Seven is optimal : a K5 selection principle for photosynthetic energy transfer

The Fenna–Matthews–Olson complex transfers excitation energy at ~99% efficiency at physiological temperature—a performance no model derives from first principles. We show that a single combinatorial fact does: the complex has seven chromophores, and seven is the unique integer that decomposes as 5+5−3, allowing two complete pentachoric cells (K₅) to share a triangular face (K₃). This K₃ junction minimises the number of inter-cell coupling edges to four—a combinatorial minimum among all 5+5−k decompositions. Applied to the measured Hamiltonian (Adolphs–Renger, 2006), the optimal decomposition places the shared face at {BChl1, BChl4, BChl6} with an intra/inter coupling ratio of 44.7, the four inter-cell edges being the four weakest in the Hamiltonian. The machinery of the companion paper P21—Ohm's law as the stationary Dirichlet state, the edge dipole as stable carrier, face saturation as the resistance mechanism—places the complex deep in the ohmic regime (40% of face saturation), where transport is governed by a monotone density gradient from antenna to reaction centre with zero saturated faces. The absolute quantum yield (~99%) is shown to be dominated by the ratio of trapping to recombination rates, not by the network topology; the K₅ contribution is the separation ratio, the structural robustness, and the identification of the transport bottleneck as catalytic rather than topological. The PE545 complex (eight chromophores, 5+5−2, K₂ junction) is tested as a falsification: its optimal ratio is 11.3, a factor 3.9× below the FMO, and none of its 280 decompositions matches the FMO value. A dipole-dipole Hamiltonian for LH2 B800 (nine chromophores, 5+5−1, K₁ junction) extends the hierarchy to K₃(44.7) > K₂(11.3) > K₁(4.5), confirmed on three independent biological systems. A final section applies the K₅ topology to the CO₂ fixation step itself: the CO₂ molecule sits at a frustration plateau (four saturated edges, zero complete faces), and the K₃ catalyst with coordinated water reduces the photon budget from three edge-flips to one—a result verified by exhaustive enumeration of all 1024 states. All results are verified by a companion script (137 tests, 0 failures) with no free parameter beyond α* = 1/(4 ln 2) and the electron mass.