Classical Limit of the Genesis Harmonic Field Equation: Reduction to the Einstein Field Equations

The Genesis Harmonic Field Equation (GHFE) proposes an extended geometric framework in which spacetime curvature is sourced by additional covariant harmonic-sector fields representing tension, rotational-phase structure, and compression–shear modes in an effective vacuum medium. Any extension of general relativity must recover the Einstein field equations in the appropriate classical limit. In this work, the classical limit of the GHFE is defined through an explicit small-parameter scaling in which harmonic-sector gradients and amplitudes are parametrically suppressed. Under this controlled decoupling limit, the harmonic-sector stress–energy contributions vanish at leading order, leaving a coarse-grained effective source that reproduces the Einstein field equations. The analysis demonstrates that the GHFE admits general relativity as a consistent macroscopic regime while allowing additional dynamical structure outside the classical domain. A weak-field Newtonian limit is also derived, showing recovery of the standard Poisson equation. Finally, the scaling behavior of harmonic-sector corrections is presented, yielding a minimal phenomenological prediction that deviations from Einstein gravity scale quadratically with the harmonic-sector amplitude. This result establishes the GHFE as a mathematically consistent extension of general relativity with a well-defined classical limit.