The 168 Monads: Derivation of the Standard Model Particle Spectrum from Fano Plane Walk-States

We derive the Standard Model particle spectrum from 168 discrete walk-states on the Fano plane, the minimal projective geometry underlying octonionic multiplication. Beginning with three fluxions {−1, 0, +1} and triadic closure, exactly 42 computational operators (glyphs) emerge from combi- natorial necessity: 7 lines × 3 strides × 2 orientations = 42. These glyphs generate 168 walk-states through four topologically distinct walk types, organizing into 14 Frobenius orbits. Fundamental constants appear as geometric eigenvalues: the fine-structure constant α−1 ≈ 137 at row 137, the gravitational boundary G−1 ∝ 168 at row 168, with ratio \( 168/137 \approx \sqrt{\pi/2} \) bridging circular and dis- crete projection basins. The framework maps all six quarks, charged leptons, electroweak bosons, the Higgs, and eight gluons to specific orbits, with mass ratios and mixing angles approximated to 0.1– 2% accuracy via transforms involving π, ϕ, and structural integers. Unmapped orbits provide dark matter candidates—states that project to 4D spacetime but do not couple to the electromagnetic channel. This approach reduces the Standard Model’s 26 free parameters to geometric consequences of octonionic projection from 8D to 4D, with residual errors within projection tolerance.