The Continuum — Part IV: The Field Equation, the Newtonian Limit, and the Three Classical Tests of General Relativity

We present, for the first time in this series, the complete derivation of the Continuum's gravity, starting from the static pressure perturbation that a single bubble of matter creates in the surrounding space: P(r) = P0(1 - GM/c^2r), from which the dimensionless potential Psi(r) = -GM/(c^2r). We show that this same perturbation admits a direct physical interpretation - the Continuum flowing toward the mass at the classical escape velocity v_flow = sqrt(2GM/r), with Bernoulli reproducing P(r) exactly - a consistency check, not a new derivation, that links the same mechanism across very different scales (from the single proton to the atomic scale, where it correctly reproduces the well-known ~ 10^-40 ratio between gravity and electromagnetism). From this perturbation we construct the Continuum's complete field equation: a generalized Poisson equation for Psi, with a dynamical term (propagation at velocity c) and a nonlinear self-interaction term (alpha*Psi*grad^2 Psi). We show that its static, weak limit exactly reproduces Newtonian gravity as a pressure gradient of P0, with the equivalence principle emerging as a structural consequence - not an added postulate. We demonstrate that the coefficient alpha of the nonlinear term is constrained to alpha = 2 by the requirement of consistency with the known time component of Einstein's metric, g00 = -(1 + 2Phi/c^2). We then present a second derivation, genuinely independent of this comparison: from the Newtonian free-fall velocity combined with the advection of a wave in the flowing fluid (equivalent to the Painleve-Gullstrand formula), which fixes the same alpha = 2 without ever referencing Einstein's metric. We further show, via the complete Euler equations for our fluid, that this second route is not an additional physical ingredient but the same physics as the nonlinear term, seen more deeply - making alpha = 2 a necessary consequence of the Continuum's equation of state alone. From this same coefficient we derive the three historical classical tests of General Relativity: light deflection (theta = 4GM/c^2b, 1.75'' for the Sun, in agreement with Eddington 1919), gravitational redshift, and the perihelion precession of Mercury (Delta phi = 6*pi*GM/[c^2a(1 - e^2)], 43''/century against 42.98'' +/- 0.04''/century observed) - the latter with the complete, step-by-step development of the modified Binet equation. We conclude by positioning the Continuum relative to three established approaches to emergent gravity. With Sakharov (1967) and Jacobson (1995) we share the vision of G and General Relativity as emergent structures, distinguishing ourselves through the classical regime and the physical specification of the underlying mechanism. With Verlinde (2011, 2016) we share the emergence of the equivalence principle, differing in mechanism (hydrodynamic pressure versus entropy), in the explicit specification of the microstructure, and in the treatment of dark matter. A comparative table across seven dimensions closes the comparison. This work constitutes Part IV of a series: the next part will address black holes and their thermodynamics (Part IVA).