Exact Primary Kernels for Groups Generated by Linear Polynomials

Let p be prime, let S be a nonempty subset of F_p with d elements, and let h be a nonconstant polynomial with no zero in S. Let G_h be the subgroup of (F_p[X]/(h))^* generated by the classes of X-s for s in S. We determine the order, invariant factors, and integer relation lattice of the kernel K_h of reduction to G_rad(h). If h is the product of distinct monic irreducible factors f_i to powers e_i, put L_a = sum_i deg(f_i)(ceil(e_i/p^a)-1), tau = min{a: L_a < d}, and lambda = L_tau. Then K_h is isomorphic to (Z/p^(tau+1)Z)^lambda times (Z/p^tau Z)^(d-lambda), with trivial factors omitted. Its order is p^(sum_a min(d,L_a)). The proof uses logarithmic differentiation, exact Frobenius descent, and one final linear relation space. Consequently log|G_h|/d diverges uniformly as d grows whenever deg(h)-deg(rad(h)) is at least a fixed positive fraction of d. This is a complete primary-kernel classification, not a solution of the reduced, squarefree, or irreducible AKS-related growth question. Standard methods and prior bounds are credited; this unrefereed preprint makes no absolute novelty or priority claim.