The Nontrivial Structure of the Transition–Event Interface: Eligibility Fibers, the Coverage Map and the Set-Valued Representation of Transitions (Structural Series, Paper I)

Abstract

This paper develops a formal mathematical description of the interface between transitions and events. The central problem is to characterize the structural conditions under which an event can be associated with a transition without assuming that the transition–event correspondence is one-to-one, exhaustive, or structurally trivial. We introduce a transition space, an event space, and an eligibility relation that determines which events are structurally compatible with a given transition. This relation induces an eligibility fiber over each transition and separates the interface into structurally distinct classes of events. The resulting construction shows that the interface is not adequately represented by a simple pointwise correspondence; it possesses an internal set-theoretic structure. We prove that relations and set-valued maps are equivalent descriptions; that the coverage map factors canonically through the quotient by fiber-equality, so that distinct transitions may collapse onto one event-side structure; that the degenerate interface is exactly the product relation, and count the nondegenerate ones in the finite case; and that every nonempty family of subsets of the event space is realizable as a coverage image. The last result shows that closure, lattice and related properties cannot follow from the interface alone. The paper provides the formal starting point for the subsequent study of structural emptiness, closure, separation, trace and information loss.