The Law of Topological Scaling The Golden Ratio as the Code of the Universe
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2026
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| author | Магтымов, Гадам |
| author_facet | Магтымов, Гадам |
| contents | <p><span lang="EN-US">This paper formulates the fundamental law of topological scaling derived from the geometry of the regular four-dimensional polytope <em>H</em><sub>4 </sub>(the 600-cell). It is shown that all physical quantities with a dimension of length to a certain power obey a discrete hierarchy:</span></p> <p><em><span lang="EN-US">Q</span></em><em><span lang="EN-US">n </span></em><span lang="EN-US">= <em>Q</em></span><span lang="EN-US">0 </span><em><span lang="EN-US">· </span></em><em><span>φ</span></em><em><span lang="EN-US">−dn</span></em><em><span lang="EN-US">,</span></em></p> <p><em><span lang="EN-US">√</span></em></p> <p><span lang="EN-US">where </span><em><span>φ</span></em><em><span> </span></em><span lang="EN-US">= (1 + 5)<em>/</em>2 <em>≈ </em>1<em>.</em>618 is the Golden Ratio, acting as the invariant scale factor of the <em>H</em><sub>4 </sub>projection operator; <em>d </em>is the dimension of the quantity, and <em>n </em>is an integer. It is proven that the discrete scaling step is an exact analytical solution to the renormalization group equations on the <em>H</em><sub>4 </sub>lattice, intrinsically guaranteeing the invariance of the speed of light <em>c </em>and the quantum of action ℏ. Experimental values of particle masses and cosmological parameters satisfy this law. Atomic matter forms at the level <em>n </em>= 120 (the number of vertices of the 600-cell), the electron is at <em>n </em>= 103, and the current epoch of the Universe’s expansion converges at <em>n </em>= 161. This allows for explaining the numerical values of the constants without introducing fitting parameters, reducing them to the pure kinematics of space-time.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19158153 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Law of Topological Scaling The Golden Ratio as the Code of the Universe Магтымов, Гадам <p><span lang="EN-US">This paper formulates the fundamental law of topological scaling derived from the geometry of the regular four-dimensional polytope <em>H</em><sub>4 </sub>(the 600-cell). It is shown that all physical quantities with a dimension of length to a certain power obey a discrete hierarchy:</span></p> <p><em><span lang="EN-US">Q</span></em><em><span lang="EN-US">n </span></em><span lang="EN-US">= <em>Q</em></span><span lang="EN-US">0 </span><em><span lang="EN-US">· </span></em><em><span>φ</span></em><em><span lang="EN-US">−dn</span></em><em><span lang="EN-US">,</span></em></p> <p><em><span lang="EN-US">√</span></em></p> <p><span lang="EN-US">where </span><em><span>φ</span></em><em><span> </span></em><span lang="EN-US">= (1 + 5)<em>/</em>2 <em>≈ </em>1<em>.</em>618 is the Golden Ratio, acting as the invariant scale factor of the <em>H</em><sub>4 </sub>projection operator; <em>d </em>is the dimension of the quantity, and <em>n </em>is an integer. It is proven that the discrete scaling step is an exact analytical solution to the renormalization group equations on the <em>H</em><sub>4 </sub>lattice, intrinsically guaranteeing the invariance of the speed of light <em>c </em>and the quantum of action ℏ. Experimental values of particle masses and cosmological parameters satisfy this law. Atomic matter forms at the level <em>n </em>= 120 (the number of vertices of the 600-cell), the electron is at <em>n </em>= 103, and the current epoch of the Universe’s expansion converges at <em>n </em>= 161. This allows for explaining the numerical values of the constants without introducing fitting parameters, reducing them to the pure kinematics of space-time.</span></p> |
| title | The Law of Topological Scaling The Golden Ratio as the Code of the Universe |
| url | https://doi.org/10.5281/zenodo.19158153 |