Dense cyclotomic subgroups of the Lorentz group and the Collins–Perez–Sudarsky obstruction

We construct a finitely generated subgroup Γ = ⟨A₅, B_J⟩ of SO⁺(3,1)⁰ from the rotational icosahedral group A₅ and a single Lorentz boost B_J of rapidity log φ, where φ = (1+√5)/2 arises from the cyclotomic unit J = 1 + ζ₅² ∈ ℤ[ζ₅]. Density of Γ in the identity component is proved in two steps. First, an explicit commutator of two conjugate boosts is shown to be a pure elliptic Lorentz element with rotation cosine c_ω = (31 + 8√5)/50, satisfying the irreducible quadratic 2500 x² − 3100 x + 641 over ℚ; Niven's theorem and a direct numerical comparison rule out rational ω/π, so the cyclic group it generates is dense in a one-parameter elliptic subgroup. Second, the Lie algebra of the identity component of the closure is Ad(Γ)-invariant; combining the irreducibility of the A₅-action on 𝔰𝔬(3,1) = 𝔨 ⊕ 𝔭 with the bracket structure forces this Lie algebra to be all of 𝔰𝔬(3,1). Topologically, every continuous Γ-invariant functional on a configuration space with continuous SO⁺(3,1)⁰-action is automatically Lorentz invariant. Under standard regularity assumptions on the dynamics, measure, regulator, and coarse-graining, the resulting Wilson effective action of any substrate field theory with Γ in its symmetry group cannot contain Lorentz-violating operators, addressing the Collins–Perez–Sudarsky–Urrutia–Vucetich (2004) radiative fine-tuning obstruction at its premise.