8.4 Exploration as an integrable path

Exploration is not merely a sequence of locally legal moves. In the DIAL setting, the current conceptual space is a model of a tensor-product repair theory \(\mathbb A\otimes \mathbb B\). A tangent probe proposes a direction; an exploratory trajectory requires those directions to compose into a finite path that remains inside the declared model space.

The relevant obstruction depends on the domain. A smooth path may encounter monodromy or fail to extend for the required time. A learned policy may be one-step improving but unstable under rollout. A local adapter law may fail when transported across linguistic charts. A causal proposal may cease to close under sequential interventions. These are all failures of finite realization, although they need not share one mathematical integrability theorem.

An explorer should therefore maintain two records: the local tangent evidence that chooses its next move and a finite-realization observer that audits the path already traversed. In this setting it is an integration observer: approximate holonomy, path-order disagreement, recurrence, multi-step regret, or held-out continuation can instantiate the latter. If the observer fails, the system should localize whether the problem is a poor move, a poor integration model, or a genuine limitation of the current theory.

. Exploratory creativity follows an admitted finite path inside a fixed theory. Local probe success chooses candidate directions; it does not certify that those directions compose into a coherent journey.