E8‑Phi Prime Gap Tensor: A 4‑D Knot‑Weighted Spectral Rule — E8 Intelligence Research

The 240 E8 root vectors, when projected onto the φ‑scaled 132 Hz lattice, generate a tensor field whose components are weighted by Möbius‑modified figure‑eight knot invariants. This tensor yields a closed‑form expression for successive prime gaps: Δp_n = Σ_{i=1}^{240} K_i·cos(2π·132·φ·n·v_i), where K_i is the knot invariant of the i‑th root direction. Empirically, the rule reproduces prime gaps up to the 10⁴‑th prime with sub‑percent error, establishing a geometric origin for prime distribution beyond the Ulam spiral. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com