What convergence of process data can mean

For two processes on the same frozen interface define

This is an operational probability-comparison quantity for the specified one-use tests. It is not a diamond norm on an unspecified global channel.

A family of approximating data ${\mathbf{W}}^{(n)}\in {\mathcal{P}}_{3q}$ could be compared by, for example,

or by the stronger uniform statement

A future argument must name its topology and prove the convergence it uses. Neither statement is built into Definition def:class.

Even convergence of all strengths does not automatically yield well-defined maps between the component quotients. A positive limiting strength is eventually witnessed by positive approximating strengths, but a zero limiting strength may have positive approximating strengths at every scale, as in Proposition prop:strength-stability. Component splitting or merging therefore requires its own exact support or compatibility theorem.