BCT Letter 26: General Relativity to All Orders from the BCT Condensate

We show that the full nonlinear Einstein field equations emerge to all orders in the metric perturbation h_μν from the Body-Centred Tetragonal (BCT) Superfluid Lattice Model's condensate action, closing the linearised-only gap left by the acoustic-metric derivation. The proof proceeds in three steps: the BCT Gross–Pitaevskii action reduces exactly to the irrotational superfluid action in the Thomas–Fermi (hydrodynamic) limit; the Madelung transformation maps this to the Jacobson–Parentani form, proven equal to the Einstein–Hilbert action to all orders for any barotropic fluid with sound speed c_s = c; and the BCT equation of state forces c_s = c exactly via the Josephson coupling condition that also fixes the crystal coupling α₀. The result holds at weak field and near black-hole horizons alike. The effective Newton constant is derived as G = c⁴/(8πK_BCT), with the bulk-modulus route matched against a second, independent numerical route via the two-loop hierarchy formula, yielding G = 6.684×10⁻¹¹ m³kg⁻¹s⁻² (0.151% error) — Prediction #100 of the programme. The BCT condensate is, in this precise sense, a UV completion of general relativity, and Bekenstein–Hawking horizon entropy is recovered using the derived G.