Axiomatics of Hysteretic Media: Hysteresis as Rule, Isomorphism as Consequence

This preprint formulates an axiomatics of hysteretic media. Five axioms (A1–A5) are introduced describing the structure of a medium with memory: threshold, hierarchy of cycles, coupling, return points, and congruence. From these, the following theorems are derived: selective wiping-out, return-point memory, power-law avalanche statistics, and universality of signatures for the class of media. The Principle of Thresholdness (Principle P) is formulated as a navigator for the directed search of new states. It is shown that the isomorphism of observed structures is the visible form of the manifested rule — hysteresis. Examples of carriers are given: magnetic systems, neuronal cascades, cosmological media, and algorithmic systems. The theory has the status of a falsifiable program with addressable predictions.