Title: The Tetrahedral Mesh and the Architecture of Matter: From Gell-Mann's Octet to the Triply Charmed and Triply Bottom Baryons Authors: Alex Bertran In March 2026, LHCb announced the observation of the doubly charmed baryon Ξ₂₂^+ with a mass around 3620 MeV/c². In June 2026, they announced Ω₂₂^+ with mass 3964. 7 ± 1. 2 MeV/c². In the Standard Model, quark masses are free parameters. In the Mesh framework, they are not. The Cosmic Mesh is a closed, self‑similar, Face‑Centered Cubic (FCC) lattice of unbreakable elastic tetrahedra at the Planck scale. Each tetrahedron can twist in discrete steps of 120°. A single deterministic rule governs all motion: a vibration moves to the neighbor that maximizes the local entanglement gradient. No probability. No collapse. Particles are standing waves trapped in topological knots. Quarks are vibrational modes in small knots. Baryons are stable triplets of colored knots (red, green, blue) that mutually stabilize via the migration rule. Using only the geometric origin of the fine‑structure constant (α⁻¹ = 135 + 2πφ/5) and the dragging correction ε = 1/169 ≈ 0. 005917 derived from the fundamental 13‑cluster (1 central tetrahedron + 12 neighbors), we predict: - Ω₂₂₂^++ (triply charmed) ≈ 4950 MeV/c² (range 4870–5070 MeV/c²) - Ω₁₁₁^-- (triply bottom) ≈ 13420 MeV/c² (range 13120–13720 MeV/c²) Both predictions are falsifiable with LHCb Run 3/4 data. Their observation would confirm that particle masses are not arbitrary constants but emerge from the geometry and elasticity of spacetime itself. This preprint applies the framework developed in “THE BREATHING UNIVERSE: A Torsional Solid Machine” (Zenodo, 2026). It shows that the same mechanical principles that give the lepton masses, the fine‑structure constant, and the gravitational constant also predict new particles with no free parameters. Keywords: particle physics, baryons, exotic hadrons, LHCb, tetrahedral lattice, discrete spacetime, topological defects, mass prediction, quark model, emergent particle masses, triply charmed baryon, triply bottom baryon, golden ratio, 13‑cluster License: Creative Commons Attribution 4. 0 International (CC BY 4. 0) Related works: Bertran, A. (2026). THE BREATHING UNIVERSE: A Torsional Solid Machine. Zenodo. Bertran, A. , Jussà, E. , Vila, M. (2025). The Universe's Geometric Inertia – The Jussà-Vila Ansatz. Zenodo. Bertran, A. (2026). The End of the Copenhagen Interpretation: Einstein Was Right — Reality Is Local, Deterministic, and Made of a Discrete Elastic Mesh. Zenodo. Updates: Update v1: "Stiffness matrix eigenvalues" suppressed. Pure geometric origin for α⁻¹ Version 2. 0 (June 2026): - Added the Voronoi cell of the FCC lattice (truncated octahedron) as the geometric origin of the baryon decuplet and octet. - Showed that the two classic Gell-Mann diagrams are 2D projections of the same polyhedron along different symmetry axes. - Introduced the distinction between hexagonal faces (stable, closed paths) and square faces (unstable, open paths) of the Voronoi cell. - Explained particle instability (short lifetimes) as energy leakage through square faces (open topology). - Added the full hierarchy of short-lived particles for charm and bottom families (Levels 1-4). - Clarified that beyond the third coordination shell (Z₃=92), no closed knots exist. Higher excitations are necessarily unstable and dissipative. - Explicitly defined the fundamental 13-cluster (1 central tetrahedron + 12 neighbors) and the Sigman pressure (13² = 169) as the elastic limit of the Mesh. - Added pedagogical boxes (pnote, conceptbox, predictionbox) for clarity. - Updated the title to reflect the geometric unification: "From Gell-Mann's Octet to the Triply Charmed and Triply Bottom Baryons". - Added references to the Voronoi cell and FCC crystallography. - All previous content (mass predictions, consistency with Ξcc⁺ and Ωcc⁺, mass ranges, decay channels) remains unchanged.
BERTRAN ALEX (Thu,) studied this question.