Worker Utility as Hysteresis: A Preisach Model of Transaction Acceptance in Gig Labour Markets

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.

Preisach Attention: A Hysteretic Model of Sequential Memory

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.