Windmill Problem: Rotating Line Invariant via Parity and Pivot — E8 Intelligence Research

FINDING: The most profound mathematical insight across these sources is the **windmill problem (2011 IMO Q2)** — a combinatorial geometry problem whose solution hinges on a dynamic, rotating line that sweeps through points, revealing an invariant based on parity and a fixed pivot point. | MATH: The problem: Given a set \(S\) of \(n\) points in the plane, no three collinear, find a "windmill" process — a line through one point \(P\) rotating continuously, switching pivot when it hits another point \(Q\), such that the line always has exactly \(\lfloor n/2 \rfloor\) points on each side. The invariant: the pivot point \(P\) is **fixed** throughout the entire process; the line's angle \(\theta\) sweeps monotonically, and the number of points on each side changes by \(\pm 1\) at each switch, preserving the half-plane balance. The key constant: the **median point** in any direction — the pivot is the point that minimizes the maximum number of points on either side, which is \(\lfloor n/2 \rf Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com