Spectral Capacity Functional and the Binary-Polyhedral Maximality Conjecture

Abstract

The bounded-flux admissibility constraint of the Cosmochrony framework assigns to each spectral sector $\rho$ of a finite Cayley graph a maximal effective amplitude $A^{\max}_{\rho} = c_{\mathrm{BI}}/\sqrt{\lambda_{\rho}}$. Aggregated over all irreducible sectors with Peter Weyl multiplicities $(\dim\rho)^{2}$, this local constraint defines a global functional $C(G,S) = \sum_{\rho} (\dim\rho)^{2}/\sqrt{\lambda_{\rho}}$ 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 $C_{\alpha}(G,S)$ 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 $C_{\alpha}$ for representative competitors at valences $d = 6$ and $d = 24$, prove binary-polyhedral maximality for $d \in \{6, 12, 24\}$ 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 $\Rightarrow$ Heisenberg structure; then, for a supplied $\mathrm{SU}(2)$ 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. Keywords: Spectral graph theory; Cayley graphs; Representation theory; $\mathrm{SU}(2)$ subgroups; Dirac Born Infeld dynamics; Laplacian spectrum; Non-abelian mode selection