Records and local apparatuses

For a finite experiment, gauge-neutral pointer and controller registers can be reserved locally. Tensoring the constrained system with these declared neutral registers does not impose a spatial tensor decomposition on the gauge degrees of freedom.

At finite matrix interfaces, a local instrument can be recorded using a local pointer and the isometry

Completeness gives the isometry identity. Recording or dephasing the pointer yields the direct-sum instrument of Proposition prop:records. For finite-dimensional local algebras, this local isometry can be completed blockwise to a unitary on a sufficiently large reserved neutral apparatus. All such apparatus factors must be retained in the locality ledger Stinespring,Watrous.

Consequently Theorem thm:onelayer applies to the complete record-retaining implementation, including later locally controlled continuations. It is not a theorem about an isolated selectively normalized branch.