5 Infinitesimal Causality

Chapter 3 supplied the categorical theory, and Chapter 4 supplied a way to discover where one of its models fails under variation. Infinitesimal causality now gives that transition a concrete scientific meaning. The tangent directions become causal only after we say how they are generated by a smooth intervention protocol on a statistical model. The protocol produces vector fields; their Lie brackets test whether the visible intervention directions close under sequential local perturbation.

In LINCS language, the protocol declares the tangent site, the visible span declares what the current model and causal vocabulary can express, and the normal bracket component is a typed obstruction to closure. It may diagnose a gap in the realized causal model, such as a missing visible direction. If the failure survives admissible model repair, it can instead motivate a theory-level proposal, such as adding a latent generator or enlarging the intervention language. The geometry locates the gap; independent evidence must still determine which explanation is admitted.

This chapter also separates two levels of intervention. Infinitesimal causality studies interventions on a modeled domain: a protocol changes a mechanism or regime and induces a tangent response. LINCS also permits interventions on the learning system that represents that domain: changing its visible span, estimator, causal vocabulary, cover, or model class. The first requires causal grounding. The second is a structural repair of the learner. A bracket residual can motivate either level, but cannot by itself decide between them.

¹1. The “Frobenius” in this chapter is the classical theorem on involutive distributions. It is distinct from the Frobenius algebras sometimes added to categorical models of classical data. ↩