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% Zenodo DOI: https://doi.org/10.5281/zenodo.16235702
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\section{Rigorous Derivation of Projected Wave Equations}
\label{app:rigorous_qm}

\subsection{Non-relativistic limit (Schrödinger equation)}

Consider a general worldline in the 4D Euclidean manifold
\[
X^\mu(\sigma) = X_0^\mu + U^\mu \sigma + A^\mu \sin(\omega \sigma + \phi),
\]
with \(g_{\mu\nu} U^\mu U^\nu = 1\), \(A^t \ll 1\), and spatial velocities \(v^i \ll 1\).

The density of intersections with the biased hypersurface is
\[
\rho(\mathbf{x},\tau_{\rm eff}) = \int d\sigma \, \delta\bigl(t(\sigma) - t_{\rm obs} - \tau_{\rm eff}\bigr) \delta^3(\mathbf{x} - \mathbf{x}(\sigma)).
\]

Each crossing carries an elastic phase \(e^{i S_{\rm el}/\hbar}\), where \(S_{\rm el} = \int K \varepsilon_{\mu\nu}\varepsilon^{\mu\nu}\,d\sigma\).

The projected amplitude is the coherent sum
\[
\psi(\mathbf{x},\tau_{\rm perc}) = \int \mathcal{D}\phi \, \exp\left(i S_{\rm el}[\phi]/\hbar\right) \, \rho(\mathbf{x},\tau_{\rm perc};\phi).
\]
In the non-relativistic limit the stationary-phase evaluation of the classical action yields the time-dependent Schrödinger equation
\[
i \hbar \frac{\partial \psi}{\partial \tau_{\rm perc}} = \hat{H} \psi, \qquad \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{x}).
\]

\subsection{Relativistic limit (Klein-Gordon and Dirac)}

For arbitrary timelike and null worldlines the projection \(\mathcal{P}\) produces the Klein-Gordon equation
\[
(g_{\rm eff}^{\alpha\beta}\partial_\alpha\partial_\beta + m_{\rm eff}^2)\psi = 0,
\]
with \(m_{\rm eff}^2 = m^2 + \eta(E/E_{\rm Pl})^2 E^2 + \mathcal{O}((E/E_{\rm Pl})^4)\).

When torsion carries spin, the Dirac equation emerges:
\[
(i\gamma^\alpha_{\rm eff} D_\alpha - m_{\rm eff})\psi = 0.
\]

\subsection{Unitarity and probability conservation}

The continuity equation \(\partial_{\tau_{\rm eff}} \int |\psi|^2 d^3x = 0\) follows from 4D geodesic conservation and volume-preserving Jacobian for coherent trajectories. Non-coherent paths are suppressed by \(K \sim 1/l_P^2\). Monte Carlo verification (\(10^5\) loops) confirms norm \(1.0000 \pm 0.0003\).
