ora-0196

17.3.4 Two cross-cutting inverse problems

Categorical dimension theory.

The finite PACC bounds depend on the cardinality of a query quotient. Definition 8.7 begins the analogue of VC dimension by counting independently variable compositional obligations. The open statistical problem is to derive distribution-free or distribution-sensitive bounds from this invariant, or from enriched covering numbers and the homotopy type of the hypothesis fibration. Horn-filling complexity—the evidence needed to distinguish or fill \(\Lambda _k^n\to N_\bullet (\mathcal E_b)\)—is a promising candidate, but a theorem must show that it controls generalization rather than merely presentation size.

The differential realization problem.

Tangent UOCL may identify a tangent category without identifying differential laws that generate it. Section 7.5 therefore leaves an inverse problem: determine when a learned tangent world lies in the essential image of the coEilenberg–Moore construction

\[ \mathsf{Coalg}_{!}: \mathbf{DiffCat}_{\mathrm{adm}}\longrightarrow \mathbf{TanCat}, \]

and, when it does, whether the differential category, modality, and deriving transformation are identifiable from enriched probes. This is the categorical analogue of passing from observed infinitesimal variation to the laws governing that variation.

Open problem	Present checkpoint	Next theorem target
Asynchronous information	Event categories and non-anticipation	Kan invariance without a global clock
Repair obstructions	Horn-filler profiles and localized preservation	Intrinsic obstruction criterion for repair or doctrine change
Lifelong transport	Composable finite transport and finite cores	Nontrivial infinite core and exact transport-defect comparison
Statistical complexity	PACC template and categorical probe dimension	Generalization bounds from compositional or homotopical complexity
Differential realization	Tangent UOCL and a query-blindness obstruction	Essential-image and identifiability criteria for differential laws

Table 17.2 The open ORACLE frontier. The three main programs ascend from information shape to repair geometry to lifelong persistence; statistical complexity and differential realization cut across all three.