Why a fixed finite amplitude interface is insufficient

This rules out a specified finite linear mode encoding, not every nonlinear finite-parameter dictionary. In particular it does not contradict Theorem thm:scaled, whose entire energy dependence outside one scalar parameter is supplied by the reference model.

A bare normalized finite-dimensional state encoding also cannot equate a fixed probability with energy at all amplitudes: normalization erases $f\mapsto cf$, whereas $Q(cf)=|c{|}^{2}Q(f)$. An amplitude register, variable gain, nonlinear response, or another specified resource is required.