Cartier Obstructions and Frattini Quotients of Jacobian-One Formal Automorphisms

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. Leading fields identify every congruence layer with the corresponding space L_d of homogeneous divergence-free polynomial vector fields of coefficient degree d. Completed Cartier flux and filtered realization lift the Lie-theoretic Cartier decomposition to actual finite jets and determine the first homogeneous action on the coefficient modules W_r=Sym^r(V*) tensor V. Writing d_r=(p-1)(n-1)+pr and ell_n=n(n-1)(n+2)/2, and denoting the Frattini subgroup by Phi(T_N), one obtains truncation-compatible isomorphisms T_N/Phi(T_N) isomorphic to L_2 direct-sum sum_{0d_2}. Thus only the constant, trace, and divergence coinvariants survive; every visible Cartier layer of index at least three lies in the Frattini subgroup.