% ========================================================================
% © 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{Introduction}

The standard model of physics, encompassing general relativity (GR) for gravity and quantum field theory (QFT) for electromagnetic, weak, and strong forces, has achieved remarkable success in describing observable phenomena. However, it faces significant challenges: the incompatibility of quantum mechanics and gravity, the origin of the arrow of time, the probabilistic nature of quantum mechanics, and paradoxes such as black hole information loss. These issues suggest that current frameworks may be effective approximations rather than the fundamental reality.

While the core ontology and projection mechanism are presented in full, some technical aspects---most notably the complete functional derivation of the projected path integral beyond the saddle-point approximation and the detailed torsional realization of the entire fermion and gauge sector---remain at a preliminary stage. These developments, together with a full quantization of the 4D manifold, are left for future work and open collaboration.

\subsection{The Emergent +t Bias and Projection onto the Locally Biased Hypersurface}
\label{sec:plus-t-bias}

Following the formation of our local coherent $+t$ cluster, particles were propelled in all four directions. Through deformations of the manifold, matter clustered spatially \emph{and} temporally, forming large groups whose collective 4-velocity is, at any given epoch, aligned along what we locally label the $+t$ direction. This bias is purely relational and dynamical—exactly analogous to how the entire Solar System moves together in its Galactic orbit, with the sign of any velocity component relative to a distant observer periodically reversing.

All our observations are confined to the finite 4D region occupied by this local coherent matter cluster and are obtained by applying the projection operator \(\mathcal{P}\) (defined in Sec.~\ref{sec:explicit_jacobian}) onto this biased hypersurface. We only observe those segments of worldlines that intersect this collectively moving region of the manifold, mediated by the projection operator \(\mathcal{P}\) and by photons whose 4D trajectories cross our local cluster region at the condition that yields the observed speed \(c\).

From now on we refer to this collectively moving region of the manifold as ``our local cluster region'' (or simply ``our local cluster'').

\subsection{Intuitive picture of the theory}

Imagine the universe as a single, eternal 4D Euclidean manifold with coordinates $(t,x,y,z)$, where the coordinate $t$ is on exactly equal footing with the three spatial directions and carries no intrinsic direction or flow. There is no fundamental ``past'' or ``future'' coexisting as separate realms; every point in the manifold is simply a location in four-dimensional space.

Particles are not points moving through time: they are complete, eternal worldlines—geometric curves extending indefinitely in all four spatial dimensions. What we observe as a particle at any instant is merely the single point where its worldline intersects our locally biased hypersurface via the projection operator $\mathcal{P}$.

Following the local clustering event that formed our coherent matter group (the analogue of the Big Bang for our cluster), matter aggregated not only spatially but also temporally. Large regions—Earth, the Solar System, our galaxy—share a collective 4-velocity that, at the present epoch, points predominantly along the direction we label $+t$. This bias is purely local, relational and dynamical, exactly analogous to how the entire Solar System is dragged along the same galactic orbit: the sign of any velocity component relative to a distant observer periodically reverses. Consequently, our collective $+t$ bias can appear as $-t$ to observers in distant clusters at other cosmic epochs.

From our perspective, confined to this collectively moving local cluster region and viewing through the projection operator $\mathcal{P}$, everything appears to advance in a single temporal direction. At the subatomic scale, however, worldlines oscillate rapidly in $t$ (in addition to $x,y,z$). An electron orbiting a nucleus is also oscillating in the real temporal coordinate; its repeated crossings of our local cluster region project via $\mathcal{P}$ as the familiar probability cloud of quantum mechanics. Apparent ``collapse'' occurs when an environmental interaction fixes the oscillation phase at a definite intersection point.

The distance we measure with a ruler is never purely spatial: the two ends are also displaced in $t$. A photon traveling between them follows a genuine 4D trajectory in the Euclidean manifold. Only those photons whose 4D velocity satisfies the exact crossing condition with our moving cluster region (enforced by the entanglement-derived elastic filter $K\sim 1/l_P^2$) remain phase-coherent and are observable—and this condition projects precisely to the observed speed $c$.

Causality is determined by physical intersections and proximity in the 4D manifold, not by the numerical order of the $t$-label. If one were to travel to coordinates we would classically call the past, one would simply arrive at a different location in the manifold where the relevant worldlines are no longer present—everything has continued moving. There are no grandfather paradoxes: the grandfather is not waiting at those coordinates; his worldline has already moved elsewhere. Each interaction creates a new state; there is no rebinding or rewinding of time.

Gravitational time dilation is attraction in the $t$-coordinate itself: isotropic 4D deformations impart an extra velocity component along $+t$ to nearby particles, reducing the density of intersections with our local cluster region when viewed through $\mathcal{P}$. Dark matter consists of coherent worldlines whose dominant drift is in the $-t$ direction: they never intersect our local cluster region (hence invisible electromagnetically) but still deform the 4D manifold isotropically, producing the observed gravitational effects.

All equations of quantum mechanics, special and general relativity, and standard cosmology emerge exactly as projections via the operator $\mathcal{P}$ of this underlying 4D Euclidean geometry onto our local cluster region. The apparent arrow of time, the probabilistic nature of quantum mechanics, and the separation between space and time are illusions created by viewing the eternal, symmetric 4D reality through the narrow window of our collectively moving slice.