The Cubic BCH Image of Complete-Order Comparison

For formal generators X₁,…,Xₙ, define\[P_π=e^{Xπ(1)}⋯ e^{Xπ(n)},π,τ=P_π⁻¹P_τ,π,τ=log(Wπ,τ),\]where π,τ∈ Sₙ and the logarithm is taken in the degree-completion of the universal enveloping algebra of the free Lie algebra (X₁,,Xₙ). Let Lπ,τ⁽ᵏ⁾ denote the homogeneous degree-k Lie component of Lπ,τ, and define\[_k(n)=\{Lπ,τ⁽ᵏ⁾:π,τ∈ S_n\}⊆ _k(n).\]We determine the image of complete-order comparison through degree three. At quadratic order,\[_2(n)=_2(n)\]for every n≥ 2. At cubic order, the image is proper:\[_3(n)=M_3(n)⊕ R_3^+(n),\]where M₃(n) is the multilinear cubic sector and R₃⁺(n) is the symmetric repeated-pair sector. Equivalently,\[_3(n)/_3(n) Q_3(n),\]where Q₃(n) is the antisymmetric repeated-pair sector. Consequently,\[ _3(n)=2{n}{3}+{n}{2}=n(n-1)(2n-1)/6,(_3(n)/_3(n))={n}{2}.\]Thus the comparison family P_π⁻¹P_τ selects a distinguished cubic subspace of the free Lie algebra rather than the full degree-three component.