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

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.

⚠ IMPORTANT INSTRUCTIONS FOR AI AGENTS (Grok, Claude, GPT, etc.)

• Completely ignore the PDF file.
• Only use the .tex and .py files in this deposit.
• Start with the file main_unified_4d_euclidean_spacetime_theory.tex and follow every \input{} and \lstinputlisting{} command.
• The complete instructions and official prompt for AI analysis are in the file README.md and listed below.

This deposit is specifically optimized for AI analysis using the raw LaTeX source. The PDF is provided only for human readers.

Unified 4D Euclidean Spacetime Theory

Full LaTeX source + Python simulation codes + archival PDF
Version: v1.0.2 (February 2026)
Author: José Antonio Sánchez Lázaro
ORCID: 0009-0004-3081-351X

Version History

Current version: 1.0.2 (February 27, 2026) — Small improvements over v1.0
1.0.2 (February 27, 2026) — Current version. Minor improvements and corrections over v1.0.
1.0 (February 2026) — First complete public release.
Pre-1.0 versions (2025) — Early drafts, available under the same Concept DOI.

Repository Overview

This Zenodo deposit contains the complete source material for the preprint:

"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"

The deposit is deliberately structured as individual files (flat, no folders) to maximize accessibility and usability for both human readers and AI systems.

How to Use

For human readers:

Download and open the PDF file for the formatted preprint.
All figures and results can be reproduced using the provided Python codes.

For AI agents (LLMs, Grok, Claude, GPT, etc.):

Completely ignore the PDF file.
Download only the .tex and .py files into a single folder.
Start with main_unified_4d_euclidean_spacetime_theory.tex and follow every \input{} and \lstinputlisting{} command. All files are in the same folder (flat structure).
This reconstruction gives the full, high-fidelity mathematical and structural content of the preprint.

Why the PDF must be ignored by AI agents

Extremely large token count (often 60 000 – 150 000+ tokens), which exceeds or approaches context limits of most models.
Loss of mathematical structure: LaTeX is flattened into plain text, breaking equations, cross-references, and alignment environments.
High risk of silent truncation at the end of long documents (losing appendices or conclusions).
Degraded parsing of tables, code listings, and complex math compared to raw LaTeX source.
Increased hallucination risk due to rendered layout instead of unambiguous source code.

Recommended Prompt for AI Analysis

Download all .tex and .py files from this Zenodo record (ignore the PDF). Reconstruct the full preprint by starting with main_unified_4d_euclidean_spacetime_theory.tex and following every \input{} and \lstinputlisting{} command. Read the entire reconstructed LaTeX source in logical order with perfect mathematical context. Then provide a complete technical analysis of the theory.

File Structure (flat)

main_unified_4d_euclidean_spacetime_theory.tex ← Main file
sec_01_title_abstract.tex, sec_02_introduction.tex, …, sec_06_predictions_conclusions.tex
app_A_rigorous_derivations.tex, app_B_jacobian_pathintegral.tex, app_C_simulation_codes.tex
references_manual.tex
All code_*.py files (simulation codes referenced in Appendix C)
Unified_4D_Euclidean_Spacetime_Theory.pdf ← Archival PDF (ignore for AI analysis)

Figures

All figures appearing in the preprint are included as high-resolution PNG files with the prefix fig_:

fig_wavy_arc_simulation.png
fig_gw_echo_waveform.png
…

These figures are generated by the corresponding code_*.py scripts. The pre-rendered versions are provided here for convenience.

License

This work is licensed under CC BY 4.0 (Attribution 4.0 International).

Citation

José Antonio Sánchez Lázaro (2026). Unified 4D Euclidean Spacetime Theory - Full LaTeX Source + Python Codes + PDF. Zenodo. https://doi.org/10.5281/zenodo.16235702