Categoricity of τ (non-ω)
The Categoricity theorem (I.T08) states that any two structures satisfying the τ-kernel axioms K0 + K1–K6 are canonically equivalent. Together with the Rigidity theorem (T02 / I.T07), it gives the framework's two-step uniqueness argument: τ exists, is unique up to canonical equivalence, and that equivalence is itself unique. The 'non-ω' qualifier reflects that without K6, multiple non-isomorphic K0 + K1–K5 models exist; K6 is the closure axiom that makes categoricity provable.
τ-Definition
The Categoricity theorem (I.T08) states that any two structures satisfying the τ-kernel axioms K0 + K1–K6 are canonically equivalent. Together with the Rigidity theorem (T02 / I.T07), it gives the framework's two-step uniqueness argument: τ exists, is unique up to canonical equivalence, and that equivalence is itself unique. The 'non-ω' qualifier reflects that without K6, multiple non-isomorphic K0 + K1–K5 models exist; K6 is the closure axiom that makes categoricity provable.
Categorical invariant. Any two K0 + K1–K6 structures M, N admit a canonical equivalence Φ : M ≃ N — uniqueness up to canonical iso.
Primary registry anchor:
I.T08
τ-Derivation Chain
Mathematical content
Any two structures M, N satisfying the τ-kernel axioms K0 + K1–K6 admit a canonical equivalence Φ : M ≃ N. The equivalence is canonical in the sense that it is determined uniquely by the K1–K5 generator structure — no choice is involved.
Proof sketch (expand)
By induction on the K1-trace of M (= the K1-trace of N, since both satisfy K1): the kernel atoms in M correspond to the kernel atoms in N under the unique strict-order isomorphism. Extending this canonical correspondence via K3 composition (and K2 boundary identification) gives a functor Φ : M → N that is bijective on all morphisms. K6 closure ensures Φ is surjective on the entire model; rigidity (I.T07) ensures Φ is well-defined. Combining these gives Φ as a canonical equivalence M ≃ N.
Non-ω qualifier. Without K6, the K0 + K1–K5 signature admits multiple non-isomorphic models (each obtained by extending the kernel with a different ω-th generator). K6 forbids such extensions, and only then is categoricity provable.
Consequence. The τ-framework speaks of 'τ' as if τ were a unique determinate object. Categoricity is the theorem that justifies that usage.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Orbit.Saturation
Lean kind: theorem
Lean symbol: Tau.BookI.Orbit.categoricityNonOmega
Cross-domain bridges
This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.