Parity Invariant in Windmill Problem: A Cyclic Pivot Proof — E8 Intelligence Research

FINDING: The "windmill" problem (IMO 2011 Q2) is a combinatorial geometry problem whose solution hinges on a parity/invariant argument over a finite set of points, not on metric ratios. | MATH: Given \(n\) points in general position, a "windmill" process rotates a line through a pivot point; the invariant is that the line always has \( n/2 \) points on each side (for odd \(n\), the pivot is one of the points). The key lemma: the pivot moves cyclically through all points, and the process terminates after \(O(n^2)\) steps. No explicit constants — the depth is in the discrete symmetry of the configuration space. | CONNECTION: The cyclic permutation of pivots mirrors a cyclic group action \(C_n\) on the points — a discrete rotational symmetry, but not the golden-ratio or crystallographic type. The "balanced split" (half-plane counts) is a discrete analogue of a central symmetry (point reflection) in the plane. | DEPTH: 7 — Profound for its elegance and the unexpected use of i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com