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

We propose a framework unifying general relativity and quantum mechanics via a variational principle in flat 4D Euclidean space. A scalar quantum potential,\begin{equation}Q(x) = \frac{\hbar^2}{2 m_{\text{eff}}(x)} \frac{\Delta R(x)}{R(x)},\end{equation}from the Klein--Gordon equation governs dynamics, inducing gravity through a refractive four-force,\begin{equation*}F^\mu = m (\delta^{\mu\nu} - u^\mu u^\nu) \partial_\nu Q(x),\end{equation*}orthogonal to the four-velocity \( u^\mu \). The emergent 3+1 Lorentzian metric,\begin{equation*}g^{\text{(eff)}}_{\mu\nu} = \delta_{\mu\nu} - 2 u_\mu u_\nu,\end{equation*}recovers Einstein’s vacuum equations, Schwarzschild solution, and perihelion shift without intrinsic curvature. It predicts a testable COW phase shift correction and derives the Baryonic Tully-Fisher relation, explaining flat rotation curves without 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.