ENTANGLEMENT AS METRIC AND THE JACOBSON VACUUM SOLUTION (VIII v6)

Jacobson’s landmark derivation of the Einstein field equations from spacetime thermodynamics (δQ = T dS) leaves an unresolved foundational tension: in pure vacuum where matter-energy vanishes (δQ → 0), the thermodynamic metric definition risks formal degeneracy. In the pregeometric framework of FCD-RBv6, we resolve this vacuum paradox by establishing that metric geometry is not a fundamental continuum, but the macroscopic coarse-graining of a pregeometric network of trans-brane Einstein-Rosen (ER) bridges. We introduce the scalar connectivity field d(x,t), defined directly from the bilateral entanglement density. In regions devoid of localized baryonic matter, the connectivity field does not vanish; rather, it relaxes asymptotically to a finite, positive baseline value d(r) → d₀ > 0. Newton’s constant emerges naturally as the inverse of this baseline connectivity (G_N = c³ / 4ħ d₀). Gravitational attraction is shown to arise as an emergent statistical depletion force proportional to ∇d(x,t), providing a non-singular, information-theoretic foundation for General Relativity.