Corollary: Record distributions

The coherent and sequential executions have the same joint distribution $\{p_r\}$ of established record values. Conditional probabilities among those record variables agree whenever the conditioning event has positive probability.

Proof. Record-value projectors belong to ${\mathcal{A}}_{\mathrm{inv}}$. Their probabilities are $p_r$ by the normal form. Classical conditioning then uses the same joint distribution.