4 Tangent Learning Sketches

The base obstruction says which structural promise has failed, but not whether the gap lies in a particular model or in the theory used to describe it. Its tangent lift asks how that failure changes under admissible infinitesimal perturbations. LINCS calls the resulting obstruction infinitesimal non-compositionality, or INC.

This chapter develops the model-level construction: the sketch is fixed while its realization varies. That restriction is methodologically important. Sensitivity within \(\operatorname {Mod}(\mathbb S,\mathcal C)\) can localize a defective map, representation, or interaction. A failure that persists across all registered model directions instead creates evidence that the available model class or sketch may be incomplete. It does not by itself license a new axiom. Theory augmentation is the higher-level repair problem developed in Chapter 22.

¹1. A tangent lift preserves the type of the declaration. It does not turn every available derivative into a legitimate learning signal. ↩