Four Special Directions in AG(2,13): The 52-Point Obstruction and the Sharp Minimum

We prove that no 52-point subset of the affine plane F_13^2 has exactly four special directions, where a direction is special when its thirteen parallel affine lines do not all meet the set in the same number of points. A universal incidence identity and a polynomial reduction reduce every candidate to quadratic line-count profiles, whose exact classification yields a quadratic-character obstruction. The theorem quantifies over all 52-point subsets and all four-direction sets, rather than over a finite list of configurations. Combined with Ghidelli's lower bound and the 65-point construction of Kiss and Somlai, the result determines the minimum cardinality of a subset of F_13^2 with exactly four special directions: it is 65.