11.2 Exact closure and typed repair

An exact mechanism study isolates the structural step before finite-sample estimation. DIAL-SKFM–1A represents infinitesimal mechanisms by integer linear vector fields and computes their commutator directly. Projection onto the declared generator span separates assimilable coefficients from a closure residual. Within the registered synthetic semantics, an observed residual support may type a missing directed mechanism; a central residual supported on one latent coordinate may type a latent common-cause bundle. Residuals requiring two latent coordinates or a noncentral extension lie outside the registered one-generator repair budget and must elicit abstention.

In this finite study, projection outside the declared span together with the exhaustive registered repair table supplies the nonrealizability gate of Chapter 10. The gate is valid only for that exact finite language: it does not prove that an arbitrary causal representation lacks an in-theory explanation.

Across eighteen admission and eighteen confirmation worlds, the typed rule recovers every registered repair, restores closure, preserves the old generators, satisfies Jacobi, produces no false extension in closed worlds, and abstains on every unsupported case. The raw-residual control is the key negative result: adding the commutator always restores closure, but yields only \(1/6\) exact typed repairs and never abstains. Closure is therefore a diagnostic and integration condition, not by itself a causal interpretation.

Experiment: DIAL-SKFM–1A. World: exact integer linear vector fields in three observed and two latent coordinates.
Obstruction: commutator residual outside the declared generator span.
Construction: directed edge or one-latent central extension, typed from residual support; otherwise abstain.
Result: 18/18 admission and 18/18 confirmation repairs, with perfect closure, preservation, Jacobi, and unsupported-case abstention.
Boundary: exact fully observed operators; no trajectory estimator or unregistered repair type.