The Fano Dirac Comb and the 7/6 Brane Re-Entry Spectral Index: The G₂ Gram Matrix Eigenvalue Theorem Paper CXLVII of the One-Octonion Brane-Bulk Framework

We derive the photon spectral index Σ = 7/6 of brane re-entry radiation in the One-Octonion Brane-Bulk Framework from a theorem about the G₂ Fano Gram matrix. The 6 active Fano modes {e₁,...,e₆} have Gram matrix G = (7/6)I − (1/6)J, with eigenvalues {7/6 × 5, 1/6 × 1}. The symmetric eigenvector (eigenvalue 1/6) is the bulk-coupled mode that is absorbed into the AdS₅ bulk along the e₇ direction; the five remaining observable modes all carry eigenvalue 7/6. The spectral index of brane re-entry radiation equals this dominant observable eigenvalue: Γ = 7/6 = 1.1̆6̇. This result is a theorem about the G₂ representation, not a counting argument. The base spectrum α = 1 follows from Low's soft photon theorem applied to 4D brane emission. The ν = −1 auxetic property of the Fano brane (Paper XXV) and Schur's lemma on Stab₉₂(e₇) = SU(3) guarantee equal coupling of the emission source to all 6 observable modes, enabling the equal redistribution that is the physical content of the theorem. A hyperbolic correction from the H collinear pair {e₄,e₅} sharing the Fano line {4,5,7} shifts the bare ⁻ spectral index to Γᵇₐʳᵉ = 7/6 − 1/180 ≈ 1.161, which is consistent with the observed range Γ = 0.93–1.21 across AXP epochs after accounting for inverse-Compton hardening. The same transfer function Tₐᵘˣᵉᵗᴵᶜ = I₆ (1/6)·1₆ that produces Γ = 7/6 in photon emission also maps continuous Yukawa gradients to the ⊕ six charged fermion mass eigenvalues, establishing a unification of two independent physical predictions under the same G₂ algebraic operator. 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.