GFRTU Processes: What You Can Apply

This document provides a whole-object workflow for using GFRTU. It does not support selecting isolated components.

Whole-object workflow

Follow this sequence only if the entry conditions from curricula/overview.md are satisfied.

Step 1: Specify the trace site

• Define a small category of traces $T$.
• Define a Grothendieck topology $J$ that encodes coverage and gluing.
• Verify that $T$ is small and that $J$ satisfies the topology axioms.

Output: a trace site $(T,J)$.

Step 2: Attach recognition fibers

• For each trace, define a finite Heyting algebra $H_t$.
• Define stabilizer ${\sigma }_t$ and drift ${\Delta }_t$ on each fiber.
• Provide reindexing maps along trace morphisms.
• Verify that reindexing preserves the Heyting structure and commutes with ${\sigma }_t$ and ${\Delta }_t$.

Output: a fibered recognition system over the trace site.

Step 3: Stabilize recognitions

• Compute fixed points of ${\sigma }_t$ and ${\Delta }_t$ at each trace.
• Check that fixed fibers assemble into a subsheaf.

Output: the stabilized layer $H^{\ast }$.

Step 4: Close the universe

• Define the closure operator $UG$ over objects in the sheaf topos.
• Iterate from $\varnothing$ to reach the least fixed point.

Output: the GFRTU as a minimal closed universe.

Step 5: Localize into cells (optional)

• Restrict the trace site to a full subcategory $U$.
• Form the induced sheaf topos $\mathbf{Sh}(U)$.

Output: a cell (local GFRTU).

Misuse warnings

• Do not apply Steps 2-4 without a valid trace site and topology.
• Do not apply stabilization or closure to data that is not fibered over $T$.
• Do not treat cells as independent universes unless they are induced from a sub-site of the original $T$.