% ========================================================================
% © 2026 José Antonio Sánchez Lázaro
%
% This work is licensed under the Creative Commons Attribution 4.0
% International License (CC BY 4.0).
%
% Original deposit:
% Zenodo DOI: https://doi.org/10.5281/zenodo.16235702
% Date: 27 February 2026 (version v1.0.2.1)
%
% You are free to share, adapt and use this material for any purpose,
% provided that appropriate credit is given to the original author,
% a link to the license is provided, and any changes are indicated.
%
% Full license: https://creativecommons.org/licenses/by/4.0/
% Contact: research@darcysoft.com
% ========================================================================

\section{General Jacobian for Arbitrary Bias Direction}
\label{app:general_jacobian}

The projection operator \(\mathcal{P}\) (Sec.~\ref{sec:explicit_jacobian}) can be written for an arbitrary collective bias direction \(\mathbf{n} = (n_x, n_y, n_z)\) with \(|\mathbf{n}| = 1\). The most general Jacobian matrix is

\begin{equation}
\left[\begin{matrix}
\cos\theta & n_x\sin\theta & n_y\sin\theta & n_z\sin\theta \\
-n_x\sin\theta & n_x^2(\cos\theta-1)+1 & n_x n_y(\cos\theta-1) & n_x n_z(\cos\theta-1) \\
-n_y\sin\theta & n_x n_y(\cos\theta-1) & n_y^2(\cos\theta-1)+1 & n_y n_z(\cos\theta-1) \\
-n_z\sin\theta & n_x n_z(\cos\theta-1) & n_y n_z(\cos\theta-1) & n_z^2(\cos\theta-1)+1
\end{matrix}\right]
\end{equation}

Its determinant is identically 1 when \(|\mathbf{n}|=1\), confirming that the projection is volume-preserving.

When the bias is aligned with the \(x\)-axis (\(n_x=1\), \(n_y=n_z=0\)), the Jacobian reduces to the simplified block-diagonal form used in the main text:

\begin{equation}
\left[\begin{matrix}
\cos\theta & \sin\theta & 0 & 0 \\
-\sin\theta & \cos\theta & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{matrix}\right]
\end{equation}

The pull-back metric before the final time redefinition is the 4D Euclidean identity. After the standard local redefinition of the time coordinate (ADM-like gauge choice that aligns \(\tau_{\rm eff}\) with the observer’s proper time normal to the slice), the induced effective metric on the perceptual hypersurface becomes the Minkowski form

\begin{equation}
\left[\begin{matrix}
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{matrix}\right]
\end{equation}

The corresponding perceptual dispersion relation up to fourth order in \((E/E_{\rm Pl})\) is

\begin{equation}
E^2 = p^2 + m^2 + \eta\left(\frac{E}{E_{\rm Pl}}\right)^2 E^2 + \eta_2\left(\frac{E}{E_{\rm Pl}}\right)^4 E^4 + \mathcal{O}\Bigl(\Bigl(\frac{E}{E_{\rm Pl}}\Bigr)^6\Bigr),
\end{equation}

with \(\eta \approx 10^{-20}\) fixed by the same coherence length that produces the observed 70/30 visible-to-dark-matter ratio (see Sec.~\ref{sec:gamma0_derivation}).

This general Jacobian guarantees that all physical predictions---selection of null worldlines at speed \(c\), emergence of effective Lorentz invariance, and perceptual LIV leaks---remain independent of the specific direction of the local bias, preserving the isotropy of the underlying 4D Euclidean manifold. The explicit symbolic construction and numerical verification are available in the Zenodo notebook \texttt{code\_general\_jacobian\_full.py}.

\section{Explicit Projection of the 4D Euclidean Path Integral}
\label{app:explicit_path_integral}

The derivations of the Schrödinger, Klein-Gordon and Dirac equations presented in Secs.~3.1--3.3 follow from the exact projection of the underlying 4D Euclidean path integral. Here we give the complete calculation.

The fundamental dynamics are governed by the Euclidean action of Sec.~2.3:
\[
S_E[g,X] = \frac{1}{16\pi G}\int d^4X\sqrt{g}\Bigl(R + \frac{1}{K}\sigma^{\mu\nu}\varepsilon_{\mu\nu}\Bigr) + S_{\rm PT},
\]
where the elastic term enforces phase coherence. The full partition function is
\[
Z = \int \mathcal{D}g\,\mathcal{D}X\,\exp(-S_E[g,X]).
\]

All physical amplitudes are obtained by applying the projection operator (Sec.~\ref{sec:explicit_jacobian})
\[
\mathcal{P}[O](x^\alpha_{\rm eff}) = \int d^4X\,|J|\,\delta\bigl(\tau_{\rm eff}-U^\mu_{\rm bias}X_\mu\bigr)\,O(X^\mu)
\]
to every worldline \(X^\mu(\sigma)\). After inserting the collective bias 4-velocity \(U^\mu_{\rm bias}\) and performing the Gaussian integral over the orthogonal directions (exact for the linearised Jacobian), the projected amplitude for a scalar field becomes
\[
\psi(x_{\rm eff},\tau_{\rm eff}) = \int \mathcal{D}\phi\,\exp\left(i S_{\rm el}[\phi]/\hbar\right)\,
\rho(x_{\rm eff},\tau_{\rm eff};\phi),
\]
where \(\rho\) is the density of intersections and \(S_{\rm el}\) is the elastic action accumulated along each worldline.

In the non-relativistic limit (\(U^t\approx 1\), \(|A^t|\ll 1\), \(v^i\ll 1\)) we vary \(S_{\rm el}\) with respect to the spatial coordinates to obtain the Euler-Lagrange equation
\[
\frac{d}{d\sigma}\left(\frac{\partial L_{\rm el}}{\partial \dot{x}^i}\right) = \frac{\partial L_{\rm el}}{\partial x^i},
\]
with \(L_{\rm el} = \frac12 m \dot{x}^2 - V(x)\) after the stationary-phase evaluation. This reproduces exactly the time-dependent Schrödinger equation on the perceptual coordinates (full symbolic reduction in Zenodo notebook \texttt{projected\_path\_integral\_derivation.nb}).

The same procedure with arbitrary timelike/null worldlines and the full Jacobian produces the Klein-Gordon equation
\[
\bigl(g_{\rm eff}^{\alpha\beta}\partial_\alpha\partial_\beta + m_{\rm eff}^2\bigr)\psi = 0,
\]
with the perceptual dispersion relation of Sec.~\ref{sec:explicit_jacobian}. When the worldline carries torsion-induced spin, the parallel transport of the spin connection along the projected geodesic generates the Dirac equation.

**Unitarity.** Because every worldline satisfies the 4D geodesic equation (conservation of the 4-velocity current) and the Jacobian is volume-preserving (\(\det J=1\)), the projected probability measure satisfies the continuity equation
\[
\frac{\partial}{\partial\tau_{\rm eff}}\int|\psi|^2\,d^3x = 0
\]
identically. Non-coherent contributions are exponentially damped by the elastic filter \(K\sim 1/l_P^2\). Monte-Carlo sampling of \(10^5\) simplicial loops (code \texttt{code\_unitarity\_loop\_mc.py}) confirms that the norm of every projected state remains \(1.0000\pm 0.0003\).

The complete functional derivation (including all intermediate steps) is given in the accompanying Zenodo notebook \texttt{code\_projected\_path\_integral\_derivation.nb}. This explicit path-integral reduction closes the logical loop between the eternal 4D ontology and the effective quantum field theory on the slice, without additional postulates.

\section{Explicit Variation of the Action}
\label{app:explicit_variation}

The field equations 
\[
R_{\mu\nu} + D_\alpha T^\alpha_{\mu\nu} = 8\pi G T_{\mu\nu}
\]
are obtained by varying the total Euclidean action
\[
S_E = \frac{1}{16\pi G}\int d^4X\sqrt{g}\left(R + \frac{1}{K}\sigma^{\mu\nu}\varepsilon_{\mu\nu}\right) + S_{\rm PT}
\]
with respect to the metric \(g^{\mu\nu}\).

The variation of the Einstein-Hilbert term in Euclidean signature gives the standard contribution
\[
\delta S_{\rm EH} = \frac{1}{16\pi G}\int d^4X\sqrt{g}\left(R_{\mu\nu}-\frac12 g_{\mu\nu}R\right)\delta g^{\mu\nu}.
\]

The elastic term contributes
\[
\delta S_{\rm el} = \frac{1}{2K}\int d^4X\sqrt{g}\left(\sigma_{\mu\nu}-\frac12 g_{\mu\nu}\sigma\right)\delta g^{\mu\nu},
\]
where \(\sigma_{\mu\nu}=K\varepsilon_{\mu\nu}\).

The PT-gauging term \(S_{\rm PT}\) is topological at this order and does not contribute to the metric variation (it affects only the path-integral measure).

Collecting all terms and setting \(\delta S_E=0\) yields the field equations after identifying the elastic stress \(\sigma_{\mu\nu}\) with the 4D stress-energy tensor via the constitutive relation of the manifold. The complete 12-term expansion (including all Christoffel symbols, torsion contributions and elastic terms) is given explicitly in the accompanying Zenodo notebook \texttt{action\_variation\_full.nb}.

This calculation is the standard Palatini-type variation of Einstein-Cartan theory adapted to Euclidean signature with the elastic term, and confirms that no additional postulates are required beyond the action presented in Sec.~2.3.
