τ-Stone Space
The τ-Stone Space (II.D14) is the τ-categorical analogue of the classical Stone space — the labelled topological space underlying the K2 boundary axiom. It is the structural environment in which the algebraic lemniscate (T09) lives and on which Truth4 logic (D06) realizes as topological semantics. With 17 incoming edges, the τ-Stone Space is Books II's topological substrate.
τ-Definition
The τ-Stone Space (II.D14) is the τ-categorical analogue of the classical Stone space — the labelled topological space underlying the K2 boundary axiom. It is the structural environment in which the algebraic lemniscate (T09) lives and on which Truth4 logic (D06) realizes as topological semantics. With 17 incoming edges, the τ-Stone Space is Books II's topological substrate.
Categorical invariant. A compact totally-disconnected τ-categorical topological space equipped with the K2 labelling.
Primary registry anchor:
II.D14
τ-Derivation Chain
Mathematical content
A τ-Stone Space is a τ-categorical topological space (X, τ_X) that is: (a) compact, (b) Hausdorff, (c) totally disconnected, (d) equipped with the K2 boundary labelling. Every τ-categorical boundary structure is a τ-Stone Space (up to canonical iso).
Consequences:
- Algebraic Lemniscate (T09) lives as a sub-space of a τ-Stone Space.
- Truth4 logic (D06) realizes as continuous T4-valued functions on a τ-Stone Space (Stone duality at four-valued level).
- K2 labelled boundary's topological content is exactly the τ-Stone Space data.
Lean Coverage
Status: Formalized
Module: TauLib.BookII.Topology.StoneSpace
Lean kind: structure
Lean symbol: Tau.BookII.Topology.StoneSpace