Fixed Layer

Definition

The fixed layer $H^{\ast }$ is the collection of recognitions that are fixed by both stabilizer and drift. At each trace $t$:

$$H_t^{\ast }=\{a\in H_t:{\sigma }_t(a)=a={\Delta }_t(a)\}.$$

Plain-language meaning

These are the recognitions that are stable and also resist drift. They are the dependable core across traces.

Minimal structure

• Stabilizer and drift on each fiber.
• Fixed points computed for each fiber.
• Reindexing maps preserve fixed points.

Entry conditions

Use the fixed layer only when:

• Stabilizer and drift commute.
• Fixed points exist and can be computed.
• Fixed points assemble into a consistent subsheaf.

Example

If $H_t=\{\perp <m<\top \}$ and ${\Delta }_t(m)=\top$, then $m$ is not fixed, but $\perp$ and $\top$ are. So $H_t^{\ast }=\{\perp ,\top \}$.