Corollary: Finite stabilization of the all-history boundary

Set ${{\mathcal{O}}_g}^{(0)}=\mathbb{C}I$ and

The family stabilizes after at most $|\mathsf{G}|(d^2-1)$ strict enlargements. Its stable value is the smallest branch-invariant observation family containing the classical effects. Agreement on its direct sum is equivalent to agreement of all finite geometric history laws.

Proof. The dimensions are nondecreasing and their sum is at most $|\mathsf{G}|d^2$, starting at $|\mathsf{G}|$. If one iteration does not enlarge the family, all subsequent iterations are identical. This proves the bound and branch invariance.

The recursively generated elements are spans of finite path effects. Conversely, every finite path effect appears at some iteration. The equivalence follows from the trace pairing with those effects.