Affine Independence Under Multiplicative Class Refinement in Finite Fields

This preprint investigates a structural phenomenon arising from the refinement of multiplicative cosets in finite prime fields. Let p be an odd prime and let Hd1 ⊆ Hd2 ⊆ F_p^× be multiplicative subgroups with d1 | d2. Each coarse coset of Hd2 contains r = d2/d1 fine cosets of Hd1, defining a refinement fiber. For each fine coset, an additive transition structure induced by x ↦ x + 1 is considered, together with a normalized transition vector. The transition counts themselves are classical objects closely related to cyclotomic numbers. The focus of this work is the internal structure of refinement fibers. A computational survey covering 18 primes, 298 refinement pairs, and 1,357 refinement fibers found maximal affine dimension r − 1 in every tested fiber. Additional algebraic audits were performed using multiplicative-character representations, determinant tests, and a controlled negative experiment. The character reconstruction reproduced the transition data to numerical precision, while the negative control produced the expected rank collapse under artificially duplicated rows. The main mathematical outcome is a computationally supported conjecture that the normalized transition vectors associated with the fine classes of every nontrivial refinement fiber are affinely independent. The result is presented as a conjecture supported by computation rather than as a universal theorem. No claim of a new prime invariant or prime signature is made.