Proposition: Finite adaptive soundness

A finite adaptive tree of isometries, unitaries, destructive basis readouts, and declared discards defines a reference-preserving CPTP channel. Every unreturned dilation factor belongs to its complete complement.

Proof. Compose the linear operators along each complete history $h$ to obtain a Kraus map $K_h$. Completeness of each readout and unitarity or isometricity of each coherent step imply, by backward induction,

The reported channel is a sum of the maps $X\mapsto K_{h}X{K}_h^{†}$, with orthogonal classical labels for reported histories and with any additional unreported indices traced out. This is CPTP. A dilation retaining every index is isometric. Tensoring with ${\mathrm{id}}_Q$ proves reference preservation.