sec-weil-profile

3.5 A Weil profile of a theory state

For a selected family \(\mathcal A\subseteq \mathsf{Weil}_1^{\mathrm{tan}}\), define the Weil profile of a theory state \(X\) schematically by

\[ \mathcal P_{\mathcal A}(X) =\{ F(A)X,\ F(u)_X\mid A,B\in \mathcal A, \ u:A\to B\} . \]

The profile contains not just individual perturbations but their relations under maps and monoidal composition. It can therefore record questions that a single gradient cannot ask:

Which first-order directions are admissible at \(X\)?
Which pairs of directions can be compared or combined?
Does changing their order alter the resulting observation?
Do locally legal variations close inside the current declaration?
Which variations are null under the chosen observer?

An empirical implementation never observes this entire family. It chooses a finite probe basis, realizes it computationally, and applies observers. Probe selection is therefore part of the creativity architecture rather than a neutral preprocessing step.