lin-0035

1.4 The semantic core

Definition 1.5 INC category

An INC category is a tangent category \((\mathcal C,T)\) equipped with:

  1. a class of tangent learning sketches;

  2. a class of admissible models \(D:J\to \mathcal C\) for each sketch;

  3. base factorization problems \(\operatorname {Fact}_{\mathbb S}(D)\);

  4. transported tangent problems \(\operatorname {Fact}_{\mathbb S}(TD)\); and

  5. when declared, an interaction signature \(\Omega _D\) acting on an admissible tangent distribution \(\mathcal A_D\).

The factorization definition of infinitesimal non-compositionality is

\[ \operatorname {INC}(D) = \mathcal O_{\mathbb S}(TD). \]

An interaction signature may additionally contain brackets, connection-based derivatives, curvature, canonical-flip comparisons, or higher jets. These refine tangent diagnosis; they do not replace the factorization problem.

Definition 1.6 LINCS structure

A LINCS structure on an INC category registers

\[ \bigl( \operatorname {Fact}_{\mathbb S}(D), \operatorname {Fact}_{\mathbb S}(TD) \bigr) \]

is treated as the basic learning datum for every admissible model. When \(\Omega _D\) is declared, the datum is refined by its associated closure or profile problems. Observation maps, scalarizations, repair languages, and admission contracts are additional registered structure.

This is deliberately a definition of structure, not yet of a new category. A category of LINCS systems additionally requires a specified class of objects—for example admissible pairs \((\mathbb S,D)\)—and morphisms that transport declarations, factorization problems, observers, and repair authority. Different applications choose different morphisms. The common structure therefore defines typed learning problems without pretending that one application-independent category of repairs has already been fixed.