Theorem: Exact recovery of the installed predecessor reports

Identify an X-H-ST residual-report or reciprocity-report interface with the same:

Then the recovered channel is exactly (eq:continuation) on all input operators and every reference extension.

If an X-K-ST finite continuation is appended on matching interfaces, its complete history channel is the same channel obtained by composing these maps.

Proof. Theorem thm:instrument identifies each subnormalized block on all operators. Applying the identical ${\Lambda }_k$ gives equality of every recorded output block. Finite induction gives equality after identical complete continuations.