Proposition: Exact measured-snapshot error

The classical flagged channel has

Thus its endpoint error at $q=0$ is $b_d$, not zero. For a pure input $|\psi ⟩={\sum }_i{\alpha }_i|i⟩$, its snapshot record entropy is

A family of snapshot entropies is not an append-only history entropy.

Proof. Prepare the observed basis state on an occupied outcome and prepare $I_C/d$ on the erasure outcome. The composite channel is

The triangle inequality gives the upper bound in (eq:measuredflag).

On the Bell input, every occupied classical branch has conditional reference state $|i⟩⟨i|$, so its contribution to Bell overlap is at most $(1-q)/d$, summed over all occupied outcomes. The erasure contribution is $q/d^2$. The Bell test gives

proving optimality. The entropy formula follows from the classical orthogonal flags.