Discrete Rational Circle Group and A1 Root Lattice via Torsion Points — E8 Intelligence Research

FINDING: The circle group \(S^1\) as a discrete subgroup via rational points, linked to the A1 root lattice and the fundamental group \(\pi_1(S^1) \cong \mathbb{Z}\). | MATH: Rational points on unit circle: \(x = \frac{2t}{1+t^2},\; y = \frac{1-t^2}{1+t^2}\) for \(t \in \mathbb{Q}\) (Pythagorean triples). Group structure: \(S^1 \cong \mathbb{R}/\mathbb{Z}\) (or \(\mathbb{R}/2\pi\mathbb{Z}\)). Discrete subgroup: \(\mathbb{Q}/\mathbb{Z} \hookrightarrow S^1\) (torsion points, roots of unity). A1 root lattice: \(\mathbb{Z} \subset \mathbb{R}\), dual lattice \(\mathbb{Z}^* = \mathbb{Z}\), norm squared \(2\) for roots \(\{\pm e_1\}\). Fundamental group: \(\pi_1(S^1) = \mathbb{Z}\). | CONNECTION: The rational points on \(S^1\) form a dense subgroup (not discrete), but the **torsion subgroup** (roots of unity \(e^{2\pi i m/n}\)) is discrete in the profinite sense and corresponds to \(\mathbb{Q}/\mathbb{Z}\). The A1 lattice \(\mathbb{Z}\) is the weight lattice of \(SU(2)\); its quotient \(\math Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com