Title: Affine Independence Under Multiplicative Class Refinement in Finite Fields II: A Continuation and Extension

We study the additive transition structure induced by the map x ↦ x+1 on multiplicative classes of a finite prime field. Let H ⊂ K ⊂ Fₚ^× be multiplicative subgroups, and let $r = [K:H]$. The K-classes are refined into H-classes. For each fine class qH ⊂ K, we associate a transition vector whose components count the images of its elements under x ↦ x+1 across the nonzero H-classes. Computational experiments first indicated a persistent relation between the number of fine classes and the affine dimension of the corresponding transition vectors. The present work establishes this relation analytically. We prove that the transition vectors are affinely independent and consequently satisfy R Aff : qH ∈ K/H\ = r-1. The proof uses an exact indicator representation of multiplicative H-classes by power sums. After summation over the subgroup H, the relevant quantities reduce to evaluations of monic polynomials of successive degrees at the distinct points qᵈ, where d=H. The resulting evaluation matrix is a Vandermonde matrix up to a triangular change of polynomial basis. The case in which a translated element reaches zero is handled directly in the construction of the proof rather than by adding zero as an observable coordinate. The computational results are retained as independent verification of the analytical result. The affine-rank relation is a structural consequence of multiplicative refinement combined with additive translation and does not, by itself, define a prime-specific signature.