Proposition: Finite center-sensitive snapshot

Every state on $\mathfrak{A}(I)$ is uniquely represented by positive matrices

Its sector probabilities and conditional states are

The complete array is affine in the state. The collection of normalized conditional states alone is not a complete snapshot.

Proof. Trace duality on each matrix summand gives the unique positive matrix ${\omega }_{\alpha }$. Evaluation on the unit gives the normalization, and evaluation on $e_{\alpha }$ gives $p_{\alpha }$. Conditional normalization introduces division by $p_{\alpha }$. Without the probabilities, distinct arrays can have the same normalized matrices.