Three convergence questions

Refinement must distinguish:

1. convergence of externally calibrated classical readings;

2. convergence of the quantum residual channel after forgetting some or all records;

3. convergence of the complete recorded experiment on a common record–residual interface.

The third implies the relevant marginal channel convergences. Neither the first nor the second, by itself, implies the third. Proposition prop:record-not-residual illustrates the residual gap even when the record laws agree exactly.

Finite classical codes also do not converge to an exact all-real record in total variation merely because their calibrated values become dense. For example, let ${\mu }_j$ be any probability measure with finite support in $[0,1]$, and let $\lambda$ be uniform Lebesgue probability. If $F_j=\mathrm{supp}{\mu }_j$, then

Hence their total-variation distance is one at every scale. A weak limit can still exist, but it is a different convergence claim. Fixed finite bins or a specified bounded-statistic interface can be used instead; the binning or testing map must be declared.

Finally, common finite samples do not select a unique interpolation, generator, or continuum limit. XIX-C's prop:two-limits and prop:interpolation record precisely this non-determination CheungXIXC. Theorem thm:refinement avoids it by freezing the target and restricting the comparison class explicitly.