Segmentation Theory: A Formal Account of Irreversible Structure

Author note (August 2026): The formal core of this preprint is under substantial reconsideration following further analysis. The paper will be revisited in light of these issues. The abstract below describes the currently posted version, which should not be regarded as the author's current final formulation. This paper develops Segmentation Theory, a formal framework for analyzing irreversibility in chains of structural interaction. Instead of treating irreversibility as a temporal property, the theory characterizes it structurally through the loss of reconstructable positions. Each structural state x is associated with a set Return(x) representing the states from which the same configuration can be reconstructed. When the structural conditions required for reconstruction expand, the set of reconstructable states necessarily shrinks under a minimal non-triviality condition. This structural contraction defines segmentation. From this relation, a segmentation order is derived, which generates segmentation cones and an Alexandrov topology on the quotient space of structural states. Segmentation Theory therefore provides a formal account of irreversibility as a structural phenomenon arising from the expansion of reconstruction conditions rather than from temporal direction alone.