Results and dependencies

The paper proves six interface statements.

1. On a prescribed substrate, block channels have an exact finite-step backwards support bound. Retiming a fixed labelled gate partial order preserves the total channel.

2. Under spectator-blind control and resources, any admissible spectator extension with the same accessible reduction preserves the geometric trace and that reduction at every step.

3. Immediate destructive readout agrees with a restricted coherent-then-readout implementation after both outputs are restricted to the same record–matter interface.

4. Record laws across evolution and label aggregation agree exactly when their effects obey a pullback identity. Their unassisted worst-case total-variation error has an exact event-norm formula.

5. Record agreement does not imply residual-instrument agreement. For sharp measurements with a coarse Lüders instrument, an exact criterion isolates the remaining intra-outcome coherence.

6. Agreement on an input observable algebra is preserved as agreement on an output observable algebra exactly when the adjoint continuation maps the latter into the former. A corresponding statement is given for classical block-diagonal state spaces.

Finite-dimensional channel conventions follow Watrous Watrous2018. The circuit principle of deferred measurement has a precise treatment in Gurevich and Blass GurevichBlass2022. Our emphasis is the additional observation boundary: geometry, control, resource tests, and accessible residual observables are all named.

The geometric certificate uses a fixed conical spacetime and the polygonal-slicing setting of DeserJackiwHooft1984,tHooft1992,tHooft1993,Kadar2005. The proofs of the interface theorems do not depend on that certificate. The certificate shows that one nontrivial change of cell incidence can meet the declared interface; it does not supply the assumptions of the interface theorems.