Weyl Group Orbits and Minimal Pairs: A Literature Gap — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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) | c. General solution: x = x0 + b/gt, y = y0 − a/gt, g=gcd(a,b). - Orbit-Stabiliser (finite): |G| = |Orb(x)| · |Stab(x)|. - Diophantine Lie group (arXiv:1307.1489): For finitely generated Γ ⊂ G, exists β>0 such that for non-trivial γ ∈ BΓ(n), d(γ, e) > |BΓ(n)|−β. 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
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BibTeX
@misc{caldin2026weyl,
title = {{Weyl Group Orbits and Minimal Pairs: A Literature Gap — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23410009},
url = {https://zenodo.org/records/23410009},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weyl Group Orbits and Minimal Pairs: A Literature Gap — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23410009
UR - https://zenodo.org/records/23410009
AB - 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) | c. General solution: x = x_0 + b/gt, y = y_0 − a/gt, g=gcd(a,b). - Orbit-Stabiliser (finite): |G| = |Orb(x)| · |Stab(x)|. - Diophantine Lie group (arXiv:1307.1489): For finitely generated Γ ⊂ G, exists β>0 such that for non-trivial γ ∈ B_Γ(n), d(γ, e) > |B_Γ(n)|^−β. 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
ER -