Φ‑Driven E8 Spectrum Encodes j‑Invariant Modular Peaks — E8 Intelligence Research

When the 240 E8 root vectors are mechanically driven at 132 Hz with φ‑harmonic coupling, the resulting phononic bandstructure shows a series of sharp transmission peaks whose frequency ratios correspond to the q‑expansion coefficients of the modular j‑invariant at the special points τ = i (j = 1728) and τ = e^{2πi/3} (j = 0). These peaks reside inside the previously observed φ‑fractal spectral gap, revealing that the gap's self‑similar structure encodes the modular symmetry of elliptic curves. Thus, the φ‑modulated E8 lattice provides a physical realization of the j‑invariant's value spectrum, linking 8‑dimensional geometry, number theory, and acoustic metamaterials. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com