Parity Invariant Explains Periodicity of the IMO 2011 Windmill Problem — E8 Intelligence Research

FINDING: The 2011 IMO "windmill" problem (P2) is a parity/invariant problem in disguise — its difficulty is combinatorial, not metric-geometric. | MATH: Setup: point P in plane, line ℓ through P; a "windmill" is a process where ℓ rotates about P, and each time it hits a point of a finite set S, that point becomes the new pivot. Core result: for any finite S and any initial line, the process is periodic. The proof hinges on a **parity invariant**: each point is used as pivot with a fixed parity of "sides," and the configuration space of (pivot, direction) is finite, forcing eventual return. No closed-form ratio; the invariant is ℤ/2. | CONNECTION: None to φ, 0.618, base-60, or crystallographic symmetry. The structure is a finite-state dynamical system on a discrete set — closer to a permutation/group-action argument than to geometric harmony. | DEPTH: 4 — pedagogically deep (invariant discovery), but mathematically it is a finiteness + parity argument, not a structural constant. FINDIN Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com