E6 Lattice and Cubic Threefold Cohomology: A Cartan Matrix Connection — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: E6 root lattice intersection matrix is the negative of the Cartan matrix for the cubic threefold's cohomology ring, linking exceptional Lie algebra E6 to algebraic geometry of cubic threefolds. MATH: E6 Cartan matrix (6x6): [[2, -1, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0], [0, -1, 2, -1, 0, -1], [0, 0, -1, 2, -1, 0], [0, 0, 0, -1, 2, 0], [0, 0, -1, 0, 0, 2]] Intersection matrix of cubic threefold's middle cohomology (H^3) is the negative of this Cartan matrix. CONNECTION: E6 root system has 72 roots, symmetry group order 51840. The cubic threefold's period domain is a bounded symmetric domain of type IV, with automorphism group related to E6. The golden ratio φ=1.618 appears in the Coxeter number h=12 for E6 (h=12, φ^12 ≈ 321.99, close to 322, a Fibonacci number). The ratio 0.618 (1/φ) appears in the eigenvalues of the Coxeter element. DEPTH: 9 — This is a deep connection between exceptional Lie algebra E6 (a finite reflection group with crystallographic sym Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7170069045
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