Bergsma--Dassios Sign Covariance Characterises Independence for Arbitrary Real-Valued Bivariate Laws

Bergsma--Dassios sign covariance $τ^*$ is a rank-based population measure of dependence. Building on zero-characterisation results under specific regularity regimes, we prove that $τ^*(X,Y)=0$ characterises independence for every real-valued bivariate distribution, including mixed and singular laws. For the unnormalised four-sample convention for $τ^*$ defined in Subsection 4.3 and the unscaled Blum--Kiefer--Rosenblatt functional $\mathscr {B}$, the proof gives the quantitative inequality $τ^*\ge 2\mathscr {B}$. This is a population identification result; no new sample-level limit theorem is claimed. The argument first encodes finite ordered distributions with rational cell probabilities by labelled path trees and applies a nonnegative sum-of-squares representation for a quartet covariance. Rational approximation and nested quantisation then remove all support and regularity restrictions. On finite uniformly weighted label sets, the tree framework also relates an edge-weighted quartet quantity to empirical distance covariance squared. As a separate combinatorial consequence, it yields the asymptotic $2/3$ upper bound for the quartet distance between binary phylogenetic trees.