Pairwise Character Geometry and the Classification of Neutral Generators in SU(2)

We study the geometry of neutral generating sets: symmetric generating sets S = S⁻¹ of a finite group G equipped with an irreducible representation ρ G → SU(2) satisfying χ_ρ(s) = 0 for all s ∈ S. The central result is that the geometry of the associated unit axes ⃗ₛ ∈ S ⊂ S² is entirely encoded in the pairwise character values: Gₛₜ:= u⃗ₛ · u⃗ₜ = -1/2χ_ρ(st). Using this identity together with the second-moment tensor M = ∑s ∈ S⁺ u⃗ₛ u⃗ₛT, we give a complete character-theoretic classification of all admissible groups: enumerate [(i)] M ∝ I ρ(G) Dicₙ, n ≥ 3; [(ii)] M ∝ I, Gₛₜ = 0 for all $s, t$ with t ≠ s± 1 ρ(G) Q₈; [(iii)] M ∝ I, Gₛₜ ∉ \0, ± 1\ for some $s, t$ ρ(G) ∈ \2O, 2I\. enumerate In particular, the binary tetrahedral group $2T$ (projective image A₄) admits no neutral generating set, a structural exclusion we prove directly. The finite subgroups of SU(2) compatible with spectral neutrality are exactly Q₈, Dicₙ (n ≥ 3), $2O$, and $2I$. This completes the Admissibility Sub-Programme of Cosmochrony.