Chapter 54: The Orbit-Set Correspondence

Part VIII built an internal set theory on a single orbit: τ-Idx, equipped with ∈_τ = ∣ . But Category τ has five orbits (the relevant theorem, Proposition [prop:orbit-disjoint]), and each orbit carries generators with distinct structural roles. This chapter extends internal set theory from single-orbit arithmetic to a five-orbit representation theory. We define an orbit-set map Set : Obj(τ) → P(Obj(τ)) that assigns to every τ-element a structurally determined set. The resulting correspondence is injective, partitions τ into “opaque” (self-containing) and “transparent” (organizing) elements, and reveals that the boundary between finite and infinite sets coincides exactly with the radial-solenoidal divide.