# Infinitesimal Causality

arXiv: https://arxiv.org/abs/2606.24621 · HTML: https://mlsanity.com/paper/2606.24621/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 27 references that are in the map.
- First posted on arXiv 23 Jun 2026.

## Builds on (most cited first, 9 of 9)

- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Causal de Finetti: On the Identification of Invariant Causal Structure in Exchangeable Data](https://mlsanity.com/paper/2203.15756/) (2022)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)
- [Do Finetti: On Causal Effects for Exchangeable Data](https://mlsanity.com/paper/2405.18836/) (2024)
- [Enumerating Markov Equivalence Classes of Acyclic Digraph Models](https://mlsanity.com/paper/1301.2272/) (2013)
- [Latent Confounded Causal Discovery via Lie Bracket Geometry](https://mlsanity.com/paper/2606.19610/) (2026)
- [Causal Density Functions](https://mlsanity.com/paper/2606.00754/) (2026)

## Built on by (most cited first, 2 of 2)

- [Learning in Infinitesimal Non-Compositional Sketches](https://mlsanity.com/paper/2607.15107/) (2026)
- [Activation-Space Order-Swap Geometry: A Site-Asymmetry Audit](https://mlsanity.com/paper/2608.25315/) (2026)

## Abstract (arXiv)

Interventions can be varied continuously in many causal models. Differentiating a specified smooth intervention protocol produces vector fields on a statistical model, and their Lie brackets describe the noncommutativity of the corresponding local perturbations. We formulate this differential geometry of interventions on smooth statistical models and call the resulting framework infinitesimal causality (IC). Given a constant-rank distribution spanned by visible intervention fields, we define the normal Lie-bracket residual and show that its vanishing is exactly the involutivity condition in the classical Frobenius theorem. We establish the coordinate invariance of the zero-residual property and characterize its dependence on the intervention protocol, visible span, and metric. Fully observed and latent-variable examples delineate the additional structural assumptions needed to interpret bracket residuals causally. We also distinguish tangent vectors on a statistical parameter manifold from derivatives of stochastic kernels. In the finite-state linearization of a Markov category, normalization and copy compatibility yield well-typed first-order defects. Normalization is automatic for differentiable paths of stochastic kernels, whereas copy compatibility characterizes a more restrictive deterministic or comonoid-preserving perturbation. Together, the geometric and kernel-level constructions make IC a precise foundation for Lie-bracket-based causal diagnostics and identify the assumptions required to pass from local intervention geometry to causal conclusions.