No Primary Source Found on A₂ Weyl Invariants or Modular Discriminant — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Search returned only pedagogical/lecture material on rank-2 root systems and Weyl groups — no primary source addressing the A₂ Weyl invariant ring or the modular discriminant Δ = g₂³ − 27g₃². | MATH: A₂ root system: roots ±α, ±β, ±(α+β); Cartan matrix A = [[2,−1],[−1,2]]; Weyl group W(A₂) ≅ S₃ (order 6). Invariant ring: ℂ[x,y]^{S₃} = ℂ[e₁,e₂,e₃] (elementary symmetric polynomials), degrees 1,2,3. Modular discriminant: Δ(τ) = g₂(τ)³ − 27g₃(τ)² = (2π)¹² η(τ)²⁴, a cusp form of weight 12. | CONNECTION: The A₂ Weyl group S₃ is the same group acting on the three half-periods of the Weierstrass ℘-function; the discriminant g₂³ − 27g₃² vanishes precisely when the elliptic curve degenerates (roots collide), i.e. when the lattice loses its S₃ symmetry. The exponent pair (4,6) for g₂,g₃ matches the A₂ invariant degrees (2,3) under the standard doubling. | DEPTH: 3 — the search surfaced no evidence bearing on the specific claim; the connection between A₂ invariants and the modular discrimi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225335278