6.9 Appearances, variations, and abductive repair

McCarthy’s distinction between appearance and reality asks how an intelligent system can move from partial observations to stable objects and mechanisms that need not be present in the observational vocabulary [ McCarthy , 2007 ] . A base database instance \(I\) records one organized appearance: observations, extracted claims, an admitted model, and their provenance. Its tangent lift \(T_{\mathcal S}I\) enlarges that record with possible first-order variations. Registered vector fields make selected variations persistent and queryable; brackets record how pairs of such variations interact.

This enlargement is important, but it is not yet a database of counterfactuals. A tangent vector is a possible local variation in the chosen semantics. It acquires an interventional or counterfactual reading only when it is tied to a protocol, a mechanism replacement, and an identification contract. Under those conditions, an infinitesimal database can retain both what was observed and how a declared family of nearby interventions would change it. In that qualified sense it supplies part of the data infrastructure for a synthetic laboratory.

The distinction also sharpens the experience–jump–axiom cycle proposed by Zahavy [ 2026 ] . The jump should not be identified with the tangent lift itself. Tangent and bracket data diagnose the limits of the current declaration; the abductive step changes that declaration. A database-level reading is:

Experience.

An admitted instance \(I:\mathcal S\to \mathcal C\) organizes observations, contexts, and provenance under the current schema.

Diagnosis.

Tangent instances, registered fields, and bracket residuals locate variations that fail to satisfy an equation, close in a visible span, or commute with a query.

Jump.

A typed repair \(j:\mathcal S\to \mathcal S^{+}\) adds or revises an object, generator, path, constraint, protocol, or latent mechanism.

Axiomatization and admission.

Constraints on \(\mathcal S^{+}\) state the enlarged theory. Old evidence is transported, new consequences are computed, and the repair is admitted only after its finite and held-out implications survive the declared tests.

Thus the database does more than store a succession of models. It retains the typed obstruction that motivated a revision, the map from the old schema to the new one, and the evidence transported across that map. The “jump” is a schema migration supported by infinitesimal evidence, not an untyped leap outside the representational language.

Kan extension gives a precise account of one part of this migration. Suppose an intervened sub-schema \(\mathcal A\) is included by \(k:\mathcal A\hookrightarrow \mathcal S^{+}\), and a locally specified replacement is represented by \(Q:\mathcal A\to \mathcal C\). When it exists,

is the universal colimit-based propagation of that local datum to the expanded schema. This construction does not manufacture the intervention or certify its causal meaning. It propagates a replacement already supplied with an admissible intervention semantics. Causal validity still depends on the protocol, consistency conditions, identification assumptions, and admission tests.

Finally, the need for bracket enrichment marks the difference between individual and interacting variations. The tangent functor supplies first-order directions; it does not by itself supply the Lie bracket needed to compare their order of action. Once bracket-admissible fields have been declared, non-closure can reveal that the current visible span is insufficient. Such a residual can motivate an abductive schema expansion, but it does not force one: misspecification, estimation error, rank change, and an incomplete protocol remain competing diagnoses.