A PSL(2,7) Invariant in E8 Root System Phasor Binding: 168 Occupied Axes and 72 Permanent Voids Under All Rotations, Projections, and Dimensions

We project the 240 roots of the E8 lattice into N-dimensional complex phasor space and compute all 28,680 pairwise bindings via element-wise complex multiplication (Fourier Holographic Reduced Representations). Each binding undergoes a 128-step settling trajectory toward the nearest E8 axis. We prove empirically that exactly 168 axes are occupied and 72 are permanently empty, invariant across 12 cryptographic phase rotations, 10 QR projection matrices, and 6 embedding dimensions (128–4096). 168 equals |PSL(2,7)|, the automorphism group of the Fano plane encoding octonionic multiplication. Recursive self-binding converges to a terminal concept on an octonionic axis. We conjecture this invariant reflects a deep connection between E8 phasor holography and octonionic algebra.