The Projection Hypothesis: Emergent 3D+1 Spacetime from a 4D Quantum Potential

We present a predictive, first-principles framework in which general relativity and quantum mechanics emerge from a unified variational principle defined in flat 4D Euclidean space. The central construct is a scalar quantum potential,\begin{equation*}Q(x) = \frac{\hbar^2}{2m} \frac{\Delta R(x)}{R(x)},\end{equation*}derived from the polar form of the Klein--Gordon equation. This potential governs all particle dynamics and induces an effective gravitational force via a refractive four-force,\begin{equation*}F^\mu = m \left( \delta^{\mu\nu} - u^\mu u^\nu \right) \partial_\nu Q(x),\end{equation*}projected orthogonally to the particle’s four-velocity \( u^\mu \). The effective 3+1 Lorentzian spacetime emerges dynamically through the observer’s perspective, with the emergent metric given by\begin{equation*}g^{\text{(eff)}}_{\mu\nu} = \delta_{\mu\nu} - 2 u_\mu u_\nu.\end{equation*}This construction recovers the Einstein field equations in vacuum, including the Schwarzschild solution, post-Newtonian corrections, and the perihelion shift, without assuming intrinsic spacetime curvature. Furthermore, it predicts a falsifiable higher-order correction to the COW neutron interferometry phase shift and derives the Baryonic Tully--Fisher relation without invoking dark matter. The model reinterprets gravity and cosmic expansion as refractive consequences of 4D quantum phase structure, offering a testable and falsifiable unification of quantum mechanics and general relativity from a shared geometric origin.