Ammann-Beenker Tiling: 8-Fold Symmetry, Silver Ratio Inflation, and Unverified CHSH Link — E8 Intelligence Research

FINDING: Ammann-Beenker tiling is a 2D aperiodic tiling with 8-fold rotational symmetry, generated by inflation factor \(1+\sqrt{2}\), and its vertex set is a projection of a 4D hypercubic lattice; the search query links it to the CHSH/Tsirelson bound, but the provided sources do **not** establish that connection. MATH: - Inflation factor: \(\lambda = 1 + \sqrt{2} \approx 2.414\) (silver ratio squared? No — silver ratio is \(1+\sqrt{2}\), so \(\lambda\) is exactly the silver ratio). - Tiles: two rhombi (acute angle \(45^\circ\), obtuse \(135^\circ\)) with edge length 1; areas ratio \(1:\sqrt{2}\). - Vertex set: projection of \(\mathbb{Z}^4\) onto a 2D plane with irrational slope, using the 8th roots of unity: \(\zeta_8 = e^{i\pi/4}\). - The tiling is a member of a one-parameter family of digitized planes in \(\mathbb{R}^4\) (arXiv:1208.3545). - No CHSH/Tsirelson equation appears in the sources. The Tsirelson bound is \(2\sqrt{2} \approx 2.828\), which is **not** \(1+\sqrt{2} Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com