A finite payload for both separations

Use a two-qubit clock for the four histories of (eq:path-table), with a declared dictionary from its basis labels to:

• the fine history and its coarse terminal slice;

• its finite branch certificate $(k,q,r,T(k,q),J_{\mu },\mathrm{loc})$;

• a selected residual audit, for example the $X$-measurement with projectors $(I\pm X)/2$.

The $X$-audit can be represented with one fresh pointer: rotate to the computational basis, copy to the pointer, rotate the surviving qubit back, and destroy the pointer. Its residual disturbance is exactly the declared Lüders instrument.

The history encoder is the channel

It can be obtained by retaining the actual source history and preparing its corresponding clock basis code. A coherent normal form

has the same final destructive readout because ${\sum }_h{L}_h^{†}{L}_h=I$. This equality is not permission to move a geometry update past a required intermediate record.

The one-qubit witness of Theorem thm:residual-obstruction is another finite setting, with one coarse outcome and an $X$-audit. It reads expectations $1$ and $0$ for the two ideal residuals.

Neither finite encoder is an acquisition map from a continuum field or an unknown metric to the register. The payload is a finite instrument and a finite certificate table, not a finite-qubit field instrument.