Proposition: Record and residual criteria

The record laws agree on every input exactly when

Agreement on a declared recorded residual algebra $\mathcal{O}$ holds exactly when

These identities are stable under untouched references.

Proof. Pair each recorded output block with $I$, or with $B$, and move the maps to the input by adjoints. States separate Hermitian effects. Linear extension gives the residual-algebra identities; reference extension tensors the corresponding maps with identity.