600-Cell Spectral Triple Series: A Complete Derivation of the Standard Model and Gravity from Noncommutative Geometry

600-Cell Spectral Triple Series: A Complete Derivation of the Standard Model and Gravity from Noncommutative Geometry

Author:
L. Morató de Dalmases

Publication Date:
2026-04-14

DOI:
https://doi.org/10.5281/zenodo.19592588

Version:
1.0

Language:
English

Abstract

This series presents a complete derivation of the Standard Model of particle physics coupled to Einstein gravity from the spectral triple of the 600-cell Coxeter group H4. We construct the spectral triple, prove uniqueness of H4, derive the internal algebra C ⊕ H ⊕ M3(C), obtain three fermion generations via a 53-cycle Möbius encapsulation, derive gauge fields from inner fluctuations, obtain gravity from the spectral action, compute the mass hierarchy and mixing matrices, fix the vacuum scale f0 = 12.8 THz via numerical diagonalization, prove spectral reconstruction and rigidity, and conclude with a final classification theorem establishing uniqueness. All physical observables are spectral invariants; the moduli space consists of a single point.

Keywords

Noncommutative geometry; Spectral triple; 600-cell; Coxeter group H4; Standard Model; Quantum gravity; Gauge theory; Fermion generations; Mass hierarchy; CKM matrix; PMNS matrix; Spectral action; Heat kernel; Yang-Mills; Einstein-Hilbert; Unification; 53-cycle

Table of Contents

Papers I-XII:

1. Paper I: Spectral Geometry Foundations of the 600-Cell

2. Paper II: Coxeter Rigidity and Uniqueness of H4

3. Paper III: Emergence of the Internal Algebra

4. Paper IV: Dirac Spectrum, Torsion, and Fermion Generations

5. Paper V: Gauge Fields from Inner Fluctuations

6. Paper VI: Spectral Action and Gravity

7. Paper VII: Fermion Mass Spectrum from 53-Cycle Geometry

8. Paper VIII: CKM and PMNS Mixing from Coherent States

9. Paper IX: Vacuum Scale f0 = 12.8 THz

10. Paper X: Spectral Reconstruction Theorem

11. Paper XI: Rigidity and No-Deformation Theorem

12. Paper XII: Final Classification and No-Existence Theorem

Appendices A-L:

• A: Minimal Axiomatic Structure

• B: Meta-Theorem of Minimal Spectral Determination

• C: Physical Predictions and Falsifiability

• D: Reserved

• E: Minimal Axiomatic Reconstruction of Physics

• F: Mathematical Gap Analysis

• G: From Spectral Triple to Quantum Fields

• H: Principle of Non-Arbitrariness

• I: Open Rigidity Problem

• J: Final Rigidity Formulation

• K: Complete Proof of the U Automorphism of Order 53

• L: Complete Solution for the Cosmological Constant and Source Code

Main Results

Geometric structure:

• Hilbert space: H = l2(V) ⊗ C4, dim H = 480

• 120 vertices, 720 edges, vertex degree 12

• Coxeter group H4 (order 14400)

Internal algebra:
AF = C ⊕ H ⊕ M3(C)

• C → U(1) hypercharge

• H → SU(2) weak isospin

• M3(C) → SU(3) color

Gauge group:
G = U(1) × SU(2) × SU(3)

Three fermion generations:

• Origin: spectral automorphism U of order 53

• 53 distinct positive eigenvalues, multiplicity 2

• Orbit decomposition: sizes 22, 8, 1

Mass formula:
mk = m0 · exp( βk · csc(π αk / 53) )

• α1 = 22, α2 = 8, α3 = 1

• mμ/me ≈ 207, mτ/mμ ≈ 16.8

Gravity:

• Spectral action: S[Λ] = Tr f(DA/Λ)

• Heat kernel expansion

• Newton's constant: G-1 = (80 f2 / π) Λ2

Vacuum scale:
f0 = 12.8 THz
E0 = h f0 = 0.053 eV (neutrino mass scale)

Rigidity:

• Moduli space: M ≅ {*}

• No free parameters

Numerical Constants Table

Fundamental Equation

Standard Model + Gravity = Spec(D600-cell) ⊕ {U}

with U53 = I, U ∉ Ggeom, and M ≅ {*}

Additional Information for Zenodo

License:
Creative Commons Attribution 4.0 International (CC BY 4.0)

Main references:

• Connes, A., Noncommutative Geometry, Academic Press, 1994

• Connes, A. & Chamseddine, A., The spectral action principle, Commun. Math. Phys. 186 (1997), 731-750

• Coxeter, H.S.M., Regular Polytopes, Dover, 1973

Contact:
morato.lluis@gmail.com

Deposit Notes

This document contains 119 pages with mathematical notation in plain text (Unicode). All formulas can be read directly without additional LaTeX processing. The complete source code for numerical diagonalization is provided in Appendix L.