Concrete example first
Take (H_t={\bot<m<\top}), with stabilizer sending (m\mapsto\top), and drift sending (m\mapsto\top) while fixing (\bot,\top).
Then:
- (m) is not fixed by either map,
- (\bot) and (\top) are fixed by both.
So the stable-at-trace subset is (H_t^*={\bot,\top}).
Formal definition
The fixed layer (H^) is assembled from fiberwise fixed sets [ H_t^={a\in H_t\mid \sigma_t(a)=a=\Delta_t(a)} ] across all traces (t).
When reindexing preserves fixed points, these (H_t^) form a subsheaf (H^\subseteq H). This is the dynamically coherent layer of recognitions in GFRTU.
Why it matters in GFRTU
- It isolates recognitions stable under both consolidation and evolution.
- It provides the base for the stability locale and defect analysis.
- It is the canonical “reliable core” used by higher constructions.