Cartier Relation Symbols and Frattini Transgression in Special Nottingham Groups

Let k=F_p with p>=5, let V=k^n with n>=3, and let T_N be a finite congruence quotient of the tangent-to-the-identity Jacobian-one formal automorphism group. Write L_d for the homogeneous divergence-free polynomial vector fields of coefficient degree d, put W_r=Sym^r(V*) tensor V, and define K_3 as the kernel of the contraction map L_2 tensor V* -> Sym^2(V*), u tensor lambda -> lambda(u). Put W=direct sum_{r>=0} W_r and let R_3=H_1(L_{>=2},W)_3 be the target-three Chevalley-Eilenberg relation space. There is a canonical surjection R_3 -> K_3 and an explicit Euler cycle lift of every element of K_3, while the homology of a divergence-free coefficient subcomplex maps onto the remaining kernel. For N>d_3=(p-1)(n-1)+3p, the canonical cross-term copy of K_3 in the homology of the Frattini quotient survives in the nonlinear group: K_3 is contained in the image of H_2(T_N,k) -> H_2(T_N/Phi(T_N),k). Consequently dim_k H^2(T_N,k) >= dim_k K_3 = n(n^3+n^2-3n-1)/2. The survival proof uses normalized Cartier representatives, the Euler correction, and finite filtered cancellation in the Frattini extension.