Concrete example first
In a fiber (H_t={\bot<m<\top}):
- stabilization (\sigma_t) sends (m\mapsto\top),
- drift (\Delta_t) also sends (m\mapsto\top),
- both keep (\bot) and (\top) fixed.
Then only (\bot) and (\top) survive both dynamics. That surviving set is (H_t^*).
Formal lesson content
For each trace (t):
- (\sigma_t:H_t\to H_t) is monotone and idempotent,
- (\Delta_t:H_t\to H_t) is monotone,
- (\sigma_t\circ\Delta_t=\Delta_t\circ\sigma_t).
The fixed fiber is [ H_t^={a\in H_t\mid \sigma_t(a)=a=\Delta_t(a)}. ] Across traces, these fixed fibers must respect reindexing so they assemble into a subsheaf (H^).
Why this lesson matters
GFRTU is not only about local logic; it is about dynamic coherence. Stabilization and drift tell you which recognitions persist under repeated consolidation and evolution.
Checks you should run
- Verify monotonicity of both maps.
- Verify idempotence of (\sigma_t).
- Verify commutation between (\sigma_t) and (\Delta_t).
- Verify reindexing preserves fixed points.