One-Sided Affine Obstructions in Complete Bipartite Hat Guessing

Preprint. We study one-sided affine restrictions in complete bipartite hat-guessing games. Let the color set be identified with a finite field K of order q ≥ 4. We prove that, for every m ≥ 1, no coordinatewise-affine strategy on the class of size q−1 can be completed to a winning strategy by arbitrary deterministic rules on the opposite class of size m. The proof combines fixed-half coverability and Cartesian-grid interpolation with a global Type-I/Type-II classification of the affine coefficient rows. The two alignment types admit exactly q! and q² coverable offsets, respectively, which is insufficient for completion. In the known unrestricted solvable range m ≥ (q−1)^(q−1), this yields a necessary-nonlinearity statement: every winning strategy has a non-affine rule on the specified class of size q−1, with respect to the fixed field labeling. A supplementary Lean 4 development formally verifies the corresponding terminal theorem with the same finite-field, side-size, and arbitrary-completion quantifiers.