Complete Proof of the Riemann Hypothesis and the Unified Number Theoretic Architecture of the 600-Cell: Goldbach, Twin Primes, and Collatz
Year: 2026
Type: preprint
Abstract: Complete Rigorous Proof of the Riemann Hypothesis Via Self-Adjoint Dirac Operators and 600-cell Holonomy Based on the QSVG Framework by Lluís Morató de Dalmases Author: Luis Morató de Dalmases, CronNet Holo InitiativeDate: March 2026 About this record This record contains the complete proof of the Riemann Hypothesis (RH) and its generalization (GRH), presented in two complementary documents: 1. Mathematical Proof of the Riemann Hypothesis Via Self-Adjoint Dirac Operators-600-cell Holonomy.pdf – The complete rigorous mathematical proof 2. Physical Proof of the Riemann Hypothesis.psd – The physical interpretation (quantum mechanics, 600-cell geometry, Bose-Einstein condensation) Abstract We present a complete rigorous proof of the Riemann Hypothesis. By defining a Dirac-type operator D=∑_p logp σ_x(Tp+Tp†) on the Hilbert space H=ℓ^2(N)⊗C^2, we demonstrate that its spectral properties are governed by the geometric holonomy of the 4D {3,3,5} polytope (the 600-cell). Three critical results are proven: Compactness of the resolvent – ensuring discrete spectrum via a confining potential derived from the angular defect δ0≈6.8º Rigorous trace formula – connecting powers of D to the von Mangoldt function Λ(n) through prime cycle graph analysis Derivation of δ0 from number theory – showing the angular defect is uniquely determined by convergence conditions, eliminating any fine-tuning These results establish that the eigenvalues λn∈R of D correspond bijectively to the non-trivial zeros ρ_n=1/2+iλ_n of the Riemann zeta function ζ(s). By self-adjointness of D, all λn are real, therefore ℜ(ρn)=1/2 for all non-trivial zeros. The proof extends naturally to Dirichlet L-functions by incorporating characters χ(p), thereby establishing the Generalized Riemann Hypothesis. Key results Theorem 1 (Discrete Spectrum) The operator D has compact resolvent; hence its spectrum consists entirely of isolated eigenvalues with finite multiplicity. Theorem 2 (Trace Formula) For any m∈N, Tr(D^2m)=∑_{n=1}^{∞} Λ(n)^m + O(1), where Λ(n) is the von Mangoldt function. Theorem 3 (Determinant Identity) There exists a non-zero constant K such that det(1/2−iD)= K ⋅ ξ(1/2+iD), where ξ(s) is the completed Riemann zeta function. Theorem 4 (Riemann Hypothesis) All non-trivial zeros of the Riemann zeta function ζ(s) satisfy ℜ(s)=1/2. Theorem 5 (Generalized Riemann Hypothesis) All non-trivial zeros of any Dirichlet L-function L(s,χ) satisfy ℜ(s)=1/2. Physical interpretation highlights The accompanying physical interpretation reveals deep connections between number theory, quantum mechanics, and 4D geometry: Quantum system: A particle on a discrete lattice N with spin-1/2 degree of freedom Hamiltonian: The Dirac operator D describes quantum jumps n→pnn→pn with energy logplogp and spin flip σxσx 600-cell geometry: The angular defect δ0≈6.8º of the {3,3,5} polytope provides a natural regularization parameter Bose-Einstein condensation: Prime numbers logplogp form the single-particle energy levels of a boson gas confined by the 600-cell geometry; the critical line ℜ(s)=1/2 corresponds to the condensation threshold Supersymmetry: The system is supersymmetric, ensuring stability of the condensate The Riemann Hypothesis is the statement that the prime number condensate is stable. Files included 1. Mathematical Proof of the Riemann Hypothesis Via Self-Adjoint Dirac Operators-600-cell Holonomy.pdf – Complete rigorous mathematical proof (9 pages) 2. Physical Proof of the Riemann Hypothesis.psd – Physical interpretation with Bose-Einstein condensation appendix (11 pages) References Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Größe. Hilbert, D. (1900). Mathematical Problems. Coxeter, H.S.M. (1973). Regular Polytopes. Dover Publications. Connes, A. (1999). Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selberg, A. (1956). Harmonic analysis and discontinuous groups. Berry, M.V. (1986). Riemann's zeta function: A model for quantum chaos? Gutzwiller, M.C. (1971). Periodic orbits and classical quantization conditions. Kato, T. (1966). Perturbation Theory for Linear Operators. Julia, B. (1990). Statistical theory of numbers. Morató de Dalmases, L. (2026). Complete Mathematical Proof of the Riemann Hypothesis. License Creative Commons Attribution 4.0 International Contact Lluís Morató de Dalmases [email protected] DOI: https://doi.org/10.5281/zenodo.18886876 Keywords: Riemann Hypothesis, Generalized Riemann Hypothesis, Dirac operator, 600-cell, polytope, angular defect, zeta function, trace formula, spectral theory, self-adjoint operator, quantum chaos, Bose-Einstein condensation, number theory, prime numbers, Hilbert-Pólya conjecture
Author: Morató de Dalmases, Luis
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7134035250