Regular Simplex Hierarchical Gravity: Part V Zero-Parameter Derivation of the Local Effective Vertex Count Force-Chain Robustness and Stress Inertia from 600-Cell Geometry
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2026
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| _version_ | 1866901983985664000 |
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| author | Sato, Ryuhei |
| author_facet | Sato, Ryuhei |
| contents | <p>As Part V of the Regular Simplex Hierarchical Gravity (RSHG) framework,<br>this paper derives the local effective vertex count Ωlocal = 100 without any free<br>parameters.<br>The derivation begins with Niven’s theorem: the impossibility of filling three-<br>dimensional space with regular tetrahedra is established as an algebraic necessity<br>rather than a numerical coincidence. The irreducible angular deficit δ = 2π −<br>5 arccos(1/3) arising from this failure serves as the sole seed of the following geo-<br>metric chain:<br>• The ideal vertex count Ωideal ≈ 103.63 from the depth structure of the 600-cell;<br>• Projection isotropy under H4 symmetry (⟨sin2 θ⟩ = 3/4, exact);<br>• Uniform cluster coefficient 5/11 guaranteeing redundancy;<br>• Stress inertia rate η = 3α/(2T ) uniquely fixed by dimensional constraints of<br>sequential propagation.<br>Integrating these four mechanisms yields Ωlocal ≈ 100.000069. The integer 100 is<br>not imposed but emerges geometrically from arccos(1/3) and the spatial dimension<br>n = 3 alone. Connections to granular physics, seismology, and the light-speed<br>resource allocation principle (RSHG Part II) are discussed.<br>Keywords: geometric frustration, 600-cell, force chains, stress inertia, jamming<br>transition, zero-parameter derivation, gravitational constant</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18830266 |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Regular Simplex Hierarchical Gravity: Part V Zero-Parameter Derivation of the Local Effective Vertex Count Force-Chain Robustness and Stress Inertia from 600-Cell Geometry Sato, Ryuhei geometric frustration regular tetrahedron packing 600-cell H4 symmetry force chains granular physics jamming transition stress inertia zero-parameter derivation gravitational constant Niven's theorem 4D projection redundancy seismology light-speed regularisation <p>As Part V of the Regular Simplex Hierarchical Gravity (RSHG) framework,<br>this paper derives the local effective vertex count Ωlocal = 100 without any free<br>parameters.<br>The derivation begins with Niven’s theorem: the impossibility of filling three-<br>dimensional space with regular tetrahedra is established as an algebraic necessity<br>rather than a numerical coincidence. The irreducible angular deficit δ = 2π −<br>5 arccos(1/3) arising from this failure serves as the sole seed of the following geo-<br>metric chain:<br>• The ideal vertex count Ωideal ≈ 103.63 from the depth structure of the 600-cell;<br>• Projection isotropy under H4 symmetry (⟨sin2 θ⟩ = 3/4, exact);<br>• Uniform cluster coefficient 5/11 guaranteeing redundancy;<br>• Stress inertia rate η = 3α/(2T ) uniquely fixed by dimensional constraints of<br>sequential propagation.<br>Integrating these four mechanisms yields Ωlocal ≈ 100.000069. The integer 100 is<br>not imposed but emerges geometrically from arccos(1/3) and the spatial dimension<br>n = 3 alone. Connections to granular physics, seismology, and the light-speed<br>resource allocation principle (RSHG Part II) are discussed.<br>Keywords: geometric frustration, 600-cell, force chains, stress inertia, jamming<br>transition, zero-parameter derivation, gravitational constant</p> |
| title | Regular Simplex Hierarchical Gravity: Part V Zero-Parameter Derivation of the Local Effective Vertex Count Force-Chain Robustness and Stress Inertia from 600-Cell Geometry |
| topic | geometric frustration regular tetrahedron packing 600-cell H4 symmetry force chains granular physics jamming transition stress inertia zero-parameter derivation gravitational constant Niven's theorem 4D projection redundancy seismology light-speed regularisation |
| url | https://doi.org/10.5281/zenodo.18830266 |