5.1 A Three-Level Hierarchy

The construction relates three mathematical levels.

1. A Markov level contains stochastic kernels together with the copy and discard operations of classical information [ Fritz , 2020 ] .

2. A statistical level contains a smooth family \(\theta \mapsto P_\theta \), its scores, and the Fisher metric [ Amari , 2016 ] .

3. An intervention level contains specified local protocols and the vector fields they induce on the statistical manifold.

This hierarchy is distinct from Pearl’s ladder of association, intervention, and counterfactual reasoning [ Pearl and Mackenzie , 2018 ] . Pearl’s hierarchy classifies the kinds of causal questions a model can answer. The hierarchy above instead separates the mathematical structures needed to give a local causal perturbation its semantics. The two classifications therefore cross-cut one another. A grounded intervention protocol can support Pearl’s interventional level, but counterfactual claims require additional structure that relates outcomes across incompatible interventions; the three levels listed here do not supply that structure by themselves.

These levels communicate through an explicit realization. A tangent vector to a parameter manifold is not itself a stochastic kernel, and a derivative of a kernel is generally a signed map rather than a Markov kernel. Likewise, an arbitrary statistical perturbation is not automatically an intervention.

This hierarchy also instantiates the declaration–realization distinction of Section 0.13. A causal vocabulary and its intervention protocols provide explicit structural commitments; learned statistical models and estimated vector fields realize them. Internal intuitionistic reasoning preserves unresolved or context-dependent causal claims, while bracket and integrability obstructions expose where the realization outruns the declared causal language. Neither the neural estimate nor the symbolic declaration is sufficient by itself.