Generative Closure (UG)

Definition

The generative closure operator $UG$ is a function on subsets of objects in the sheaf topos $R=\mathbf{Sh}(T,J)$. It adds whatever objects are forced to exist by the recognition structure and logical requirements. The GFRTU is the least fixed point of $UG$ above $\varnothing$.

Plain-language meaning

$UG$ is the “do we need to add more?” operator. You start with nothing, then repeatedly add everything that must exist. When nothing new is forced, you have the minimal universe.

Minimal structure

• A sheaf topos $R$ based on the trace site.
• A specification of which objects must exist for recognitions and logic.
• A closure rule $UG$ that adds required objects.

Entry conditions

Use generative closure only when:

• You can define what objects are required for recognitions and logic.
• You can iterate $UG$ to a fixed point.

Example

If $UG$ says “include the terminal object and recognition object,” then $UG(\varnothing )$ adds those. If applying $UG$ again adds nothing, the universe is stable.