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. All results are verified by a companion script (113tests, 0 failures) with no free parameter beyond α* = 1/(4 ln 2) and the electron mass.