Quantum Gravity, Holography, and Graviton Scattering Hilbert Space from G2 Representation Theory, Black Hole as Brane Puncture, Graviton Scat- tering, and the BekensteinHawking Coecient from the G2 Partition Function

This paper proposes a G2/Fano framework for quantum spacetime geometry and develops one explicit eective derivation Hawking temperature from T ∗ 2 dephas- ing while outlining corresponding interpretations for the area spectrum, singularity resolution, black hole information, and quantum tunnelling. The central proposal is that quantum gravity, in the brane-bulk octonionic framework, is modelled as the represen- tation theory of G2 acting on the Fano lattice. The area observable is identied with a mode-weighted face-count operator whose eigenvalue is derived in three steps: (i) the hyperbolic face area in the local H2 lattice geometry reduces to ℓ2 P at Planck-scale curva- ture saturation; (ii) the relaxation uncertainty at the T ∗ 2 hyperbolic-to-planar transition provides a framework-native correction; (iii) degree-of-freedom restriction in asymmetric gravitational wells gives a 2 : 1 Schwarzschild-to-Kerr area ratio per quantum number j, consistent with the known metric result and proposed as a new prediction (P31). Among all correspondences developed here, the Hawking/T ∗ 2 identication is the most explicit: it links a horizon coherence time directly to an observable temperature, re- covering the Hawking scaling and exact coecient from three stated assumptions. The BarberoImmirzi parameter γ = ln 7/(2π √ 3) is suggested by the G2 orbit structure; the 1/4 in the BekensteinHawking entropy is motivated by a (1/2)DOF × (1/2)bc counting argument. Comparisons with loop quantum gravity, string theory, and causal dynamical triangulations are oered as structural observations rather than verdicts. The dynamical structure of the quantum gravity theory is fully derived: Paper LXXVIII showed the principle of least action is a theorem of ν = −1 auxetic brane mechanics (|1 + ν| = 0 makes boundary terms in δU vanish identically, leaving EulerLagrange as a mechanical theorem); Papers XLIII and XLVI derived the path integral as the sum over the α- spectrum of brane-bulk excursion sequences; Paper L derived the Born rule from brane uid incompressibility. The gravitational path integral is the sum over brane-bulk ex- cursion sequences with Fano UV completion at kmax = π/ℓP , with every component derived rather than postulated. Four new predictions are made (P27P29, P31); P30 is listed as a conjectural future program and P32 as an appendix prediction. 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.