Proposition: Finite compression limit

Suppose a sequence of admitted smooth data packets converges on common compact dependency regions, with the same observation tests and prescriptions. For the sourced family, require common compact scalar-data support and the same Newton branch. Then the finite table entries converge.

If ${A_k}^{(n)}$ are the resulting diagonal matrices and

the complete instruments formed with a common sufficiently large $B$ satisfy

Proof. Lemma lem:wave, smooth multiplication and differentiation, and compact integration prove continuity of the coordinate and scalar evaluations. For a common compact spatial source support,

The locally integrable Newton kernel bounds this expression uniformly on each compact target by a source smooth seminorm. Thus the declared initial preparation, its wave evolution, and the sourced evaluations are continuous as well. The gap evaluation is another compact smooth pairing.

Choose $B$ so that all limiting and approximating effects under consideration have spectrum in $[1/4,3/4]$. Because the matrices are diagonal,

using $|\sqrt{a}-\sqrt{b}|\le |a-b|$ on that interval. For contractions $L,\tilde{L}$,

Sum this estimate over the eight recorded Kraus branches and divide by two. This gives (eq:instrument-limit).