Concrete example first
Let (H_t={\bot<m<\top}). Define
- (\sigma_t(\bot)=\bot),
- (\sigma_t(m)=\top),
- (\sigma_t(\top)=\top).
Applying (\sigma_t) once pushes uncertain recognition (m) to a stable value (\top). Applying it again changes nothing. That is stabilization.
Formal definition
For each trace (t), a stabilizer is an endomorphism (\sigma_t:H_t\to H_t) that is:
- monotone: (a\le b \Rightarrow \sigma_t(a)\le \sigma_t(b)),
- idempotent: (\sigma_t(\sigma_t(a))=\sigma_t(a)).
In GFRTU, stabilizer maps are part of the primitive recognition dynamics and must be compatible with reindexing across traces.
Why it matters in GFRTU
- It models convergence to stable recognitions.
- It defines one half of the fixed-layer condition (\sigma_t(a)=a).
- Together with drift, it determines the stable core that survives dynamics.