Emergent Gravity from the Phi_MDL Field: Einstein Equations, Kink Sources, and Quantum Gravity Scale

Starting from the -symmetric Klein–Gordon field with V = (m_φ^2/49)(1- 7Φ) and m_φ = m_τ = 1776.86MeV, we derive six gravitational results. (1) T_μν = tial_μΦ tial_νΦ - η_μνL is machine-certified in Lean~4 (zero ; symmetry, vacuum-vanishing, and gravity prerequisites in .lean). (2) The linearized Einstein equations G_μν = 8π G_N T_μν follow from MDL-Lovelock correspondence (P35) combined with minimal coupling. (3) The BPS kink is a gravitational source: $∈t T_00dx = M_ kink = 290.10MeV (relative error 1.4×10^-6$); conservation holds to 6×10^-12. (4) The classical cosmological constant vanishes at all seven vacua; the one-loop correction Δ V_ CW ≈ -1.7×10^7MeV^4 exceeds the observed dark-energy scale by 10^42 — identified but not resolved. (5) The quantum gravity scale is M_ Pl^ tot = π/√(3) ≈ 1.81 lattice units from the Nyquist residual ε_0(M) = π^2/(3M^2). (6) The three-tape CMCA τ_c gradient-kick mechanism recovers Newtonian gravity F = G_ effM/(4π b^2) in the continuum limit ( , P45~). Newton's constant G_N is derived via the gravitational hierarchy formula ( ).