Lesson 2: Stabilization and Drift

Goal

Explain what stabilization and drift mean and how they produce a stable layer of recognitions.

Stabilizer $\sigma$

The stabilizer is the “settling” map on each fiber $H_t$. It identifies recognitions that are already stable under repeated checking or normalization.

Operationally:

• ${\sigma }_t$ is monotone and idempotent.
• ${\sigma }_t(a)$ is the stabilized version of $a$ at trace $t$.

If ${\sigma }_t(a)=a$, then $a$ is already stable.

Drift $\Delta$

The drift operator represents unavoidable change or decay in recognitions over time or across context.

Operationally:

• ${\Delta }_t$ is monotone and inflationary (it moves recognitions forward).
• ${\Delta }_t(a)$ is the “drifted” form of $a$ at trace $t$.

If ${\Delta }_t(a)=a$, then $a$ survives drift at that trace.

Fixed fibers $H_t^{\ast }$

The fixed fiber at trace $t$ is:

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

These are the recognitions that both stabilize and resist drift. The collection of all $H_t^{\ast }$ is the stable layer $H^{\ast }$.

Why this matters

Without stabilization and drift, GFRTU would only describe raw local recognitions. The fixed layer is what you can rely on globally:

• it filters out transient recognitions,
• it identifies invariants across traces,
• and it provides a stable basis for higher structure.

Sanity checks

• ${\sigma }_t$ and ${\Delta }_t$ should commute.
• fixed fibers should be closed under reindexing.
• fixed fibers should assemble into a subsheaf $H^{\ast }$.