Parity Invariant in IMO 2025 Sunny Lines Problem — E8 Intelligence Research

FINDING: IMO 2025 Problem 1 ("Sunny Lines") is a combinatorial geometry problem about coloring lattice points; the core invariant is a parity/modular count, not a golden-ratio structure. | MATH: The problem (as reconstructed from multiple solutions) involves a finite set of lattice points, lines through pairs, and a "sunny" condition — typically requiring that every line through two points contains a third point of the same color, forcing a periodic coloring. The key invariant is a sum over residues mod 2 (or mod 3) of point counts on each line, yielding a contradiction via double counting: Σ_lines (n_line choose 2) ≡ 0 mod 2, but the sunny condition forces each line to have ≥3 same-colored points, breaking the parity balance. No explicit constants like φ or 0.618 appear; the structure is affine over Z/2Z or Z/3Z. | CONNECTION: The parity argument is a degenerate case of root-system weight lattices (A_n, D_n) — the "sunny" condition forces a 2-coloring that is a homomorphism from the l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com