Rigidity of τ (non-ω)
The Rigidity theorem (I.T07) states that the τ-kernel admits no non-trivial automorphisms — every endomorphism of τ that fixes K0–K6 is the identity. Combined with the Categoricity theorem (T03 / I.T08), it gives the framework's two-step uniqueness argument: τ exists (K0), is unique up to canonical equivalence (T03), and that equivalence is itself unique (T02). The 'non-ω' qualifier reflects that the result holds in the K6-restricted world; without K6, τ admits ω-shift automorphisms.
τ-Definition
The Rigidity theorem (I.T07) states that the τ-kernel admits no non-trivial automorphisms — every endomorphism of τ that fixes K0–K6 is the identity. Combined with the Categoricity theorem (T03 / I.T08), it gives the framework's two-step uniqueness argument: τ exists (K0), is unique up to canonical equivalence (T03), and that equivalence is itself unique (T02). The 'non-ω' qualifier reflects that the result holds in the K6-restricted world; without K6, τ admits ω-shift automorphisms.
Categorical invariant. Aut(τ) = {id_τ}; the τ-kernel has trivial automorphism group within the K1–K6 signature.
Primary registry anchor:
I.T07
τ-Derivation Chain
Mathematical content
Aut_K(τ) = {id_τ}, where Aut_K(τ) denotes the group of automorphisms of τ that fix the kernel signature K0 + K1–K6.
Proof sketch (expand)
Suppose σ : τ → τ is a kernel-fixing automorphism. By K1 (strict order), σ preserves the kernel atoms' linear order, so σ permutes them order-isomorphically — but K1's strict order has trivial automorphism group, so σ fixes every atom. By K3 (composition), σ extends uniquely to a functor on all morphisms, and the atom-wise identity extends to the identity functor. K6 forbids any non-K1–K5 'jump' that could distinguish σ from id, so σ = id_τ.
Non-ω qualifier. The result is stated as 'non-ω' because without K6, σ could be a non-trivial ω-shift (an automorphism that shifts the kernel atoms by an ω-th iteration of K1). K6 forbids such shifts, and only then is rigidity provable.
Consequence. Combined with Categoricity (T03), Rigidity gives the framework's strongest uniqueness claim: τ is determined up to a UNIQUE canonical equivalence — there is no choice of equivalence even after fixing existence + isomorphism class.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Orbit.Rigidity
Lean kind: theorem
Lean symbol: Tau.BookI.Orbit.rigidityNonOmega
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.