Pauli Matrices and Mod 2 Topology in Spin(7,8,9) and F₄ Insulators — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Pauli matrices form the su(2) basis for 2×2 Hermitian observables, with eigenvalues ±1 (mod 2 structure) that underpin Z₂ topological invariants in insulators; mod 2 cohomology of Spin(7,8,9) and F₄ classifying spaces mirrors this parity symmetry. | MATH: Pauli matrices σ₁,σ₂,σ₃ satisfy σᵢ² = I, {σᵢ,σⱼ}=2δᵢⱼI, eigenvalues {+1,−1} ≡ {0,1} mod 2; su(2) ≅ so(3) via adjoint map; Z₂ index = (−1)^ν where ν ∈ {0,1} counts occupied band parity; mod 2 Steenrod algebra cohomology H*(BSpin(n);F₂) for n=7,8,9 and H*(BF₄;F₂) are isomorphic as algebras — a direct Z₂-graded ring identity. | CONNECTION: Eigenvalues ±1 are the two-element group ℤ₂, the same parity group governing crystallographic root systems (e.g., Bₙ, Dₙ have ℤ₂ quotients); the ratio 1:1 (equal magnitude eigenvalues) is the degenerate case of 0.618/1.618 — the golden ratio emerges when eigenvalues are {φ, −1/φ} (trace = φ−1/φ = 1), but Pauli matrices have trace 0, giving the symmetric pair {+1,−1} — this is the *additive* ha Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204114408
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