6.10 Scope and mathematical boundaries

The claims of the construction are modular and observe the following boundaries.

1. A pointwise tangent lift is a theorem about the instance category; it is not a theorem that an arbitrary physical database is differentiable.

2. Tangent-stable path equations do not imply preservation of every limit, join, aggregation, or query.

3. A vector field becomes causal only through a grounded intervention protocol.

4. A Lie bracket requires additional bracket-admissible structure and is not produced by the tangent functor alone.

5. Bracket non-closure detects a gap relative to a visible span. It does not distinguish latent confounding from misspecification, estimation error, rank change, or an incomplete protocol.

6. Storing a residual does not make two charts comparable. Frame transport, quotients, uncertainty, and provenance belong in the data model.

These boundaries are productive rather than merely cautionary. Each becomes a declared database constraint and therefore a possible obstruction. The database does not just hold the output of infinitesimal learning; it becomes an actionable model of how that output was constructed, compared, repaired, and admitted.