Abstract

Why do such different systems — quantum measurements, thermal ensembles, Bayesian updates and large language models — keep producing probability mappings with strikingly similar structure? We introduce relational probability fields: distributions over discrete possibilities shaped by context-dependent relational potentials and resolved by collapse, sampling, or flow. We show that the Born rule, Boltzmann statistics and softmax sampling can be expressed within a single template: potentials induce an outcome distribution, and resolution instantiates particular outcomes. The central contribution of this paper is a set of interface style descriptors — concentration, context sensitivity and memory — that characterise how resolution behaves across domains. These descriptors do not replace domain-specific formalisms; they form a comparative layer that makes cross-domain differences and similarities tractable. Two systems may share the same abstract template while exhibiting very different interface styles: one may generate sharp, low-sensitivity distributions with little memory; another diffuse, high-sensitivity distributions with strong history dependence. Together, these descriptors operationalise what we call a system’s relational mode: its characteristic way of turning contextual relations into realised outcomes. We provide explicit definitions and an alignment map that shows how the template and descriptors apply in quantum measurement, statistical mechanics, Bayesian updating and next-token prediction. Built on this structural foundation, we sketch a speculative coherence-based interpretation: probabilities as measures of relational fit between configurations and their context, and collapse as a form of coherence optimisation. The paper is primarily conceptual: it does not propose a new microphysical theory, nor does it derive the Born rule or Einstein’s equations from a single principle. Instead, it offers a compact cross-domain language designed to generate tractable questions at the interface of quantum theory, statistical mechanics and machine learning — with possible extensions discussed in the outlook.