Declared finite data

Choose a finite catalogue

and finitely many real linear diagnostics ${\ell }_j$ on an appropriate real subspace. Allowed examples include:

• the real or imaginary part of a coefficient pairing with a fixed smooth joint test;

• evaluation of a fixed directional descendant at a fixed smooth configuration;

• a boundary pairing ${\int }_{\Sigma }\chi \cdot {\mathfrak{b}}_{\Psi ;\mathbf{\eta }}(F)$;

• the normalized separator ${\mathcal{Q}}_f$.

These maps are continuous by the preceding theorems.

Each finite table entry ${\ell }_j(F_x)$, or a certified enclosure for it, is supplied classical data. No general computability theorem for arbitrary smooth inputs is assumed. In particular, continuity alone is not an algorithm for evaluating an arbitrary real number exactly.

Choose a finite bound $M>0$ such that all selected entries have absolute value at most $M$, and put

for the installed normalized score selected by clock code $a$. A finite report word $r(a,x)$ may additionally store prescribed rounded values and exact catalogue flags. It is a report of installed data, not a universal admission test.

For the catalogue containing $B_f$ and any finitely many regular approximants, ${\mathcal{Q}}_f$ gives exactly the entries one and zero. Its existence and nonzero normalization were proved, not inferred from numerical sampling.