Overview
This article develops the global layer of the spectral admissibility programme. The local bounded-flux constraint assigns each irreducible sector an admissibility window proportional to $1/\sqrt{\lambda_\rho}$. The spectral capacity functional aggregates those windows over all representation sectors with Peter--Weyl multiplicities.
The naive aggregate can be distorted by constructive alignment: sectors with $\mu_\rho>0$ have artificially lowered Laplacian eigenvalues and therefore inflated capacity. The paper introduces interference-penalised functionals $C_0$ and $C_\alpha$ as the global form of the no-premature-selection admissibility principle.
The main theorem proves binary-cover dominance for the verified valences $d \in \{6,12,24\}$ by explicit character-table computation. The general binary-polyhedral maximality statement remains formulated as a precise conjecture.
Core contributions
- Global capacity functional: the paper defines $C(G,S)=\sum_{\rho,\lambda_\rho>0}(\dim\rho)^2/\sqrt{\lambda_\rho}$ as the Peter--Weyl aggregate of local admissibility windows.
- Global form of A3: constructive alignment with $\mu_\rho>0$ is identified as artificial spectral inflation, and the penalised functional $C_\alpha$ suppresses it.
- Binary-cover dominance: for $d\in\{6,12,24\}$, the binary groups dominate their canonical SO(3)-quotient competitors at the level of unnormalised $C_\alpha$.
- Bosonic identity: the bosonic sectors of a binary group reproduce exactly the capacity of its SO(3) quotient.
- Fermionic surplus: the strict dominance comes entirely from positive spinorial contributions unavailable in the SO(3) quotient.
- Dicyclic exclusion: binary dihedral groups are excluded structurally through anisotropic half-turn axis geometry and the adjoint identity $\widehat A_{\rm adj}=2M-I$.
Verified maximality mechanism
The algebraic proof separates each binary group into bosonic and fermionic sectors. The bosonic part descends to the SO(3) quotient and contributes exactly the same capacity. The difference between the binary group and its quotient is therefore carried entirely by the fermionic surplus.
- At $d=6$, $Q_8$ is fixed by Gram rigidity and is the unique orthogonal neutral case.
- At $d=12$, $2O$ dominates $S_4$, with a constant fermionic surplus $C_\alpha^{\mathrm{ferm}}(2O)=4\sqrt{3}$.
- At $d=24$, $2I$ dominates $A_5$, with positive surplus $C_\alpha^{\mathrm{ferm}}(2I)=24/\sqrt{18+6\alpha}+36/\sqrt{28}$.
Interpretation
The result shows that global admissibility cannot be read from raw spectral softness alone. A low eigenvalue caused by constructive alignment is not genuine admissibility; it is an artefact of the generating-set representation.
Penalisation restores the structural hierarchy: local bounded-flux admissibility, Heisenberg structure, quaternionic minimality and SU(2) selection all point toward the same binary polyhedral chain.
The Ramanujan property is not a direct maximiser of spectral capacity. Its role is to act as a consistency condition, preventing pathological eigenvalue collapse and keeping the capacity ordering structurally meaningful.
Relation to the Cosmochrony programme
This paper globalises the one-mode spectral admissibility analysis. The earlier local result identified which representation sectors are most robust under bounded flux; this work asks which whole finite group structures remain favoured after all sectors are aggregated.
The conclusion is that the admissibility principle is not only local. It induces a global ordering on admissible group structures, with binary covers gaining a strict advantage from spinorial sectors while preserving the bosonic capacity of their SO(3) quotients.
References
Jérôme Beau. Spectral Capacity Functional and the Binary-Polyhedral Maximality Conjecture. Preprint, Zenodo.