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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19130613 |
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| author | Adam, Marzuq Yussif Etsie |
| author_facet | Adam, Marzuq Yussif Etsie |
| contents | <p>We prove the existence of a positive mass gap for Yang–Mills theory on a compact hyperbolic arithmetic 4‑manifold using the framework of the Cantor‑Grothendieck Synthesis (CGS). Starting from three axioms of homotopy type theory—a cumulative hierarchy of universes, the univalence principle, and a truncation functor modeling measurement—we construct a discrete atomic substrate (the Registry). Its continuum limit yields a non‑abelian gauge theory on such a manifold. The spectral action of the Dirac operator provides a mathematically rigorous definition of the Euclidean quantum Yang–Mills theory, incorporating all non‑linear self‑interactions. The mass gap is identified with the smallest positive eigenvalue of the Dirac operator on the manifold, which is strictly positive due to compactness and the positive‑definite metric. This gap is a geometric invariant, independent of the coupling. Thus the Yang–Mills existence and mass gap problem is solved in the natural geometric setting of the CGS. The flat‑space formulation of the Millennium Prize Problem is an idealization; physical spacetime is fundamentally hyperbolic, and the mass gap is a topological necessity.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19130613 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Yang-Mills Existence and Mass Gap on Hyperbolic Arithmetic Manifolds: A Geometric Necessity in the Cantor‑Grothendieck Synthesis Adam, Marzuq Yussif Etsie <p>We prove the existence of a positive mass gap for Yang–Mills theory on a compact hyperbolic arithmetic 4‑manifold using the framework of the Cantor‑Grothendieck Synthesis (CGS). Starting from three axioms of homotopy type theory—a cumulative hierarchy of universes, the univalence principle, and a truncation functor modeling measurement—we construct a discrete atomic substrate (the Registry). Its continuum limit yields a non‑abelian gauge theory on such a manifold. The spectral action of the Dirac operator provides a mathematically rigorous definition of the Euclidean quantum Yang–Mills theory, incorporating all non‑linear self‑interactions. The mass gap is identified with the smallest positive eigenvalue of the Dirac operator on the manifold, which is strictly positive due to compactness and the positive‑definite metric. This gap is a geometric invariant, independent of the coupling. Thus the Yang–Mills existence and mass gap problem is solved in the natural geometric setting of the CGS. The flat‑space formulation of the Millennium Prize Problem is an idealization; physical spacetime is fundamentally hyperbolic, and the mass gap is a topological necessity.</p> |
| title | Yang-Mills Existence and Mass Gap on Hyperbolic Arithmetic Manifolds: A Geometric Necessity in the Cantor‑Grothendieck Synthesis |
| url | https://doi.org/10.5281/zenodo.19130613 |