3.9 The declaration card

Before differentiation or optimization, a learning-sketch declaration should answer:

1. What are the formal objects and generating arrows?

2. Which parallel paths are required to agree?

3. Which cones and cocones are designated?

4. What counts as an admissible realized model?

5. What object records failure of the factorization problem?

6. Which observation maps make that failure measurable?

7. Which variations are merely presentational?

8. What evidence would justify a repair rather than abstention?

9. Which parts of the sketch may be augmented, and what would make such an extension conservative?

10. Which edits count as structural interventions, what do they target, and which non-target obligations must they preserve?

A failed factorization can reveal either of two gaps. The declared theory may be adequate while the candidate model realizes it poorly, or the persistent failure may indicate that the sketch itself lacks a generator, relation, or universal construction needed to express the phenomenon. The obstruction alone does not decide between these explanations. We must ask how it changes under admissible variations and where those changes are supported.

The next chapter supplies this sensitivity analysis by applying tangent structure to the declaration. The result is not simply a derivative of the final scalar loss: it is a lifted factorization problem whose obstruction records how compositional failure moves. At first this diagnoses gaps in a model of a fixed sketch. Later, the same localization logic will help identify explanatory pressure on the sketch and motivate a carefully admitted theory augmentation.