Gravity, Newton's Constant, and the Cosmological Constant from G2 Compactification

This paper addresses the gravitational sector of the G2 algebraic framework developed in Papers I–IV. We show that: (1) through the MacDowell-Mansouri formulation of gravity as a gauge theory, the EinsteinHilbert action emerges from the Yang-Mills action of the unity sector, giving Newton's constant in the structural form G = /(w_unity × M_GUT²), where = 1/42 is the structural coupling and w_unity = ε ε 0.6685 the unity-sector weight; (2) the ratio w_unity/w_info = 13.73 ≈ 14 = dim(G2) is suggestive of a bulkto-boundary relation of holographic type; (3) the compact G2 manifold has a two-scale moduli structure with radius ratio R /R = 50 = val(N); and (4) a flux quantum n = 19 combined with val(N) = 50 gives a ₂ ₁ proton-mass-scale estimate 19 × 50 = 950 MeV, agreeing with the observed 938.3 MeV at the 1.28% level — an order-of-magnitude relation, not a precision result. We present the gravitational derivation as structurally complete and the compactification results as partially established, stating explicitly that moduli stabilisation and the absolute Planck scale are not fixed by the framework and remain open. Structural integers cited in the body are drawn from the corpus database of Paper I; their sources are tabulated in the Appendix. Keywords: gravity, Newton's constant, MacDowell-Mansouri, cosmological constant, G2 holonomy, M-theory compactification, holographic principle, moduli stabilisation