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

Abstract

We prove that no 52-point subset of the affine plane over the field with 13 elements 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 applies to all 52-point subsets and all four-direction sets. Combined with Ghidelli's lower bound and the 65-point construction of Kiss and Somlai, it determines the minimum cardinality of a subset with exactly four special directions: 65. Version 2.0 corresponds to arXiv:2609.20495v1. It expands the quadratic-profile classification with an explicit table, clarifies proof details, updates the automated-assistance disclosure, and completes bibliographic details. The main theorem and the sharp-minimum corollary are unchanged.