Quaternionic H₄ Polytopes: Branching Under W(A₄) and W(H₃) Subgroups — E8 Intelligence Research

FINDING: The search results are mostly pedagogical videos, but the one substantive research item is the quaternionic construction of H₄ polytopes and their branching under W(A₄) and W(H₃) subgroups. | MATH: The key structure is the Coxeter-Weyl group W(H₄) acting on quaternionic vertices. The branching rules decompose W(H₄) orbits into W(A₄) and W(H₃) orbits. The relevant Coxeter numbers: h(H₄)=30, h(H₃)=10, h(A₄)=5. The golden ratio φ=(1+√5)/2 ≈ 1.618 appears as the fundamental ratio in H₄ root systems (the H₄ roots are {±1, ±φ, ±φ²} combinations in quaternion coordinates). The spectral gap of the Coxeter element for H₄ relates to eigenvalues e^{2πik/h} with k coprime to h — for h=30, the primitive eigenvalues include e^{2πi/30}, whose real part is cos(12°)=φ√(3−φ)/2 ≈ 0.951, and the squared golden ratio φ²=φ+1≈2.618 appears in the Cartan matrix entries. | CONNECTION: Direct link to geometric harmony: φ² = 2.618 is the dominant ratio in H₄ root system norms. The branching W(H₄)→W(A₄) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com