Spectral Capacity Functional and the Binary-Polyhedral Maximality Conjecture

The one-mode admissibility analysis of Steps~1--5 singles out the spin-12 sector of every finite subgroup of SU (2) as the most admissible representation channel. We now ask whether this advantage is global: does the binary icosahedral group 2I (or more generally any binary polyhedral group) maximise a natural aggregate of per-sector admissibilities over all finite groups of comparable order and generating-set size? We introduce the spectral capacity functional C (G, S) and its interference-penalised variants C₀ (G, S) and C_ (G, S), establish their Peter--Weyl foundations, and show that the naive functional C does not suffice to discriminate the quaternionic chain. The penalised variant C_ restores the hierarchy. We compute C_ for a representative set of competitors at valences d=6 and d=24, and prove the binary-polyhedral maximality conjecture: the adjoint identity A₀₃₉ = 2M-I translates the geometric isotropy of icosahedral axis distributions into a strict capacity inequality, excluding binary dihedral groups analytically and confirming 2I as the unique maximiser for all > 0. The role of the Ramanujan property is clarified: it acts as a spectral consistency condition, not as a direct variational maximiser of C.