The Critical Cartier Diagonal in the Lower p-Central Series 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 in n variables. Write P_1=T_N, P_{i+1}=P_i^p[P_i,T_N], and d_r=(p-1)(n-1)+pr. For every r>=2 and N>d_r, put F_{i,r}^{(N)}=gr^{d_r}P_i. The corresponding critical diagonal layer satisfies F_{r-1,r}^{(N)}=L_{d_r} and F_{r,r}^{(N)}=(Pi_r^{eq})^{-1}(delta_1 tensor E_r^(1)), where delta_j=(det V*)^{tensor(p^j-1)}, Pi_r^{eq}:L_{d_r}->delta_1 tensor W_r^(1) is the rational GL(V)-equivariant Cartier map, W_r=Sym^r(V*) tensor V, and E_r is the coefficient exact subspace. Hence C_{r-1,r}^{(N)}=gr^{d_r}(P_{r-1}/P_r) isomorphic to delta_1 tensor (W_r/E_r)^(1) and is nonzero. The commutator subgroup already realises the denominator, while restricted powers satisfy gr^{d_r}[P_{r-1},T_N]=(Pi_r^{eq})^{-1}(delta_1 tensor E_r^(1)) and 0!=gr^{d_r}(P_{r-1}^p) is contained in E_{d_r} and in gr^{d_r}[P_{r-1},T_N], so restricted powers occur at the critical physical degree but vanish in the critical quotient. We also determine the rational GL(V)-module structure, including the four-term exact sequence at the double-Cartier resonance indices. Quadratic commutators induce natural surjections between successive critical layers; the first transition kernel is delta_1 tensor K_3^(1), where K_3 is the kernel of the contraction map L_2 tensor V* -> Sym^2(V*), u tensor lambda -> lambda(u), and K_3 is the target-three contraction kernel.