Sakharov-Induced Einstein Gravity from the Null Worldtube Condensate (Paper 18)

This paper derives Einstein's equations as the long-wavelength dynamics induced by the Null Worldtube Theory (NWT) condensate. Starting from the minimal three-field Lagrangian of Papers 15–16 — a complex scalar coupled to a U(1) gauge field and an S2-valued Skyrme–Faddeev field at the BPS critical coupling — integrating out matter at one loop on an arbitrary curved background generates the Einstein–Hilbert action plus controlled higher-curvature corrections, via Sakharov's induced-gravity mechanism. Key results: Newton's constant matches CODATA to -11 ppm, inside the ±22 ppm experimental error bar. The Schwarzschild metric is verified as a vacuum solution by direct symbolic computation: all 10 independent components of the Ricci tensor vanish identically. Linearisation reproduces the gravitational-wave equation with two transverse-traceless polarizations and Newton's law in the non-relativistic limit. The 1045 Phase 5 mismatch reported in Paper 16 is structurally identified with the K7 Wilson amplitude that Paper 17 supplied. The Sakharov fermion-sign issue dissolves at the substrate level: the heat-kernel sum is over substrate fields (bosonic three-field Lagrangian plus optional so(10) coset), not IR-projected SM fermions and gauge bosons, which are themselves topological projections of the substrate. The graviton is shown to be an emergent variable rather than a member of the topological vortex zoo. The gravitational coupling that emerges from the NWT condensate at the classical level (Papers 14–17) is now verified at the quantum-loop level (Sakharov), at the structural level (K7 amplitude), and at the dynamical level (Einstein equations on arbitrary backgrounds). Code: the complete nwt-substrate library accompanying this paper is at https://github.com/JimGalasyn/nwt-substrate, archived on Zenodo (concept DOI 10.5281/zenodo.20012027 resolves to the latest release; this paper cites version v0.1.1, DOI 10.5281/zenodo.20012121). v2 (2026-05-04) numerical correction. The companion library 'nwt-substrate' (v0.1.1) had a stray '3 alpha^2' where the m_e/M_Pl bracket should have had '(21/8) alpha^2' (the structural derivation in Paper 17 always gives dim(Adj)/dim(S) = 21/8 in the bracket; the '3' is the prefactor-form coefficient that arises after distributing the leading 8/7). The library bug propagated into Paper 18 v1's numerical reports. In v2: G prediction is -11 ppm of CODATA (was reported as +29 ppm), now correctly inside the +-22 ppm experimental band; (m_e/M_Pl) deviation is -5.6 ppm; and the (M_Pl/m_e)^2 hierarchy deviation flips sign (arithmetically, since hierarchy = 1 / (m_e/M_Pl)^2) from -29 ppm to +11 ppm. No structural conclusions change. The companion library is bumped to v0.1.2.