The Breathing Universe - A Torsional Solid Machine: A Mechanical Foundation for Spacetime and Cosmology

Title: The Breathing Universe - A Torsional Solid Machine: A Mechanical Foundation for Spacetime and Cosmology modified in 2026 March 17th version Authors: Alex Bertran (Institut d'Astrofísica de Badalona, IABDN) Abstract: This preprint presents a complete mechanical re-foundation of physics, proposing that spacetime is a discrete, elastic, fractal tetrahedral Mesh at the Planck scale. From this single substrate, we derive the emergence of spacetime geometry, quantum phenomena, and cosmic dynamics. The Mesh evolves through deterministic cycles of topological entanglement and relaxation, with the "Great Unknotting" replacing inflation, generating baryons via fractal friction, and imprinting Weyl Curvature Imprints (WCIs) that replace dark matter. Cosmic acceleration emerges as residual elastic pressure from incomplete relaxation. Formalized via the Jussà-Vila Ansatz (geometric inertia from WCIs) and the Scale Bridge Hypothesis (linking local curvature to global expansion), this framework eliminates singularities, explains quantum non-locality mechanically, reinterprets black holes as knot-traps, and resolves cosmological tensions (e. g. , Hubble tension, JWST early galaxies) without new particles or fields. All established physics—General Relativity, Quantum Mechanics, the Standard Model—emerges seamlessly from the Mesh's dynamics. Keywords: topological determinism, discrete tetrahedral mesh, Weyl Curvature Imprints (WCIs), Jussà-Vila Ansatz, Scale Bridge Hypothesis, fractal spacetime, quantum gravity, cosmology, dark matter alternative, dark energy alternative, Great Unknotting, Planck-scale mechanics, emergent spacetime. Description: This is an updated preprint version (v2) of the paper, building on the original framework to provide a unified mechanical substrate for physics. The tetrahedral Mesh ontology replaces the continuum manifold, offering solutions to quantum-gravity incompatibilities and cosmological anomalies. Key innovations include: derivation of physical constants from Mesh elasticity; quantum mechanics as statistical excitation propagation; and cosmology as topological relaxation cycles. Testable predictions include anisotropic Hubble expansion, decaying dark energy, and discrete spacetime signatures in high-energy observations. This work is licensed under a Creative Commons Attribution 4. 0 International License (CC-BY 4. 0). For the previous version, see DOI: 10. 5281/zenodo. 18394459 (or insert the actual OLD DOI if different). Feedback and collaborations welcome at alex. bertran@iabdn. cat. Files Included: TheBreathingUniverse_ATorsionalSolidMachine. tex (LaTeX source) Compiled PDF (if uploading) Figures and supplementary materials (if any) Related Works: Bertran (2025): The Weyl Curvature Imprint (WCI) Framework (DOI: 10. 5281/zenodo. 15270535) Bertran et al. (2025): The Universe's Geometric Inertia – The Jussà-Vila Ansatz (DOI: 10. 5281/zenodo. 17117607) Bertran (2026): The Scale Bridge Hypothesis (DOI: 10. 5281/zenodo. 17352947) What's new in this version 2026/02/26? This updated preprint significantly expands and formalizes the original conceptual framework into a more rigorous, mathematically grounded theory. Key improvements and new content include: Much deeper mathematical foundation New dedicated section: "Mathematical Foundations of the Tetrahedral Mesh" with formal definitions of the simplicial complex, primal-dual structure (Voronoi cells), discrete exterior calculus (cochains, exterior derivative, Hodge star, discrete Laplace-Beltrami operator), elastic dynamics equations, and explicit recovery of continuum limits. Visual and geometric clarification Introduction of a new figure illustrating the Primal-Dual Minimum Cluster (tetrahedron + circumsphere + Voronoi cell), making the core geometric unit much clearer. Emergence of fundamental physics from first principles Detailed mechanical/geometric origin of: electric charge (120° torsional phase shift in tetrahedral/FCC packing) spin-1/2 (720° double cover due to lattice connectivity) the fine-structure constant α⁻¹ ≈ 137 (impedance ratio between volumetric breathing and torsional modes) exactly three lepton families explained via knot theory on the tetrahedral lattice (unknot → electron, trefoil → muon, figure-eight → tau; no fourth family possible due to coordination number 4) Emergence of Maxwell's equations Explicit derivation as geometric/torsional flux identities on the dual lattice. Twelve Fundamental Principles New appendix summarizing the complete ontology of the Tetrahedral Mesh Universe in 12 clearly enumerated principles with corresponding mathematical expressions. Elastic foundation of particle masses and families New appendix deriving lepton mass hierarchy and the number 137 from FCC lattice resonance modes and topological knot stability. Enhanced predictions, falsifiability better sectioning, cross-references, and table summaries. Overall length and depth The manuscript is now ~3× longer, shifting from a mainly conceptual sketch to a self-contained theoretical framework with derivations, while preserving accessibility. These changes make the work substantially more rigorous, visually supported, and testable — transforming it from an exploratory idea into a comprehensive alternative paradigm for spacetime, matter, and cosmology. March 17th update: Title changed The Fine-Structure Constant from the Mesh topology