Weyl Group Orbits and Minimal Pairs: A Literature Gap — E8 Intelligence Research

FINDING: The search results are pedagogically shallow (YouTube tutorials on linear Diophantine equations and orbit-stabilizer) with one substantive research paper on Diophantine properties of nilpotent Lie groups — no direct result on "Weyl group orbit termination" or "minimal pair" is found. MATH: - Linear Diophantine: \( ax+by=c \) solvable iff \( \gcd(a,b) \mid c \). General solution: \( x = x_0 + \frac{b}{g}t,\; y = y_0 - \frac{a}{g}t \), \( g=\gcd(a,b) \). - Orbit-Stabiliser (finite): \( |G| = |\mathrm{Orb}(x)| \cdot |\mathrm{Stab}(x)| \). - Diophantine Lie group (arXiv:1307.1489): For finitely generated \( \Gamma \subset G \), exists \( \beta>0 \) such that for non-trivial \( \gamma \in B_\Gamma(n) \), \( d(\gamma, e) > |B_\Gamma(n)|^{-\beta} \). The group is Diophantine if this holds for all \( k \) (likely all generating sets). CONNECTION: - The Diophantine property for Lie groups is a metric analogue of *lattice spacing* — it quantifies how far discrete subgroups Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com