arXiv:2611.14645v1 Announce Type: new Abstract: The average rectangular volume ($\ARV$) of a bivariate aggregation function is the average of its volumes over all axis-parallel rectangles of the unit square; for a copula, it is the expected copula volume of the rectangle spanned by two independent uniform points. We determine the exact region of the pairs $(\ARV(C),\rho(C))$, where $C$ ranges over all bivariate copulas and $\rho$ is Spearman's rho. The region is compact, convex and symmetric, and its upper boundary consists of three analytically parametrized arcs. A previously proposed family of shuffles of Min generates one arc, but only on part of its range; the other two arcs are generated by singular copulas whose supports contain hyperbolic branches. The boundary is continuously differentiable away from its left endpoint $\ARV=\frac{11}{120}$, where it has a vertical tangent. The proof uses a folding transformation and explicit Kantorovich potentials for a family of optimal transport problems. Finally, we show that the arachnitude, a rank correlation used in actuarial practice, is an affine transformation of $\ARV$, so that the result also describes the joint region of Spearman's rho and arachnitude.
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