lin-0032

1.1 What an axiomatization must distinguish

Let

\[ \mathbb S=(S,\mathcal D,\mathcal L,\mathcal K) \]

be a learning sketch, let \(J=\operatorname {Path}(S)\), and let

\[ D:J\longrightarrow \mathcal C \]

be an admissible realization in a tangent category \((\mathcal C,T)\). The commutativity data induce

\[ q_{\mathcal D}:J\longrightarrow J/{\sim _{\mathcal D}}. \]

The designated cones and cocones add limit and colimit obligations. We write \(\operatorname {Fact}_{\mathbb S}(D)\) for the combined factorization problem.

Four levels must remain separate.

  1. The declaration specifies what is required to compose.

  2. The realization supplies a candidate model of that declaration.

  3. The observation makes some aspect of failure detectable.

  4. The admission rule decides whether evidence licenses a change.

Collapsing these levels makes an axiomatization circular. If a proposed repair may silently rewrite its observer, or if an observer may silently redefine the declaration, the system can remove evidence of failure without repairing the failure.

Boundary

The axioms define structural learning relative to a declaration. They do not declare the sketch correct, the cover complete, the observers sufficient, or the admitted policy normatively acceptable. Those are explicit modeling and validation commitments.