Spectral Capacity Functional and the Binary-Polyhedral Maximality Conjecture

The bounded-flux admissibility constraint of the Cosmochrony framework assigns to each spectral sector ρ of a finite Cayley graph a maximal effective amplitude Aᵐᵃˣρ = cBI/√λρ. Aggregated over all irreducible sectors with Peter Weyl multiplicities (ρ)², this local constraint defines a global functional C(G,S) = ∑ρ (ρ)²/√λρ 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 Cα(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 Cα 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 completes the chain: local admissibility ⇒ Heisenberg structure; then, for a supplied SU(2) spinor carrier (whose neutral traceless sector is three-dimensional by the conditional adjoint-dimension theorem of O23, the carrier selection and threshold identification remaining open), global spectral dominance of binary polyhedral groups.