Multifractality in the Tree of Life: A Branching-Process RIFS Proof

Abstract: We study a branching-process random iterated function system (RIFS) that formalizes the foundational principles of nestedness, duality, and randomness in the living tree of life (Hudnall & D'Souza, 2025). In this construction, each leaf of a branching process generates a subtree at a strictly smaller contraction scale, thereby unifying classical branching processes and random IFS theory in a single framework. We prove rigorously that this branching-process RIFS is multifractal under explicit, mild assumptions. Two variants are analyzed: a non-anchored case with a nontrivial compact attractor, and a biologically motivated anchored case in which the invariant set collapses to a point while tangent measures obey the same multifractal law. Thus, multifractality emerges as a necessary mathematical consequence of nestedness, duality, and randomness, yielding a minimal-condition theorem that explains the ubiquity of multifractal signatures in biological data.