The Inverted Sun:√2 Geometry as a Universal Optimum for Multi-Scale Energy Transport. From Photosynthetic Excitons to Mitochondrial Solitons

Photosynthesis and mitochondrial respiration are textbook biochemical inverses. This paper identifies a deeper symmetry: they share the same geometry. Both systems exhibit √2 scaling in their energy transport architectures. In photosynthesis, LHCII inter-pigment distances follow a √2 geometric series (mean error 4.76%, p < 0.014). In mitochondria, cristae dimensions follow √2 ratios (1% deviation). Both systems reject their "middle energy" input: chlorophyll rejects green light (the geometric mean of the visible spectrum), and Complex II of the electron transport chain pumps zero protons despite accepting electrons — the only non-pumping complex. In both cases, the middle rejection maintains the energy gradient that drives the system. The paper proposes that this convergence reflects a universal optimization principle: √2 is the optimal scaling ratio for multi-scale energy transport networks because it provides two simultaneous properties. Noncollision between adjacent levels (no resonance trapping — energy doesn't get stuck) and octave connectivity between alternate levels (enabling trans-scale transport — energy can cross multiple scales efficiently). The energy carriers differ — excitons in LHCII, Grotthuss proton solitons in mitochondria — but the network geometry is the same because the transport optimization problem is the same. The structural analogy is presented quantitatively as two "spectral batteries": photosynthesis operates between red and blue poles with the green pivot rejected, respiration operates between NADH and O₂ reduction potentials with the succinate (Complex II) pathway bypassing the gradient. The noncollision property is derived from Diophantine approximation theory (the same Parity Lemma proved in the companion √2 Noncollision Foundation paper). Four falsifiable predictions are proposed. THz spectroscopy of intact mitochondrial membranes should reveal collective vibrational modes at √2-spaced frequencies, absent in disrupted cristae. ATP synthesis rate should correlate (r > 0.5) with √2-compliance of cristae morphometry across conditions (health vs disease, young vs aged). Cross-system universality: both LHCII and mitochondrial supercomplexes within the same plant cell should follow √2 scaling with comparable precision. Complex II inhibition should NOT reduce proton-motive force per electron, confirming its role as gradient bypass rather than gradient contributor.