Overview
This article develops the second step of the spectral admissibility programme. The starting point is the one-mode result of Spectral Admissibility: on Cayley graphs of finite SU(2) subgroups, bounded flux gives each irreducible sector an admissibility window proportional to $1/\sqrt{\lambda_\rho}$, and the spin-$1/2$ sector is always the most admissible one locally.
The central question of the present paper is whether this advantage remains true globally. To answer it, the paper introduces the spectral capacity functional $C(G,S)$, a Peter--Weyl weighted aggregate of all per-sector admissibility windows. It then shows that the naive aggregate is contaminated by constructive alignment: sectors with $\mu_\rho>0$ can appear artificially favoured simply because the generating set resonates with them.
Once this artefact is removed through the penalised functionals $C_0(G,S)$ and $C_\alpha(G,S)$, the binary chain re-emerges as the structurally preferred one. At valence $d=6$, $Q_8$ is the unique exactly neutral case. At valence $d=24$, the binary icosahedral group $2I$ overtakes its rotational competitor $A_5$ and becomes the preferred substrate.
Core contributions
- Global functional: the paper defines the spectral capacity $C(G,S)=\sum_{\rho,\lambda_\rho>0} (\dim\rho)^2/\sqrt{\lambda_\rho}$ as the natural Peter--Weyl aggregate of per-sector admissibility windows.
- Interference diagnosis: it shows that the naive functional is inflated by sectors with positive adjacency eigenvalue $\mu_\rho>0$, which reflect constructive alignment rather than genuine admissibility.
- Penalised variants: the hard-threshold capacity $C_0$ and the soft family $C_\alpha$ are introduced to suppress this artificial enhancement and recover structural preference.
- Valence-$6$ comparison: among the main competitors at $d=6$, $Q_8$ is the unique group for which the naive and penalised capacities coincide, because its non-trivial sectors are all neutral or negatively aligned.
- Valence-$24$ result: at $d=24$, the penalised capacity selects $2I$ over $A_5$, with the advantage increasing as the penalisation strengthens.
Interpretation
The article shows that global admissibility cannot be read directly from raw spectral softness. It must distinguish true structural preference from accidental resonance between a generating set and a representation sector.
- Peter--Weyl multiplicity gives the correct weighting for aggregating local admissibility windows into a group-level quantity.
- Constructive alignment can make rotational competitors appear stronger than they really are under bounded flux.
- Interference penalisation restores the binary-polyhedral advantage and turns local low-spin preference into a global selection principle.
Within this perspective, the Ramanujan property is not the direct maximiser of capacity. Its role is instead to act as a spectral consistency condition, preventing pathological spectral compression and protecting the spinorial hierarchy from accidental blow-up.
Relation to the Cosmochrony program
This paper is the global completion of Spectral Admissibility. The first paper showed which individual sectors are most robust under bounded flux. The present one asks which whole groups remain preferred once those admissibility windows are aggregated across all sectors.
Its conclusion is that the binary polyhedral chain is not just locally favoured but globally selected once constructive interference is accounted for correctly. This prepares the next step of the programme, where neutral-generator geometry is analysed and the admissible subgroup list is reduced structurally before generation counting and hierarchy construction.
References
Jérôme Beau. Spectral Capacity Functional and the Binary-Polyhedral Maximality Conjecture. Preprint, Zenodo.