Proposition: A residual audit enlarges the boundary

The initial diagonal algebra is sufficient for all geometric histories in the two-step model. If the fine-history residual observable $X$ is additionally audited on every branch, the minimal initial observation space is the full algebra $\mathcal{B}({\mathbb{C}}^2)$.

Proof. The diagonal sufficiency was proved above. For the two middle histories,

are $4/25$ times rotations of $X$ through opposite angles $\pi /4$ in the $X,Y$ plane. Their sum and difference span $X,Y$. The history effects already span $I,Q$. The minimal backward construction of Proposition prop:minimal therefore gives all four qubit matrix directions.