Finite data do not enumerate all key protocols

Theorem thm:statistics concerns one named observable and specified preparations. It does not validate all protocol outputs for every $n$.

For example, if complete single-copy input preparations differ by trace distance at most $r$, the elementary tensor telescoping bound is

This does not leave a nonzero uniform gap for all $n$ at a fixed positive preparation tolerance. If Eve's purification or side information changes, even the complete comparison interface must first be supplied.

A finite proof of a universally quantified theorem can use structural lemmas. A finite empirical sample cannot replace those lemmas merely by being called a certificate.