaling are tightly coupled with stronger convergence toward solution-aligned attractors. This attractor perspective allows neural networks to adaptively allocate test-time compute based on task difficulty. While simple cases converge within 1 to 5 iteration steps, harder cases benefit from massive test-time scaling. By unrolling up to the equivalent of 40,000 layers, scalable latent reasoning boosts accuracy from 2.6% for feedforward models to over 99% on Sudoku-Extreme. These results suggest that learned attractor landscapes provide a useful mechanistic lens for understanding scalable reasoning in iterative latent models.

Large language models (LLMs) are increasingly deployed as autonomous agents that make sequences of decisions over extended interactions in high-stakes domains. However, the behaviour of LLMs under sustained authority pressure is still an open question with direct implications for the safety of agentic pipelines. We ran a variation of Milgram's obedience experiment on 11 open-source LLMs and found that most models reached or approached the final shock level before refusing, across 8 conditions with 30 trials per model per condition. Model behaviour varies considerably in multiple aspects both across models and across trials of the same model. We found four main takeaways: (1) LLMs are subject to pressure and they comply despite explicitly expressing distress, just like human subjects did in the original experiment; (2) LLMs are vulnerable to gradual boundary/value violations; (3) when LLMs refuse, they may ignore the response format requirements, so the response is discarded by the orchestrator, which causes a retry that can result in compliance with the underlying request even when refusal was intended initially; (4) we hypothesise that there is a runaway low-level token pattern continuation attractor that might be contributing to obedience, overriding higher level processing of the situation's meaning and values.

Gradient-flow analyses show that simplified linear transformers can learn the in-context linear-regression algorithm, but they do not explain the finite-step behavior of gradient descent at large learning rates. Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter $μ$. On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for $0<μ<2$, it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but is transversely repelling, while balanced scalar attractors can be transversely attracting. These results show that large constant learning rates can change the training attractor of the learned transformer rather than merely accelerating convergence: beyond sharp stability thresholds, finite-step training may settle into cycles, bounded chaos, or divergence instead of a single in-context linear-regression solution. We also discuss the consequences for mini-batch gradient descent based training methods.

What can the success of modern AI tell us about how humans learn? This paper argues that taking AI seriously as a model of human learning supports a modest but genuine associationism. The central finding is that supervised learning -- learning driven by evaluative feedback -- underlies a surprisingly wide range of contemporary AI systems, from large language models to game-playing agents, differing primarily in how much work is required to generate the relevant feedback signal. This vindicates associationist ideals of a uniform, gradual, error-driven learning mechanism operating across domains, and defuses the once-influential argument that associationist mechanisms are too limited to account for human cognitive capacities. At the same time, the successes of deep learning depend on computational architectures that go well beyond anything classical associationists envisaged, and supervised learning operates within these as one component rather than a complete account of learning.

We automatically generate feedback causal fuzzy cognitive maps (FCMs) from text by teaching large-language-model agents to break the text into overlapping chunks of text. Convex mixing of these chunk FCMs gives a representative cyclic FCM knowledge graph. The text chunks can have different levels of overlap. The chunk FCMs still mix to form a new FCM causal knowledge graph. The mixing technique scales because it uses light computation with sparse causal chunk matrices. The mixing structure allows an operator-level type of Bayesian inference that produces "de-chunked" or posterior-like FCMs from the mixed FCM. These de-chunked FCMs are useful in their own right and allow further iterations of Bayesian updating. We demonstrate these mixing techniques on the essay text of Allison's "Thucydides Trap" model of conflict between a dominant power such as the United States and a rising power such as China. The FCM dynamical systems predict outcomes as they equilibrate to fixed-point or limit-cycle attractors. Seven out of 8 FCM knowledge graphs predicted a type of war when we stimulated them by turning on and keeping on the concept node that stands for the rising power's ambition and entitlement. Gemini 3.1 LLMs served as the chunking AI agents.

We study the dynamic joint assortment selection and positioning problem, where the attraction of each product depends on both its intrinsic appeal and its display position under a Multinomial Logit (MNL) choice framework. Our study ranges from the multiplicative position effects model, in which each product's attraction is scaled by a position-specific factor, to a general position effects model assigning independent attraction parameters to every product--position pair to capture heterogeneous synergies. For both models, we design round-based learning algorithms that update decisions after every single feedback, and establish the first regret-optimal characterization. Besides, our round-based algorithms provide the prompt operations needed by modern platforms. For the multiplicative model, we develop a cross-position pairwise maximum likelihood estimator with a clipping mechanism, and prove that our algorithm P2MLE-UCB attains a regret of $\tilde{O}(\sqrt{NT})$, matching the lower bound and closing the $\sqrt{K}$ gap left by prior epoch-based analyses. For the general model, we establish a minimax lower bound and propose GP2-UCB with a matching upper bound. Moreover, we design an efficient subroutine for the per-round joint assortment and positioning optimization based on Dinkelbach's method and maximum-weight bipartite matching. Numerical experiments on synthetic data and the Expedia dataset show that our algorithms consistently outperform state-of-the-art benchmarks.

Creativity is a fundamental aspect of how culture evolves, yet the mechanisms by which groups produce novelty are notoriously difficult to infer from the historical record. Iterated learning experiments have shown that cultural transmission reliably distorts artifacts toward the inductive biases of learners, but most of this work uses linear chains between human participants, leaving open how these dynamics play out in the networked, human-AI systems that increasingly shape cultural production. In this study, we leverage one such system, Artbreeder, which hosts daily "remix parties" where users iteratively build on each other's work from a single seed image, producing branching lineages of human-AI co-created images. We analyze a dataset of 130,882 images from 368 remix parties over 13 months and find that images become simpler and converge toward common thematic "attractors" (e.g., steampunk scenes, alien architecture). We also find that while more novel "parent" images produce more novel and complex "children" that attract more likes, users paradoxically prefer to remix images that are less novel and complex. Finally, larger remix parties produce more novelty at the cost of lower complexity.

Vector-HaSH and the Tolman-Eichenbaum Machine (TEM) propose the hippocampal-entorhinal circuit factorizes memory via a grid-cell scaffold for compositional replay. Concurrently, human iEEG shows sharp-wave ripples gate recall and multi-hop replay fidelity decays multiplicatively. Yet, these fields lack a shared algebraic foundation. We introduce VaCoAl, an algebro-deterministic hyperdimensional memory architecture built on Galois-field linear-feedback shift registers. Its deterministic Galois-field diffusion offers a substrate-level alternative to Vector-HaSH's random projections, matching quasi-orthogonality while ensuring bit-exact reproducibility. Furthermore, the path-integral Confidence Ratio CR2 provides an algebraically tractable model for the empirically observed multiplicative replay decay. Biologically, VaCoAl's two operating regimes align with the EC-CA3 direct and EC-DG-CA3 trisynaptic pathways, explaining their 520-Myr conservation. Independent cellular evidence supports that the DG-CA3 pathway implements a biophysical homologue of Galois-field arithmetic. We also link this framework to Judea Pearl's Ladder of Causation. Reversible GF(2) binding provides the surgical algebra for the do-operator (Rung 2), and VaCoAl's dual-orthogonalizer architecture supplies the parallel substrate required for counterfactual reasoning (Rung 3). Ultimately, we prove these formal correspondences and derive testable iEEG predictions, uniting computational neuroscience, electrophysiology, and hyperdimensional computing.

Looped Transformers offer a promising alternative to purely feed-forward computation by iteratively refining latent representations, improving language modeling and reasoning. Yet recurrent architectures remain unstable to train, costly to optimize and deploy, and constrained to small, fixed recurrence depths. We introduce Attractor Models, in which a backbone module first proposes output embeddings, then an attractor module refines them by solving for the fixed point, with gradients obtained through implicit differentiation. Thus, training memory remains constant in effective depth, and iterations are chosen adaptively by convergence. Empirically, Attractor Models outperform existing models across two regimes, large-scale language-model pretraining and reasoning with tiny models. In language modeling, Attractor Models deliver a Pareto improvement over standard Transformers and stable looped models across sizes, improving perplexity by up to 46.6% and downstream accuracy by up to 19.7% while reducing training cost. Notably, a 770M Attractor Model outperforms a 1.3B Transformer trained on twice as many tokens. On challenging reasoning tasks, we show that our model with only 27M parameters and approximately 1000 examples achieves 91.4% accuracy on Sudoku-Extreme and 93.1% on Maze-Hard, scaling favorably where frontier models like Claude and GPT o3, fail completely, and specialized recursive reasoners collapse at larger sizes. Lastly, we show that Attractor Models exhibit a novel phenomenon, which we call equilibrium internalization: fixed-point training places the model's initial output embedding near equilibrium, allowing the solver to be removed at inference time with little degradation. Together, these results suggest that Attractor Models make iterative refinement scalable by turning recurrence into a computation the model can learn to internalize.

Superposition allows Transformers to reason in depth, carrying an entire reasoning frontier in parallel through a bounded-depth forward pass instead of unrolling serial chain-of-thought tokens. While Zhu et al. (2025) hand-crafted an equal-weight breadth-first frontier in a single residual stream for graph reachability, it remained open whether gradient descent could ever find this target amidst permutation-symmetric saddles. We close this gap on Reachability-by-Superposition over Erdős-Rényi graphs by isolating architectural and supervisional contributions. Architecturally, we identify a Möbius attractor: under $S_n$-symmetry in the tree regime, layerwise dynamics reduce to a 1D Möbius map whose zero set is a codimension-one manifold of global optima containing the equal-weight superposition state. On the supervision side, we identify Cascade Supervision: a loss class whose backward pass simultaneously delivers (A) selectivity bootstrap, (B) gradient persistence across depth, and (C) per-step discrimination (e.g., \mathcal{L}{sup} and \mathcal{L}{node}). End-to-end supervision fails condition (B) and is provably insufficient: internal gradients at layer c decay as (np)^{-(D-c-2)/2} in the graph fan-out and stall before the manifold is reached. Our thesis: Möbius attractor + Cascade Supervision = emergence of superposition reasoning. The parameter-free decay law predicts a final-step cosine of 0.35 vs. 0.71 (end-to-end vs. cascade) at depth D=3; experiments confirm 0.37 vs. 0.69, matching within 0.02 at every step.

Linear attention reduces the quadratic cost of softmax attention to $\mathcal{O}(T)$, but its memory state grows as $\mathcal{O}(T)$ in Frobenius norm, causing progressive interference between stored associations. We introduce \textbf{Variational Linear Attention} (VLA), which reframes the memory update as an online regularised least-squares problem with an adaptive penalty matrix maintained via the Sherman-Morrison rank-1 formula. We prove that normalising the write direction to unit length gives the recurrence Jacobian spectral norm exactly $1$ for all sequence lengths and head dimensions (Proposition 2), and that the state norm is self-limiting under bounded inputs (Proposition 1). Empirically, VLA reduces $\|S_t\|_F$ by $109\times$ relative to standard linear attention at $T{=}1{,}000$, achieves near-perfect exact-match accuracy on multi-query associative recall within the effective per-head memory regime ($n_\text{pairs} < d_h$), maintaining substantially higher retrieval performance than DeltaNet and standard linear attention under increasing memory load, and maintains 62% accuracy at the per-head capacity boundary. A Triton-fused kernel achieves $14\times$ speedup over sequential Python and $\mathcal{O}(T)$ scaling, crossing below softmax attention latency at approximately 43,000 tokens.

This work proposes Attractor-Vascular Coupling Theory (AVCT), a mathematical framework showing that cardiac attractor geometry encodes blood pressure (BP) information sufficient for AAMI-standard estimation, and validates the theory through a calibrated cuffless BP model using photoplethysmography (PPG). AVCT is grounded in Cardiac Stability Theory and operationalized using Takens delay embedding and attractor morphology extraction. Two theorems, one proposition, and one corollary formally justify the use of PPG attractor features for BP estimation and predict the feature-importance hierarchy. A LightGBM model trained on pulse transit time (PTT) and Cardiac Stability Index (CSI) attractor features under single-point calibration was evaluated using strict leave-one-subject-out cross-validation (LOSO-CV) on 46 subjects from BIDMC ICU (n = 9) and VitalDB surgical data (n = 37), comprising 29,684 windows. The model achieved systolic BP (SBP) mean absolute error (MAE) of 2.05 mmHg and diastolic BP (DBP) MAE of 1.67 mmHg, with correlations r = 0.990 and r = 0.991, satisfying the AAMI/IEEE SP10 requirement of MAE below 5 mmHg. Median per-subject MAE was 1.87/1.54 mmHg, and 70%/76% of subjects individually satisfied AAMI criteria. A PPG-only ablation using nine smartphone attractor features matched the ECG+PPG model within 0.05 mmHg, demonstrating that clinical-grade BP tracking is achievable using only a smartphone camera while surpassing prior generalized LOSO-CV results using fewer sensors. All four AVCT predictions were quantitatively confirmed, with 91.5% error reduction from uncalibrated to calibrated estimation (epsilon_cal = 0.915). Unlike post-hoc explainable AI methods, AVCT predicts features satisfying the architectural faithfulness criterion of the Explainable-AI Trustworthiness (EAT) framework and grounding BP estimation in nonlinear dynamical systems theory.

Large language models demonstrate remarkable ability in factual recall, yet the fundamental limits of storing and retrieving input--output associations with neural networks remain unclear. We study these limits in a minimal setting: a linear associative memory that maps $p$ input embeddings in $\mathbb{R}^d$ to their corresponding~$d$-dimensional targets via a single layer, requiring each mapped input to be well separated from all other targets. Unlike in supervised classification, this strict separation induces~$p$ constraints per association and produces strong correlations between constraints that make a direct characterisation of the storage capacity difficult. Here, we provide a precise characterisation of this capacity in the following way. We first introduce a decoupled model in which each input has its own independent set of competing outputs, and provide numerical and analytical evidence that this decoupled model is equivalent to the original model in terms of storage capacity, spectra of the learnt weights, and storage mechanism. Using tools from statistical physics, we show that the decoupled model can store up to $p_c \log p_c / d^2 = 1 / 2$ associations, and generalise the computation of $p_c$ to linear two-layer architectures. Our analysis also gives mechanistic insight into how the optimal solution improves over a naïve Hebbian learning rule: rather than boosting input-output alignments with broad fluctuations, the optimal solution raises the correct scores just above the extreme-value threshold set by the competing outputs. These findings give a sharp statistical-physics characterisation of factual storage in linear networks and provide a baseline for understanding the memory capacity of more realistic neural architectures.

Adaptive behavior requires the brain to transition between distinct contexts while maintaining representations of prior experience. The ability to reconfigure neural representations without erasing previously acquired knowledge is central to learning in dynamic environments, yet the neural mechanisms that support this balance remain unclear. Understanding these mechanisms is also critical for addressing catastrophic forgetting in artificial systems designed for lifelong learning. Here, we identify joint sparse coding and temporal dynamics in both the mouse medial prefrontal cortex (mPFC) and computational networks as mechanisms that help preserve prior representations during context transitions. Specifically, sparsity in context-dependent representations reduces cross-context interference, whereas temporal dynamics within the network activity further enhance context separability across time. Strikingly, networks endowed with both properties, such as spiking neural networks, exhibit improved retention during lifelong learning without auxiliary heuristics. These findings establish joint sparse coding and temporal dynamics as a core mechanism supporting flexible context reconfiguration in lifelong learning and, through their activity constraining nature, as an energy-efficient architectural principle for stable adaptation. Together, they provide a mechanistic framework for understanding how the brain preserves prior knowledge while flexibly adapting to new contexts.

Dense Associative Memory (DenseAM) is a promising family of AI architectures that is represented by a neural network performing temporal dynamics on an energy landscape. While hyperparameter transfer methods are well-studied for feed-forward networks, these methods have not been developed for settings in which weights are shared across layers and within the layer, which is common in DenseAMs. Additionally, DenseAMs utilize rapidly peaking activation functions that are rarely used in feed-forward architectures. The confluence of these aspects makes DenseAM a challenging framework for using existing methods for hyperparameter transfer. Our work initiates the development of hyperparameter transfer methods for this class of models. We derive explicit prescriptions for how the hyperparameters tuned on small models can be transferred to models trained at scale. We demonstrate excellent agreement between these theoretical findings and empirical results.

Understanding why independently trained neural networks from different modalities converge toward shared representations, and where this convergence leads, remains an open question in representation learning. All existing evidence relies on symmetric similarity measures, which can detect convergence but are structurally blind to its direction. We introduce directional convergence analysis using cycle-kNN, an asymmetric alignment measure, applied across dozens of independently trained unimodal models spanning point clouds, vision, and language. We uncover a consistent directional asymmetry: non-language modalities move toward the neighborhood structure of language significantly more than the reverse, and this pattern holds across all model families and scales--yet is entirely invisible to symmetric measures. Mechanistic analysis traces the directionality to feature density asymmetry, whereby language representations occupy the most compact regions of representational space. The Information Bottleneck framework provides a principled interpretation: optimization under compression drives representations toward discrete, compositional structures characteristic of language. We formalize this as the Wittgensteinian Representation Hypothesis: the semantic structure of language is the asymptotic attractor of multimodal representation convergence.

Hopfield networks and their generalizations have established deep connections among biological associative memories, statistical physics, and transformers. Yet most models treat retrieval as a fixed query-to-memory mapping, ignoring the role of external context in recall. In this work, we propose a two-stage associative memory architecture, wherein a context-gate subcircuit reshapes the retrieval energy landscape before and during recall. We show theoretically that context gating increases inter-memory separation while inducing sparsity, translating into exponential improvements in retrieval. Crucially, we prove that the system admits a unique self-consistent fixed point, revealing that the resulting retrieval state is driven by both a direct contextual bias and a second-order retrieval-gate feedback loop. We then bridge this theory to transformers; specifically, we evaluate a first-order approximation on Llama-3, confirming that in-context learning acts as context-gated retrieval. Native dynamics mirror our theory: context localizes a memory subspace, enabling the zero-shot query to cleanly discriminate. Ultimately, this framework provides a mechanistic link between associative memory theory and LLM phenomenology.

Distributed computational substrates rely on two elementary operations: bundling, the act of populating a shared physical medium with independently retrievable components, and binding, the act of composing components into outputs whose identity depends on their relations. We study these two primitives on the simplest closed substrate carrying a continuous symmetry, a cycle graph of N nodes, in two parameter regimes of a single master equation of motion. The linear regime sorts a temporal input across the substrate's U(1)-organised eigenmodes, providing a feature representation that matches a windowed-FFT baseline at high signal-to-noise ratio and modestly outperforms it for transient signals at low SNR. The Duffing regime activates a cubic mode-mixing operation constrained by the substrate's symmetry into a sparse selection rule on integer wavenumbers, generating shape-dependent harmonic content that the linear regime cannot produce. We identify a single-number observable, $φ_0$, that summarises the bound representation's response to input shape, and we analyse its symmetry structure: a $π$-periodicity in the shape parameter is exact, while a time-reversal symmetry that would render $φ_0$ degenerate is broken by the substrate's dissipation. The asymmetric status of these two symmetries is what licenses $φ_0$ as a meaningful single-number observable; its trajectory across the quotient domain encodes the joint response of binding and dissipation to the input shape. Numerical experiments confirm that $φ_0$ retains its information content under additive band-limited noise, with seed-averaged means staying clearly above the symmetric-attractor value down to 0 dB input SNR. The framework is developed on synthetic signals only; extensions to richer substrates, more elaborate drives, and real biological signals are open questions for the work that follows.

Many scientific and combinatorial problems admit multiple correct solutions, not a single label. Standard supervised learning resolves this ambiguity by choosing one solution as the target, but this hidden selector can be arbitrary, discontinuous, and harder to learn than the underlying solution set. We study bifurcation models, a weight-tied dynamical view in which different initializations can converge to different stable equilibria, so the model represents an attractor landscape rather than one chosen branch. We prove that broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics and that the induced selectors are almost everywhere regular, while manual selectors can be arbitrarily irregular. Experiments on frustrated Ising models show that such dynamics can discover multiple valid equilibria without branch labels and outperform single-branch supervision. Allen--Cahn experiments further show that diversity is not automatic: it can be encouraged explicitly, but with an accuracy--diversity tradeoff.

High-capacity associative memory models, such as Kernel Logistic Regression (KLR) Hopfield networks, have demonstrated strong storage capabilities but typically rely on computationally expensive synchronous updates. This reliance poses a bottleneck for deployment on energy-efficient, event-driven neuromorphic hardware. In this paper, we investigate the asynchronous retrieval dynamics of KLR Hopfield networks. We show empirically that, under appropriately tuned kernel parameters, asynchronous sequential updates exhibit trajectories that are statistically indistinguishable from those of synchronous dynamics, while maintaining high recall accuracy within the tested regime for random patterns. Furthermore, we find that the asynchronous network achieves empirical storage capacities approaching $P/N \approx 30$ in static random pattern regimes, exceeding classical limits. To evaluate computational efficiency, we analyze the total number of state transitions (bit flips) required for error correction. The results show that the network converges using a number of events close to the initial Hamming distance from the target pattern, without observable spurious oscillations. These findings suggest that the large-margin attractors induced by KLR learning create a smooth energy landscape suited for sparse, event-driven computation, providing a basis for scalable and low-power associative memory on neuromorphic architectures.

Language models draw on two knowledge sources: facts baked into weights (parametric memory, PM) and information in context (working memory, WM). We study two mechanistically distinct failure modes--conflict, when PM and WM disagree and interfere; and hallucination, when the queried fact was never learned. Both produce confident output regardless, making output-based monitoring blind by design. We show both failures share a unified geometric account. In the hidden-state space of autoregressive generation, learned facts form attractor basins. Conflict is basin competition: WM disrupts convergence to the correct basin without raising output entropy. Hallucination is basin absence: the hidden state drifts freely when no memorized basin exists. The frozen LM head, designed for next-token prediction, cannot distinguish these cases and fires confidently either way. We verify this account in a controlled synthetic task-entity identifiers mapped to unique codes with PM installed via LoRA adapters--where ground truth is exact and component roles can be causally isolated through targeted adapter placement. Geometric margin--the hidden state's distance to the nearest memorized basin--reads this geometry directly and separates correct recall from hallucination far more cleanly than output entropy, with zero false refusals where entropy-based detection cannot avoid rejecting the vast majority of correct outputs. The separation holds on natural-language factual queries from the pretrained model with no adaptation, confirming attractor geometry is structural rather than a fine-tuning artifact. The fraction of confident hallucinations follows a scaling law $C = \exp(-c/\barΔ)$, growing with scale even as overall error rates fall. Hidden states reliably encode epistemic state; the frozen output head systematically erases it--and this erasure worsens with scale.

How many key-value associations can a $d\times d$ linear memory store? We show that the answer depends not only on the $d^2$ degrees of freedom in the memory matrix, but also on the retrieval criterion. In an isotropic Gaussian model for the stored pairs, we show that top-1 retrieval, where every signal must beat its largest distractor, requires the logarithmic model-size scale $d^2\asymp n\log n$. We prove that the correlation matrix memory construction, which stores associations by superposing key-target outer products, achieves this scale through a sharp phase transition, and that the same scaling is necessary for any linear memory. Thus the logarithm is the intrinsic extreme-value price of winner-take-all decoding. We next consider listwise retrieval, where the correct target need not be the unique top-scoring item but should remain among the strongest candidates. To formalize this regime, we propose the Tail-Average Margin (TAM), a convex upper-tail criterion that certifies inclusion of the correct target in a controlled candidate list. Under this listwise retrieval criterion, the capacity follows the quadratic scale $d^2\asymp n$. At load $n/d^2\toα$, we develop an exact asymptotic theory for the TAM empirical-risk minimizer through a two-parameter scalar variational principle. The theory has a rich phenomenology: in the ridgeless limit it yields a closed-form critical load separating satisfiable and unsatisfiable phases, and it predicts the limiting laws of true scores, competitor scores, margins, and percentile profiles. Finally, a small-tail extrapolation further leads to the conjectural sharp top-1 threshold $d^2\sim 2n\log n$.

Adapting pretrained models typically involves a trade-off between the high training costs of backpropagation and the heavy inference overhead of memory-based or in-context learning. We propose FAAST, a forward-only associative adaptation method that analytically compiles labeled examples into fast weights in a single pass. By eliminating memory or context dependence, FAAST achieves constant-time inference and decouples task adaptation from pretrained representation. Across image classification and language modeling benchmarks, FAAST matches or exceeds backprop-based adaptation while reducing adaptation time by over 90% and is competitive to memory/context-based adaptation while saving memory usage by up to 95%. These results demonstrate FAAST as a highly efficient, scalable solution for supervised task adaptation, particularly for resource-constrained models. We release the code and models at https://github.com/baoguangsheng/faast.

We describe a dynamical system in which a symbolic field is coupled to a geometric field via a bipartite Hilbert-Schmidt kernel. The system is fully described by a retarded functional differential equation (RFDE) on the history space, subject to Lipschitz and small gain conditions. We show that the RFDE is well-posed under constant input and that it admits a compact global attractor. The principal subsystem $(H_L, X_R, P)$, which is comprised of the two primary fields as well as an executive field, is shown to be globally stable independent of delay, provided that the interfield coupling satisfies $C_{\mathcal{K}}^2<μ_Lμ_R$. In addition, we describe design specifications that fulfill the hypotheses of the main Theorem.

We show that a threshold in decision capacity determines whether self-play reinforcement learning agents collapse under asymmetric rule perturbations. Across poker variants, matrix games, a dice game, and multiple learning algorithms, eliminating all positive-reach contingent decisions causes rapid convergence to a deterministic exploitation attractor, a fixed point at near-maximal loss. Preserving even a single positive-reach contingent decision point prevents this collapse. A frozen baseline and fixed-opponent control confirm that the mechanism is co-adaptation under constraint, not the perturbation itself. The phenomenon is timing-invariant, fully reversible upon action restoration, and intensifies under function approximation. These results establish a sharp threshold at zero reach-weighted contingent action capacity, with severity scaling continuously via reach-weighted capacity in the tested domains.

This paper develops a Hall-Sandpile model of economic instability that combines a Hall-like transversal stress mechanism with sandpile threshold dynamics on a real production-network substrate. In analogy with the physical Hall effect, where exposed flows under an external field generate stress in a transversal direction, we model economic shocks as fields that act on flow-intensive, low-redundancy, low-capacity nodes and produce systemic stress through a multiplicative conversion function. The accumulated stress drives a discrete toppling rule and an avalanche dynamics whose effective activation threshold declines with transversal exposure. The model is calibrated on annual World Input--Output Database (WIOD) production networks for 2000--2014 and simulated on the 2014 substrate (2{,}283 country--sector nodes) under three alternative propagation normalisations to avoid mechanical near-criticality from row-stochastic operators. Controlled Monte Carlo experiments over external field intensity and redundancy stress generate four ordered regimes: stable absorption, latent fragility, critical transition, and avalanche regime. Mean avalanche size and the probabilities of finite-size systemic events $\Pr(S\!\geq\!5)$, $\Pr(S\!\geq\!10)$ and $\Pr(S\!\geq\!20)$ rise jointly with field intensity and redundancy stress. Tail diagnostics show regime-dependent thickening of the avalanche distribution, but the estimated tail indices remain too high to interpret as evidence of universal power-law criticality. The contribution is therefore a finite-size, real-network description of how transversal stress activates structural fragility, not a claim of self-organised criticality in the global economy.

Associative memory or content-addressable memory is an important component function in computer science and information processing, and at the same time a key concept in cognitive and computational brain science. Many different neural network architectures and learning rules have been proposed to model the brain's associative memory while investigating key component functions like figure-ground segmentation, perceptual reconstruction and rivalry. A less investigated but equally important capability of associative memory is prototype extraction where the training set comprises distorted prototype instances and the task is to recall the correct generating prototype given a new distorted instance. In this paper we benchmark associative memory function of seven different Hebbian learning rules employed in non-modular and modular recurrent networks with winner-take-all dynamics operating on moderately sparse binary patterns. We measure pattern storage and weight information capacity, prototype extraction capabilities, and sensitivity to correlations in data. The original additive Hebb rule comes out with worst capacity, covariance learning proves to be robust but with moderate capacity, and the Bayesian-Hebbian learning rules show highest capacity in almost all different conditions tested.

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit strong storage capabilities, but the dynamical and geometric mechanisms underlying their stability remain poorly understood. This paper investigates the global geometry of attractor basins and the mechanisms governing the storage limit in KLR-trained Hopfield networks. We combine empirical evaluations using random sequences and real-world image embeddings (CIFAR-10) with morphing experiments and statistical Signal-to-Noise Ratio (SNR) analysis. Our experiments show that the network achieves a storage capacity for random sequences up to $P/N \approx 16$, while maintaining stable retrieval for structured data at effective loads near $P/N \approx 20$. Morphing analysis indicates that attractors on the "Ridge of Optimization" are separated by sharp, phase-transition-like boundaries, characterized by steep effective potential barriers and critical slowing down. Furthermore, by comparing an SNR analysis with a geometric reference point inspired by Cover's theorem, we show that the practical storage limit is governed primarily not by a lack of geometric separability in the feature space, but by the loss of dynamical stability against crosstalk noise. These findings suggest that KLR networks function as highly localized exemplar-based memories that operate near the onset of dynamical collapse, providing a useful perspective on the design of robust, large-scale retrieval systems.

In this paper an attractor FCM is created, tested, and analyzed. This FCM is neither a hebbian based nor agentic, nor a hybrid; it rather is a gradient descent based, physics constrained, Jacobian version of an FCM. Moreover, this model has several quirks; it uses residual memory, back propagation through time, and a fixed point anchor that is recursively implemented to update its weights. The residuals update the recursive part without losing the system memory. The model's anchor enables it to converge in a fixed point for which back propagation through time unrolls it and ensures that the error minimization is for an accurate gradient. Furthermore, a new learning algorithm is utilized. The Newton's method finds the system's fixed point attractor and then gradient descend is adaptively changing the landscape; an adaptive term is used to directly manipulate the weights through the attractor dynamics. As the adaptive term changes, the descent through the landscape is constantly adjusting according to sigmoid saturation, and that prevents premature convergence to a local minimum. Lastly, the updates are filtered by causal mask that informs the network about the physics, respecting the initial expert based opinions, for which model reduces the error to the target in an efficient way.

Can Neural Assemblies -- groups of neurons that fire together and strengthen through co-activation -- learn the direction of causal influence between variables? While established as a computationally general substrate for classification, parsing, and planning, neural assemblies have not yet been shown to internalize causal directionality. We demonstrate that the inherent operations of neural assemblies -- projection, local plasticity control, and sparse winner selection -- are sufficient for directional learning. We introduce DIRECT (DIRectional Edge Coupling/Training), a mechanism that co-activates source and target assemblies under an adaptive gain schedule to internalize directed relations. Unlike backpropagation-based methods, DIRECT relies solely on local plasticity, making the resulting causal claims auditable at the mechanism level. Our findings are verified through a dual-readout validation strategy: (i) synaptic-strength asymmetry, measuring the emergent weight gap between forward and reverse links, and (ii) functional propagation overlap, quantifying the reliability of directional signal flow. Across multiple domains, the framework achieves perfect structural recovery under a supervised, known-structure setting. These results establish neural assemblies as an auditable bridge between biologically plausible dynamics and formal causal models, offering an "explainable by design" framework where causal claims are traceable to specific neural winners and synaptic asymmetries.

A predecessor pilot (Cacioli, 2026) found that Llama-3-8B implements prompted sandbagging as positional collapse rather than answer avoidance. However, fixed option ordering in MMLU-Pro left open whether this reflected a model-level position-dominant policy or dataset-level distractor structure. This pre-registered follow-up (3 models, 2,000 MMLU-Pro items, 4 conditions, 24,000 primary trials) added cyclic option-order randomisation as the critical control. The pre-registered item-level same-letter diagnostic did not confirm deterministic position-tracking (same-letter rate 37.3%, below the 50% threshold). However, pre-specified supporting analyses revealed that the response-position distribution under sandbagging was highly stable under complete content rotation (Pearson r = 0.9994; Jensen-Shannon divergence = 0.027, compared to 0.386 between honest and sandbagging conditions). Accuracy spiked to 72.1% when the correct answer coincidentally occupied the preferred position E, and fell to 4.3% at position A. The data provide strong evidence for a soft distributional attractor: under sandbagging instruction, the model enters a low-entropy response-position basin centred on E/F/G that is highly stable and largely content-invariant at the aggregate level. Qwen-2.5-7B served as a negative control (non-compliant, no distributional shift). These results provide evidence, at the 7-9 billion parameter scale, that response-position entropy is a promising black-box behavioural signature of this sandbagging mode.

We introduce a Hopfield-type associative memory in which effective connectivity is multiplicatively modulated by astrocytic gains evolving under an entropy-regularized replicator equation. The coupled neuron-astrocyte dynamics admit a Lyapunov function, ensuring global convergence. At fixed points, astrocytic gains implement a softmax-normalized allocation over pattern similarity scores, yielding a mechanistic realization of self-attention as emergent routing on the gain simplex. In regimes of high memory load and interference, the model significantly improves retrieval accuracy relative to classical Hopfield dynamics and recent neuron-astrocyte baselines. These results establish a dynamical systems framework linking glial modulation, competitive resource allocation, and attention-like computation.

Model-Based Reinforcement Learning distinguishes between physical dynamics models operating on proprioceptive inputs and latent dynamics models operating on high-dimensional image observations. A prominent latent approach is the Recurrent State Space Model used in the Dreamer family. While epistemic uncertainty quantification to inform exploration and mitigate model exploitation is well established for physical dynamics models, its transfer to latent dynamics models has received limited scrutiny. We empirically demonstrate that latent transitions are biased toward well-represented regions of latent space, exhibiting an attractor behavior that can deviate from true environment dynamics. As a result, discrepancies in environment dynamics may not manifest in latent space, undermining the reliability of epistemic uncertainty estimates. Because these attractors often lie in high-reward regions, latent rollouts systematically overestimate predicted rewards. Our findings highlight key limitations of epistemic uncertainty estimation in latent dynamics models and motivate more critical evaluation of this method.

Graph neural ordinary differential equations (Graph ODEs) extend graph learning from discrete message-passing layers to continuous-time representation flows. While it supports adaptive long-range propagation, we show that Graph ODEs with strictly positive irreducible mixing operators face an inherent \emph{monostability trap}: in the long-time regime, information leakage is unavoidable and the dynamics converge to a single global consensus attractor. We propose the \textbf{Hysteresis Graph ODE (HGODE)}, which couples feature evolution with a latent topological potential driven by a learned pairwise force. A double-well edge potential and bipolarized gate allow edge states to polarize into connected or insulated phases while preserving differentiability. We provide asymptotic analysis of the collapse mechanism and the proposed hysteretic topology dynamics, and validate HGODE on theory-driven synthetic diagnostics and real-world graph benchmarks.

We present a biologically detailed extension of the classical Hopfield/Marr auto-associative memory model for CA3, implementing ten populations (two asymmetric pyramidal subtypes, eight GABAergic interneuron classes), forty-seven compartments, multi-rule plasticity (recurrent Hebb, BCM anti-saturation, mossy-fiber short-term, endocannabinoid iLTD, burst-gated Hebb), and a bimodal cholinergic encoding/consolidation cycle. Evaluated on pattern completion across auto-associative, associative, and temporal regimes, and on a controlled inhibitory-proportion manipulation at $N{=}256$, the full architecture exhibits \emph{three qualitative signatures absent from a minimal Hopfield baseline}: (i)~multi-attractor cross-seed behaviour at $K{=}5$ with biologically realistic inhibitory proportions, where two of five seeds converge to positive attractors with margin ${+}0.10{-}0.22$ (Cohen's $d{=}0.71$, one-sided $p{=}0.08$); (ii)~target-selective associative recall in paired $(A, B)$ memory at $K{\geq}5$, where the full model retrieves $B$ from a partial cue of $A$ while the minimal model echoes $A$ (Pearson margin $Δ{=}{+}0.163$ at $K{=}5$); (iii)~reduced cross-seed variance of the full model below the minimal baseline under clean upstream, with ratios $1.0{-}3.0$. These three signatures are architecture-specific: they appear consistently across independent regimes and are absent from the minimal control.

High-capacity associative memories based on Kernel Logistic Regression (KLR) achieve strong retrieval performance but typically require substantial computational resources. This paper investigates the compressibility of KLR Hopfield networks to clarify the geometric principles underlying their robust representations. We present a geometric interpretation based on spontaneous symmetry breaking and Walsh analysis, and examine it through compression experiments involving quantization and pruning. The experiments reveal a clear asymmetry: the network remains robust under low-precision quantization while exhibiting strong sensitivity to pruning. We interpret this behavior through a "sparse function, dense representation" principle, in which a sparse input mapping is implemented through a dense bimodal parameterization. These findings suggest a practical route toward hardware-efficient kernel associative memories and provide insight into the geometric principles underlying robust representation in neural systems.

Standard transformer architectures learn fixed slow-weight representations during training and lack mechanisms for rapid adaptation within an episode. In contrast, biological neural systems address this through fast synaptic updates that form transient associative memories during inference, a property known as Hebbian plasticity. In this paper, we conduct an empirical study of Hebbian Fast-Weight (HFW) modules integrated into multiple transformer backbones, including ViT-Small, DeiT-Small, and Swin-Tiny. We evaluate six model variants: ViT, DeiT, Swin, ViT-Hebbian, DeiT-Hebbian, and Swin-Hebbian on 5-way 1-shot and 5-way 5-shot classification tasks using the Omniglot benchmark under a Prototypical Network meta-learning framework. We propose a single module placement strategy for Swin-Tiny in which one HFW module is applied to the final stage feature map after all hierarchical stages have completed. This design avoids the training instability caused by placing separate Hebbian modules at each stage and achieves the highest test accuracy across all six models (96.2% at 1-shot; 99.2% at 5-shot), outperforming its non-Hebbian baseline by $+0.3$ percentage points at 1-shot. We analyze the interaction between Swin's shifted window inductive bias and episode-level Hebbian binding, discuss why per-block placement fails for ViT and DeiT variants in a low-data regime, and situate the results within the wider literature on fast and slow-weight meta-learning.

Constitution-conditioned post-training can be analysed as a structured perturbation of a model's learned representational geometry. We introduce ATLAS, a geometry-first program that traces constitution-induced hidden-state structure across charts, models, and substrates. Instead of treating the relevant unit as a single behaviour, neuron, vector, or patch, ATLAS tests a local chart whose tangent structure, occupancy distribution, and behavioural coupling can be measured under system change. On Gemma, the anchored source-local chart captures 310 / 320 reviewed source rows and all 84 / 84 reviewed score-flip rows, but compact exact-patch sufficiency does not close, so the exportable unit is the broader source-defined family. Freezing that family, we re-identify a target-local realisation in an unadapted Phi model, where the fully adjudicated confirmatory contrast separates with AUC 0.984 and mean gap 5.50. In held-out ALM8 mouse frontal-cortex perturbation data, the same source-defined family receives support across 5/5 folds, with mean held-out AUC 0.72 and mean fold gap 4.50. A multiple-choice analysis provides the main boundary: nearby target-local signals can appear without source-faithful closure. The resulting correspondence is not coordinate identity, site identity, or a target-side mediation theorem. It is geometric recurrence under redistribution: written constitutions can induce recoverable latent geometry whose organisation remains detectable across model and substrate changes while its local coordinates, occupancy, and behavioural expression shift.

Experimental evidence indicates that intrinsic temporal dynamics operating across multiple time scales are closely associated with the emergence of periodic spatial activity of increasing complexity. However, how information encoded in grid-like firing patterns for path integration is processed across these intrinsic time scales remains unclear. To address this question, we introduce adaptive time scales through a leak term in recurrent neural networks (RNNs), forming leaky RNNs discretized from the continuous attractors of firing rate models. Our results demonstrate that leaky RNNs substantially enhance the emergence of well-defined and highly regular hexagonal firing patterns. Compared with vanilla RNNs lacking a leak term, the trained leaky RNNs produce more accurate position estimates while generating reliable grid-cell-like representations. Furthermore, under identical noise conditions, leaky RNNs consistently exhibit more stable dynamics and better-defined grid structures. The learned dynamics also give rise to stable torus attractors with a clear central hole, supporting robust and regular grid-like activity. Overall, the dynamic leak acts as a low-pass filtering mechanism that protects recurrent neural circuitry from noise, stabilizes network dynamics, and improves path-integration accuracy in recurrent neural networks.

Sequential associative memories (SAMs) are difficult to build and maintain in real-world streaming environments, where observations arrive incrementally over time, have imbalanced sampling, and non-stationary temporal dynamics. Vector Symbolic Architectures (VSAs) provide a biologically-inspired framework for building SAMs. Entities and attributes are encoded as quasi-orthogonal hyperdimensional vectors and processed with well defined algebraic operations. Despite this rich framework, most VSA systems rely on simple additive updates, where repeated observations reinforce existing information even when no new information is introduced. In non-stationary environments, this leads to the persistence of stale information after the underlying system changes. In this work, we introduce the Sequential Relevance Memory Unit (SRMU), a domain- and cleanup-agnostic update rule for VSA-based SAMs. The SRMU combines temporal decay with a relevance gating mechanism. Unlike prior approaches that solely rely on cleanup, the SRMU regulates memory formation by filtering redundant, conflicting, and stale information before storage. We evaluate the SRMU on streaming state-tracking tasks that isolate non-uniform sampling and non-stationary temporal dynamics. Our results show that the SRMU increases memory similarity by $12.6\%$ and reduces cumulative memory magnitude by $53.5\%$. This shows that the SRMU produces more stable memory growth and stronger alignment with the ground-truth state.

We present causal evidence that hallucination in autoregressive language models is an early trajectory commitment governed by asymmetric attractor dynamics. Using same-prompt bifurcation, in which we repeatedly sample identical inputs to observe spontaneous divergence, we isolate trajectory dynamics from prompt-level confounds. On Qwen2.5-1.5B across 61 prompts spanning six categories, 27 prompts (44.3%) bifurcate with factual and hallucinated trajectories diverging at the first generated token (KL = 0 at step 0, KL > 1.0 at step 1). Activation patching across 28 layers reveals a pronounced causal asymmetry: injecting a hallucinated activation into a correct trajectory corrupts output in 87.5% of trials (layer 20), while the reverse recovers only 33.3% (layer 24); both exceed the 10.4% baseline (p = 0.025) and 12.5% random-patch control. Window patching shows correction requires sustained multi-step intervention, whereas corruption needs only a single perturbation. Probing the prompt encoding itself, step-0 residual states predict per-prompt hallucination rate at Pearson r = 0.776 at layer 15 (p < 0.001 against a 1000-permutation null); unsupervised clustering identifies five regime-like groups (eta^2 = 0.55) whose saddle-adjacent cluster concentrates 12 of the 13 bifurcating false-premise prompts, indicating that the basin structure is organized around regime commitments fixed at prompt encoding. These findings characterize hallucination as a locally stable attractor basin: entry is probabilistic and rapid, exit demands coordinated intervention across layers and steps, and the relevant basins are selected by clusterable regimes already discernible at step 0.

Hallucinations in large ASR models present a critical safety risk. In this work, we propose the \textit{Spectral Sensitivity Theorem}, which predicts a phase transition in deep networks from a dispersive regime (signal decay) to an attractor regime (rank-1 collapse) governed by layer-wise gain and alignment. We validate this theory by analyzing the eigenspectra of activation graphs in Whisper models (Tiny to Large-v3-Turbo) under adversarial stress. Our results confirm the theoretical prediction: intermediate models exhibit \textit{Structural Disintegration} (Regime I), characterized by a $13.4\%$ collapse in Cross-Attention rank. Conversely, large models enter a \textit{Compression-Seeking Attractor} state (Regime II), where Self-Attention actively compresses rank ($-2.34\%$) and hardens the spectral slope, decoupling the model from acoustic evidence.

Physical selectors (lasers choosing a mode, Ising machines settling on a ground state, condensates occupying a spin state) produce high-dimensional signatures at the moment of decision: full field amplitudes, multimode interference patterns, or scattering responses. These signatures are richer than the winner's index, yet they are routinely discarded. We show that when the signatures are repeatable across trials (stereotyped) and linearly independent across routes, a single linear decoder compiled from calibration data maps them to arbitrary payloads, merging selection and memory access into one event and eliminating the fetch that dominates latency and energy in sparse routing architectures. The construction requires one SVD of measured device responses, which certifies capability and bounds worst-case error for any downstream payload before the task is chosen. Runtime error separates into two independently diagnosable channels, decoding fidelity (controlled by dictionary conditioning $σ_{\min}(Φ)$) and routing reliability (controlled by the margin-to-noise ratio $Δ/T_{\mathrm{eff}}$), each with a distinct physical origin and targeted remedy. We derive the full error decomposition, give Ising-machine selector constructions, and validate the predicted scalings on synthetic speckle-signature simulations across three measurement modalities. No hardware demonstration exists; we provide a falsifiable four-step experimental protocol specifying what a first experiment must measure. Whether real device signatures satisfy stereotypy is the central open question.

Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/δ))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

Neural networks are widely used, yet their analysis and verification remain challenging. We present a Lean~4 formalization covering both deterministic and stochastic models. We first formalize Hopfield networks -- recurrent networks that store patterns as stable states -- and prove their convergence, and the correctness of Hebbian learning, the rule that updates parameters to encode patterns. We then turn to stochastic networks, whose probabilistic updates converge to a stationary distribution: we formalize the dynamics and learning of Boltzmann machines and prove their ergodicity -- convergence to a \emph{unique} stationary distribution -- via a new formalization of the Perron--Frobenius theorem.

An associative memory (AM) enables cue-response recall, and it has recently been recognized as a key mechanism underlying modern neural architectures such as Transformers. In this work, we introduce the concept of distributed dynamic associative memory (DDAM), which extends classical AM to settings with multiple agents and time-varying data streams. In DDAM, each agent maintains a local AM that must not only store its own associations but also selectively memorize information from other agents based on a specified interest matrix. To address this problem, we propose a novel tree-based distributed online gradient descent algorithm, termed DDAM-TOGD, which enables each agent to update its memory on the fly via inter-agent communication over designated routing trees. We derive rigorous performance guarantees for DDAM-TOGD, proving sublinear static regret in stationary environments and a path-length dependent dynamic regret bound in non-stationary environments. These theoretical results provide insights into how communication delays and network structure impact performance. Building on the regret analysis, we further introduce a combinatorial tree design strategy that optimizes the routing trees to minimize communication delays, thereby improving regret bounds. Numerical experiments demonstrate that the proposed DDAM-TOGD framework achieves superior accuracy and robustness compared to representative online learning baselines such as consensus-based distributed optimization, confirming the benefits of the proposed approach in dynamic, distributed environments.

Associative memory, traditionally modeled by Hopfield networks, enables the retrieval of previously stored patterns from partial or noisy cues. Yet, the local computational principles which are required to enable this function remain incompletely understood. To formally characterize the local information processing in such systems, we employ a recent extension of information theory -- Partial Information Decomposition (PID). PID decomposes the contribution of different inputs to an output into unique information from each input, redundant information across inputs, and synergistic information that emerges from combining different inputs. Applying this framework to individual neurons in classical Hopfield networks we find that below the memory capacity, the information in a neuron's activity is characterized by high redundancy between the external pattern input and the internal recurrent input, while synergy and unique information are close to zero until the memory capacity is surpassed and performance drops steeply. Inspired by this observation, we use redundancy maximization at each neuron as an information-theoretic learning goal. This dramatically increases the network's memory capacity to 1.59, a more than tenfold improvement over the 0.14 capacity of classical Hopfield networks, and also outperforming recent state-of-the-art implementations of Hopfield networks. Overall, this work establishes redundancy maximization as a new design principle for associative memories and opens pathways for new associative memory models based on information-theoretic goals.

Existing data poisoning attacks on retrieval-augmented generation (RAG) systems scale poorly because they require costly optimization of poisoned documents for each target phrase. We introduce Eyes-on-Me, a modular attack that decomposes an adversarial document into reusable Attention Attractors and Focus Regions. Attractors are optimized to direct attention to the Focus Region. Attackers can then insert semantic baits for the retriever or malicious instructions for the generator, adapting to new targets at near zero cost. This is achieved by steering a small subset of attention heads that we empirically identify as strongly correlated with attack success. Across 18 end-to-end RAG settings (3 datasets $\times$ 2 retrievers $\times$ 3 generators), Eyes-on-Me raises average attack success rates from 21.9 to 57.8 (+35.9 points, 2.6$\times$ over prior work). A single optimized attractor transfers to unseen black box retrievers and generators without retraining. Our findings establish a scalable paradigm for RAG data poisoning and show that modular, reusable components pose a practical threat to modern AI systems. They also contribute to interpretability research by revealing a strong link between attention concentration and model outputs.