Proposition: Meaning of the finite encoding

On payload input $|a⟩⟨a|$ at setting $k$, the scored outcome $By$ has expectation $b_{ka}$. More generally, on a payload density matrix ${\rho }_X$, its mean is $\mathrm{Tr}(A_k{\rho }_X)$.

Gauge-related source jets have the same first three table entries. The fourth row records the nonzero plateau mismatch rather than eliminating it.

Proof. The score identity (eq:score) gives the mean. Gauge independence of the first three rows follows from Theorem thm:descent; linearity in the witness scalar gives the fourth row.