Derivation of Kleiber-Type 3/4 Scaling from Optimal Space-Filling Transport Networks in Three Dimensions

Abstract. Kleiber's Law, commonly expressed as B ∝ M³ᐟ⁴, has long been treated as an empirical regularity in biological scaling. Here I show that the exponent β = 3/4 arises necessarily within a broad and physically important class of three-dimensional, space-filling transport networks subject to geometric packing and flow-optimisation constraints. The argument does not require organism-specific biological fitting; instead, it depends on a small set of explicit structural assumptions: hierarchical branching, terminal-unit invariance, three-dimensional space filling, and dissipation minimisation under conserved transport. Three mutually reinforcing routes are presented: dimensional scaling, constrained flow optimisation, and numerical network sampling. All converge on the same scaling exponent in the admissible 3D regime. The result does not claim that every conceivable biological system must obey exact β = 3/4 scaling, but rather that 3/4 is the mathematically preferred exponent for optimised volume-serving transport networks embedded in three-dimensional space. In that sense, Kleiber-type scaling is not an arbitrary biological coincidence, but the expected consequence of transport geometry.