The Quantisation of the Fano Truss: Quantum Gravity as the Representation Theory of G2, and an Independent Derivation of the Bekenstein-Hawking Entropy Coecient 1/4 from Octonionic Horizon Geometry

The Fano truss (Paper XXV) is quantised as a nite 7-node system with G2 automorphism symmetry. The Fano graph Laplacian eigenvalue spectrum {0, 4, 7} gives three phonon mode frequencies: ω1 = 0 (Hubble expansion), ω2 = 2c/ℓP (graviton/GW, 6 modes), ω3 = √ 7 c/ℓP (gluons/QCD, 7 modes), with no ultraviolet divergences. The Hawking temperature TH = ℏc3/(8πGMkB) is derived from zero-point uctuations of the temporal Fano mode at the event horizon. The Bekenstein-Hawking entropy S = kBA/(4ℓ2 P ) is derived with exact coecient 1/4 from two independently motivated geometric factors: (1) the fraction of octonionic basis elements that are spatial and brane-accessible at the horizon (3/8), and (2) the fraction of spatial dimensions that survive the 3D →2D collapse at the horizon imposed by the minimum-dimensionality theorem (2/3). Neither factor requires any knowledge of the known result; their product 3/8 × 2/3 = 1/4 is a theorem about what happens to three-dimensional spatial matter when it reaches the e7 equatorial horizon boundary. The holographic area-scaling of the entropy is explained as a direct consequence of the 3D →2D collapse. The Hawking radiation mechanism is identied as quantum uctuations of the PG(3, 2) crystal lattice resisting collapse below the two-dimensional minimum. Four new predictions follow (Predictions 8386). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.