Bergsma--Dassios sign covariance with ties: a nonnegative decomposition and sharp bounds

We construct tie-symmetrised extensions $\widetilde D$ and $\widetilde R$ of the unnormalised Hoeffding and Blum--Kiefer--Rosenblatt functionals $D$ and $R$ and prove, for every real-valued bivariate law, the exact nonnegative decomposition $ τ^{*}(X,Y)=12\widetilde D(X,Y)+24\widetilde R(X,Y), \qqua