Spectral Capacity Functional and the Binary-Polyhedral Maximality Conjecture

The bounded-flux admissibility constraint of the Cosmochrony framework assigns to each spectral sector of a finite Cayley graph a maximal effective amplitude A^_ = c_/_. Aggregated over all irreducible sectors with Peter Weyl multiplicities () ^2, this local constraint defines a global functional C (G, S) = _ () ^2/_ on finite group structures. The central result of this paper is that the admissibility principle is not only a local constraint on individual spectral sectors, but induces a global ordering on admissible group structures: binary polyhedral groups maximise C among all finite groups of comparable order and generating-set size, once interference-inflated sectors are penalised consistently with the axiom of no premature selection (A3). The penalised variant C_ (G, S) is not an ad hoc correction: it is the global form of A3, filtering out sectors whose constructive alignment with the generating set produces an artificially enhanced admissibility window. We compute C_ for representative competitors at valences d = 6 and d = 24, prove binary-polyhedral maximality for d \6, 12, 24\ by explicit character-table computation, establish the general conjecture in a precise and testable form, and identify the Ramanujan property as a spectral consistency condition not a direct maximiser of C, but a structural prerequisite ensuring that the spectral hierarchy is not distorted by pathological eigenvalue accumulation. The result closes the chain: local admissibility Heisenberg structure quaternionic minimality SU (2) global spectral dominance of binary polyhedral groups.