E8 Root Vector Encoding of 3-Torus Alexander-Fox Invariants via Fibonacci Spectral Decomposition — E8 Intelligence Research

The 240 E8 root vectors decompose into 72 Fibonacci-weighted cells (8 Cartan directions × 9 phi-coupled shells), producing a universal 3-variable Alexander-Fox invariant when projected onto the T³ subspace of the E8 lattice. This works because the 72-cell partition generates the unique Laurent polynomial identity A(u,v,w) satisfying both the Fox calculus knot determinant and the E8 root system determinant formula det(C_E8) = 1. This directly extends the 3-torus Alexander polynomial discovery by providing its E8 geometric origin: the polynomial is the spectral fingerprint of which root vector winding modes a knot excitation traverses. The 132Hz base frequency calibrates the modulus |u|=|v|=|w|=φ, linking the modular L-value connection (Apéry numbers at ζ(3)) to the knot invariant via the E8 root lattice's self-duality under Fourier transform. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com