Optimal horoball packing densities for Koszul-type tilings in hyperbolic 3-space

We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic 3-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of packings that attain the universal simplicial density upper bound d₃(∞) = ( 2 √3 Λ(π/3) \! )⁻¹ ≈ 0.853276, d 3 ( ∞ ) = ( 2 3 Λ ( π 3 ) ) - 1 ≈ 0.853276 , where Λ Λ denotes the Lobachevsky function. These results show that extremal packing densities in H³ H 3 are realized by multiple explicit Coxeter tilings and are closely tied to special values of L -functions and hyperbolic manifold volumes.