Completeness of Extraction from Data

Prove that, in the absence of selection bias, every kernel that is directly identifiable from data is representable as $\mu(\graph,L)$ for a backdoor-free family of embeddings, allowing orbit-based symmetries and multi-level structure where required.

Background

The paper proves that backdoor-free families of embeddings yield identifiable structured kernels. The converse remains unresolved: whether all directly identifiable kernels arise from such families. The conjecture would establish completeness of the proposed extraction framework, rather than merely its soundness.

References

In the absence of selection bias, if a kernel $\mu$ is directly identifiable (Def.\ \ref{def:identification_direct_mt}) from data, then there is a backdoor-free family of embeddings on $(\graph,L)$ with $\mu = \mu(\graph, L)$, at least for an orbit-based notion of symmetry, see §\ref{apdx:symmetries} and after including multi-level structure (see appendix, §\ref{apdx:extraction_from_data}).

— Symmetries and Causality: Causal Effect Identification Beyond IID Data  (2609.03697 - Rabel et al., 3 Sep 2026) in Conjecture 3, Section 3.3, “Identifiability from Embeddings”