Proposition: Minimal backward boundary

For a fixed horizon and terminal observation spaces, define

These are the smallest observation spaces satisfying (eq:history-closure). Agreement on their initial direct sum is equivalent to agreement of all complete history-conditioned terminal expectations.

For a stationary table and geometry-only reports, iteration from ${{\mathcal{O}}_g}^{(0)}=\mathbb{C}I$ stabilizes after at most $|\mathsf{G}|(d^2-1)$ strict enlargements. The stable space characterizes agreement of all finite geometric history laws.

Proof. Adjoints preserve adjoints, so the recursively defined spans are observation spaces. Any closed family must contain these spans, proving minimality. The recursively generated elements are precisely spans of pathwise pullbacks. The inserted identity contributes no independent condition because ${\sum }_h{\Psi }_{h|g}^{\ast }(I)=I$.

For stationary iteration, the sum of dimensions starts at $|\mathsf{G}|$ and is at most $|\mathsf{G}|d^2$. An iteration without enlargement is already fixed. Path effects generate exactly the stable space.