What is encoded

Fix a finite catalogue of smooth configurations, tests, and prescription labels. For each catalogue entry $i$ and coefficient order $j=0,1,2$, the classical number

is mathematically defined by the preceding construction. An installed table stores finitely many rational approximations, intervals, or exact values that have been supplied independently.

Neither existence of these numbers nor a finite lookup circuit is an algorithm for acquiring arbitrary smooth fields. No universal finite encoding of real numbers or smooth configurations is assumed.

The catalogue is nonempty without any numerical execution: take $m=0$, $h=k=0$, ${\varphi }_0=0$ or $1$, ${\varphi }_1={\varphi }_2=0$, and a fixed test with integral one. The exact coefficient triples are respectively $(0,0,0)$ and $(1,0,0)$. These are evaluations of the classical functional, not measurements on a continuum quantum field.