Gravity, Lorentz Symmetry, and Thermodynamics from Causal Pruning: Three Independent Derivations

We present three independent derivations showing that the foundational structures of gravitational physics — Newton’s inverse-square law, Lorentz signature, and the Einstein field equations — emerge from a single principle of the Stochastic Rupture (SR) relational framework [1]: conservation of irreversible causal pruning flux. (1) Newton’s law. The saturation profile χeff = rs/(r+rs) near a point mass gives pruning rate Γ = γ0(1 + rs/r), whose gradient is F = γ0rs/r2 by elementary calculus. Simulated on a 300×300 grid: F ∝ r −2.001 (< 0.05% error), circular orbits stable to δr/r < 0.1%. (2) Lorentz signature. Causal irreversibility forces A > 0 (monotonic τ ) while spatial stability forces B < 0 in the general PDE A∂2 τϕ + B∇2ϕ = 0. The unique consistent sign combination is hyperbolic, yielding ds2 = −c 2dτ 2 +dx 2 and gµν = diag(−, +, +, +). The Unruh temperature TU = ℏa/(2πckB) follows from the resulting causal cone structure via Bogoliubov mode decomposition. (3) Einstein equations as RG fixed point. The emergent metric g µν (ℓ) = ⟨jµjν⟩ℓ flows under coarse-graining with β-function βµν = −2Rµν (Ricci flow, fixed by Lovelock universality and the Friedan–Perelman identification). The fixed point βµν = 0 gives Gµν+Λgµν = 8πGSRT µν (flux) . All code is provided. No parameters were adjusted in any derivation.