The Emergent Field Equations of Non-Linear Spacetime Mechanics: A Substrate Framework for Quantum Gravitational Phenomenologies

We derive the full non-linear field equations of Non-Linear Spacetime Mechanics (NLSM), demonstrating that General Relativity emerges as an effective low-energy continuum limit of a deeper viscoelastic substrate. The cosmological constant Λ(t) ≈ 3/R_H(t)^2 is identified as a dynamical pre-stress anchored at the cosmic horizon scale. In galactic regimes, the elastic strain response yields a modified Newtonian dynamics (MOND) limit without extra dark matter particles. For high-energy cluster collisions, a substrate viscoelastic relaxation time τ ≈ 1.8 Gyr produces a spatial offset (≈ 78.4 kpc) between baryonic matter and gravitational potential peaks, explaining Bullet Cluster observations. Rigorous proofs for diffeomorphism invariance and perturbative stability (m_eff^2 ≥ 0) confirm the mathematical integrity of the field equations.