Exact Natarajan Dimension of the Natural PSL(2,q) Action

For every prime power q, this note classifies the Natarajan dimension of the natural PSL(2,q) action on the projective line: 1 for q=2; 2 for q=3,4,5,7,11; and 3 for every other prime power. The odd-characteristic proof uses a normalized quadratic-character cube criterion, Paley graphs, and Paley tournaments. Unreviewed preprint. Model, proof, and computations were rechecked with AI in another session of the same workflow. This is not independent third-party confirmation or human peer review. References remain PARTIAL and novelty remains INCONCLUSIVE. No priority claim is made; known ingredients and overlapping small cases are identified. The verification package uses only the Python standard library. It includes 15 finite-field checks and a separately implemented check for nine prime fields. Repository: https://github.com/yonezaemon1-hub/no42-psl2-natarajan-dimension