Hysteresis

Humans have always externalized thought onto tools, from the tally and the abacus to the map and, now, large language models. I model the agent, the tool, and the task as one dynamical system in which competence (what the user retains) and reliance (what the user outsources) co-evolve, and find that the outcome is bistable. Above a critical tool availability the competent state is destroyed and competence collapses toward a low dependent floor as the user outsources completely. Lowering availability does not reverse the collapse until a far lower threshold, so history of practice rather than the current tool fixes the state. Two users with the same present access can therefore occupy opposite and lasting states, one competent and one dependent, decided only by which they built first. The collapse threshold depends jointly on the competence a user brings to a task and on the tool's transparency, the fraction of its working a user can reconstruct. In the case where an agent faces an uncertain goal, a tool can cause agency itself to transfer to the tool and the human-agent becomes an agentic-instrument, irreversibly, because the tool's model is too large to internalize. The model is tested against several independent data sets, including GPS and map use, arithmetic expertise, and language models. These results reframe how tools should be built, how artificial intelligence is deployed, and what a tool-resistant education might require.

This study introduces a particle-based computational framework to investigate the scalability of morality and the systemic decoupling of formal law from decentralized social ethics in expanding populations. While micro-societies reinforce ethical conduct through local reciprocity, macroscale systems introduce anonymity that strains cognitive memory limitations. We model individual agents as discrete particles with finite memory capacities ($L$) and dynamically evolving, stochastic choice profiles ($μ$) regulated by non-linear social pressure switches. Monte Carlo ensemble simulations demonstrate a distinct, non-linear phase transition as the population scales ($N \to \infty$). When the population metric outpaces memory capacity ($N \gg L$), the local re-encounter probability drops as $\mathcal{O}(L/N)$. This structural dilution neutralizes decentralized peer-to-peer accountability, causing global behavioral norms to decouple from moral baselines and drift toward a minimalist legal floor. Furthermore, cyclic scale experiments expose a prominent, path-dependent hysteresis loop, mathematically formalizing the non-Markovian inertia and irreversible nature of moral decay in self-organizing social systems.

Classical theories of cognition describe problem solving as exploratory search through structured problem spaces in which repeated interaction gradually compresses search into efficient representational structures. Predictive artificial intelligence systems introduce a distinct regime in which stabilization may occur before exploratory diversification unfolds, supplying solutions and decision trajectories prior to internally generated search. This paper develops a geometric dynamical framework in which attention evolves over a landscape of strategies shaped by stabilizing drift, endogenous exploratory perturbation, and responsiveness-gated learning. Predictive assistance is modeled as a process of exogenous exploratory compression that stabilizes trajectories before self-generated exploration broadens the accessible regions of strategy space. The framework yields three main results. First, sustained predictive stabilization reduces exploratory responsiveness by attenuating the effective influence of intrinsic perturbations even when exploratory variability remains present. Second, curvature accumulates and relaxes asymmetrically, producing hysteresis and delayed recovery of exploratory mobility after assistance withdrawal. Third, developmental outcomes depend critically on the timing of stabilization, with early intervention narrowing future exploratory traversal before broad representational diversification has occurred. The framework generates empirically testable predictions concerning exploratory entropy, premature convergence, and delayed recovery following predictive stabilization. More broadly, the results suggest that predictive systems may reshape the geometry of exploratory cognition itself.

Worker utility is not observed -- only its consequence is. Each gig transaction produces a single bit: accepted or rejected. We argue this structure points directly to the Preisach hysteresis model as the natural representation of latent worker preferences. The Preisach operator models aggregate output as an integral over a population of binary threshold elements -- precisely the structure that emerges when heterogeneous workers each carry a private acceptance wage. We estimate two latent utility surfaces: acceptance utility U_1(X) and rejection utility U_0(X), via a dual-output neural network (shared layers 256->128, margin loss enforcing U_1 >= U_0). Classification reduces to the Preisach gap U_1(X) - U_0(X), passed into an XGBoost classifier alongside clip-stabilised price-to-threshold encodings. On 36,891 gig transactions, this pipeline achieves Jaccard = 0.827 and ROC AUC = 0.799. The price-to-threshold encoding accounts for +11.0 pp AUC over raw utility features. The model confirms the directional asymmetry hysteresis predicts: price decreases depress completion rates more than equivalent increases raise them. Applied to the full dataset, the model's recommendations simultaneously reduce the total wage bill by 21.3% and increase expected fill rate by 9.7 pp. For 74.2% of transactions, P(accept) already exceeds 0.80; reducing the wage keeps it above threshold (mean post-cut P = 0.972), releasing cost savings (median 31%). For the remaining 25.4%, a median 7% wage increase recovers +43 pp acceptance. A model without an explicit indifference zone cannot execute both moves simultaneously.

We introduce the Preisach Attention Layer (PAL), a novel sequence modelling architecture grounded in the classical Preisach hysteresis operator from mathematical physics. PAL replaces the softmax attention mechanism with a binary relay operator parameterised by learned activation and deactivation thresholds, maintaining a stack of local extrema as its internal state. A single-layer PAL-Transformer with O(1) depth is Turing-complete under arbitrary precision arithmetic, achievable through simulation of a two-stack pushdown automaton -- in contrast to the O(log n) depth required by standard hard-attention transformers. Second, we prove that the function classes computable by PAL and by the transformer are incomparable: PAL computes historical range statistics in O(1) layers that require O(log n) layers for transformers, while transformers support random-access retrieval that PAL cannot perform without auxiliary state. The separating property is rate-independence -- PAL responds only to the sequence of local extrema, not to absolute token positions or temporal spacing. Third, we show that the extremum stack constitutes a minimal sufficient statistic of the input history for all rate-independent functionals, providing a formal analogue of the wiping property in classical hysteresis theory. PAL is thus an efficient architecture for tasks with long episodic memory and weak positional dependence, with O(n log n) total inference cost versus O(n^2) for standard attention.

We prove that the extremum stack of a discrete sequence is a minimal sufficient statistic for the class of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish K(Pi_n) - O(1) <= K_R(u_{0:n}) <= K(Pi_n) + O(1), where K_R(u_{0:n}) is the length of the shortest program answering every query in the class R, and the O(1) overhead is independent of both the sequence length n and the stack depth k. Sufficiency follows from the classical wiping property of the Preisach hysteresis operator. Minimality is established via a finite indicator family whose rate-independence is verified explicitly. Any compression of a hysteresis-driven stream that preserves the full class R must therefore retain at least K(Pi_n) - O(1) bits; the stack-based compression algorithm implied by the result carries a Kolmogorov optimality guarantee that none of the standard time-series compression methods provide.

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 monostability trap: in the long-time regime, information leakage is unavoidable and the dynamics converge to a single global consensus attractor. We propose the 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.

Operational disruptions in retail payments can induce behavioral responses that outlast technical recovery and may amplify liquidity stress. We propose a multi-agent model linking card payment outages to trust dynamics, channel avoidance, and threshold-gated withdrawals. Customers and merchants interact through repeated payment attempts, while customers additionally influence one another on a Watts-Strogatz small-world network. Customers update bounded memory variables capturing accumulated negative experience (scar) and perceived systemic risk (rumor), with merchants contributing persistent broadcast signals that may lag operational recovery. We prove that, under mild conditions on memory persistence and threshold gating, aggregate withdrawal pressure can peak strictly after the outage nadir, including during the recovery phase. Simulations reproduce behavioral hysteresis and confirm delayed peaks of outflows. We further study payment substitution via instant transfer: substitution consistently reduces peak avoidance, yet its effect on cumulative outflows is non-monotonic under realistic merchant broadcast persistence. Robustness experiments across random seeds show stable qualitative behavior. The model highlights why "status green" is not equivalent to risk resolution and motivates incident response strategies that address perception, merchant messaging, and post-recovery communication in addition to technical remediation.