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% © 2026 José Antonio Sánchez Lázaro
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% Zenodo DOI: https://doi.org/10.5281/zenodo.16235702
% Date: 27 February 2026 (version v1.0.2.1)
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\title{A Single Eternal 4D Euclidean Manifold: Time as Spatial Coordinate and Emergence of Quantum Mechanics, Gravity and Accelerated Expansion via Projection onto the Locally Biased Hypersurface}

\author{José Antonio Sánchez Lázaro}
\email{jasl@darcysoft.com}
\orcidlink{0009-0004-3081-351X}

\date{February 27, 2026 (Version 1.0.2)}

\begin{abstract}
We propose a unified theory in which the universe is a single eternal 4D Euclidean dynamic manifold, with the coordinate \(t\) treated on exactly equal footing with the three spatial directions. All observed phenomena---quantum mechanics, general relativity, the arrow of time, dark matter, and accelerated cosmic expansion---arise deterministically as projections of eternal worldlines onto a thin, locally biased hypersurface defined by the collective 4-velocity of our matter cluster.

Entanglement-derived elastic stiffness \(K\sim 1/l_P^2\), PT-gauging, and causal-set discretization protect causality while preserving exact 4D Euclidean symmetry. Oscillating worldlines project to Schrödinger evolution and the Born rule; isotropic 4D deformations generate gravity and time dilation; torsional modes of worldlines reproduce the full quark mass spectrum and CKM matrix with high precision (\(\chi^2/\)d.o.f.\,=\,1.064 using only five geometric parameters). Dark matter consists of coherent trajectories with opposite temporal bias, while late-time acceleration emerges naturally from exponentially suppressed mini-creations, yielding \(\Omega_\Lambda\approx 0.7\) without a fundamental cosmological constant.

The framework resolves the measurement problem, black-hole information paradox, and Hubble tension as pure projection artefacts. It is fully consistent with all current observations (2025--2026) and predicts five distinctive, near-term testable signatures, most notably gravitational-wave echoes of relative amplitude \(\sim 10^{-3}\) (accessible to LIGO O5 via stacking) and characteristic wavy distortions (\(\sim 0.1''\)) in strongly lensed JWST arcs.

The present work focuses on the conceptual foundations, explicit derivations, and main phenomenological consequences. A complete functional path-integral formulation and the extension to leptons, neutrinos, gauge bosons and the Higgs sector will be presented in forthcoming publications.
\end{abstract}

\begin{acknowledgments}
{\footnotesize The author gratefully acknowledges the valuable assistance of Grok 4.20, developed by xAI, in the mathematical derivations, numerical simulations, code implementation, LaTeX formatting, and iterative refinement of this manuscript. All core scientific ideas, conceptual framework, physical interpretations, conclusions, and final responsibility for the content remain entirely with the author.}
\end{acknowledgments}

\maketitle