8.3 Local exploration with Weil probes

The Weil profile from Chapter 3 turns the local neighborhood of a theory state into an explicit experimental object. For a finite probe family \(\mathcal B\subseteq \mathsf{Weil}_1^{\mathrm{tan}}\),

records selected directions and relations among them. Exploration chooses a probe, observes a response, updates its local picture, and chooses again.

Different probe shapes support different questions. A first-order probe asks for a sensitive direction. Paired probes reveal interactions. Repeated or higher-order probes test curvature-like change, order effects, or failure of a linear approximation. Null probes identify variations invisible to the current observer. The collection is a typed experimental vocabulary, not a claim that all creative search is literally continuous.

The local-to-finite passage requires care. A tangent direction may not integrate, and a locally promising direction may terminate at a boundary or enter a region where the approximation fails. An exploratory trace should therefore separate:

1. the probe that proposed a direction;

2. the finite operation used to realize it;

3. the observations collected along the path; and

4. the admission test applied at the endpoint.