Paper CCXXIX: General relativity as an emergent theorem: the G2 auxetic equilibrium of the Klein quartic and the 14=14 identity We prove three theorems that together establish general relativity as an emergent theorem of the

We prove three theorems that together establish general relativity as an emergent theorem of the One-Octonion Brane-Bulk Framework, not as a postulate. THEOREM 1 (14=14): The equality dim(G2) = 14 = (triangles per heptagon of K(7)) is not numerological; both 14s arise from the single group PSL(2,7) ≅ GL(3,2) acting on PG(2,2) [the Fano plane = Im(O)] in two different representations. The bijection maps each of the 6 root pairs (E+α, E-α) to one heptagon edge with its two flanking triangles, and the Cartan pair (H1, H2) to the e0 scalar edge — the same e0 that appears as Sign 342 in the Tamil-Indus G2 grammar. THEOREM 2 (GR Emergence — Auxetic Equilibrium): The Einstein field equations Gµν = (8πG/c4) Tµν are the unique consistency condition ensuring that the G2-equivariant bijection of Theorem 1 is preserved under all brane diffeomorphisms. The proof identifies the Jacobi identity of G2 with the Bianchi identity, making energy-momentum conservation a theorem, not an assumption. THEOREM 3 (8π and Λ): The factor 8π in 8πG traces to χ(K(7)) = -4 via Gauss-Bonnet (∫K dA = -8π), and the cosmological constant Λ is the brane pre-stress from T1 relaxation (Λopp(t)), completely separate from the 14=14 auxetic equilibrium.

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.