Generic distinctness of DAT conditional-independence constraints

Establish that, for generic choices of the noise densities f1,…,fM used in the Differentiable Adjacency Test (DAT) soft-subset representation defined by \tilde Z_{\psi,m} = \psi_m Z_m + (1-\psi_m) N_m, the family of conditional-independence constraints p(X in A, Y in B | \tilde Z_\psi) = p(X in A | \tilde Z_\psi) p(Y in B | \tilde Z_\psi) for all measurable sets A and B and for all values of \tilde Z_\psi remain distinct and therefore cannot all be satisfied for any parameter vector \psi that is not close to the indicator vector of a separating set SepSet(X, Y).

Background

The paper introduces the Differentiable Adjacency Test (DAT), which relaxes the discrete separating set selection problem into a differentiable separating representation search by mixing each candidate conditioning variable Z_m with independent noise N_m via a parameter \psi_m. This creates a soft-subset representation \tilde Z_\psi used to test conditional independence X ⟂ Y | \tilde Z_\psi.

Theorem 1 shows equivalence between the separating set selection and separating representation search when the noise distributions f_m have sufficiently thick tails, and recovering a separating set by selecting variables with \psi_m=1. However, away from this thick-tail regime, it is unclear whether the relaxed problem avoids spurious separating representations.

The conjecture asks to show that for generic choices of the noise densities f_m, the infinitely many conditional-independence equalities defining X ⟂ Y | \tilde Z_\psi remain distinct, so they cannot all hold unless \psi is close to a binary selector corresponding to an actual separating set. Proving this would clarify identifiability and robustness of the DAT relaxation beyond the specific thick-tail conditions.