The mitochondrion produces ~10¹⁴ ATP molecules per day in a human body. The classical chemiosmotic model explains this through stochastic proton diffusion — but given the extreme macromolecular crowding (~500 mg/mL protein concentration), random diffusion appears insufficient to explain observed power densities. This paper proposes that the answer lies in geometry: the mitochondrion is a precision-tuned proton waveguide whose fractal architecture is governed by √2. Three independent lines of evidence converge. First, the mammalian ATP synthase c-ring comprises exactly 8 subunits, yielding a proton step angle of 45° where cos(45°) = √2/2. This is the unique integer stoichiometry producing a simple √2 expression, and it achieves the highest bioenergetic efficiency among all characterized ATP synthases (2.67 H⁺/ATP for c₈ versus 3.33–5.67 for other organisms). The c₈ ring is simultaneously a Fibonacci number and √2-optimal — a unique intersection suggesting evolutionary convergence on a mathematical optimum. Second, cryo-electron tomography reveals that cristae dimensions follow √2 scaling with 10% deviation in Parkinson's, Alzheimer's, Leigh syndrome), and ATP efficiency correlating with cristae geometry (r > 0.7).
Thierry Marechal (Sun,) studied this question.