A Substrate Independent Scaling Law

Metabolic rate scales as body mass to the 3/4 power. This holds across 27 orders of magnitude — from bacteria

to whales — and is one of the most universal quantitative patterns in biology. It is called Kleiber's law.

West, Brown, and Enquist (1997) derived this exponent from the geometry of vascular networks. Blood vessels

branch, fill space, and that branching pattern produces 3/4. The argument is elegant, but it has two problems.

First, it requires blood vessels. Bacteria, fungi, and unicellular organisms have no vascular networks — yet they

obey the same scaling law. The derivation cannot apply to them.

Second, the derivation contains a circular step. The resonance velocity condition — a key piece of the proof —

already assumes the 3/4 exponent before deriving it. The answer is embedded in the setup. This was identified

by Dodds (2010) and Banavar et al. (2002).

Banavar, Maritan, and Rinaldo (1999, 2010) provide a cleaner route through optimal transport theory. For any

system distributing resources from sources to N sinks embedded in d-dimensional space, the minimum-cost

network scales as C ~ N(1+1/d). Set d=3 and the 3/4 exponent follows directly. No vascular geometry required.

But the Banavar result still needs a reason why d=3 is the relevant dimension. For organisms with circulatory

systems, you can point at physical space. For bacteria and fungi, that answer is unexplained. The question this

paper addresses: can the dimension of the constraint network be derived from first principles, without

reference to any physical substrate?