Determine whether natural corpora contain sampler-independent conditional dependencies

Determine whether natural language or other natural-data corpora contain conditional dependencies that prevent independent-update diffusion samplers from selecting exact multi-position write groups using their available per-position distributions.

Background

Theorem 2 establishes that per-position distributions cannot certify whether a selected group is conditionally independent, although a particular target distribution may nevertheless permit the groups selected by a sampler. The ScanAndAdd experiment closes this gap for a synthetic distribution by explicitly exhibiting dependent command and answer groups and showing that confidence rankings encounter them.

The authors explicitly leave unresolved whether analogous dependencies, with comparable consequences for exact parallel decoding, occur in any natural corpus. This is distinct from the general theoretical impossibility because it asks for empirical or mathematical characterization on natural data distributions.