Geometric Derivation of Planck Mass: π⁴⁵ Formula with 0.574 ppm Precision
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| Lenguaje: | inglés |
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2025
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| _version_ | 1866901528227348480 |
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| author | Novgorodtsev, Aleksei |
| author_facet | Novgorodtsev, Aleksei |
| contents | <p>We derive M_Pl/m_e = π⁴⁵ × (1 + 2α + α/13 − (8/9)α²) from pure geometry.</p> <p>ZERO free parameters. ALL coefficients from sphere packing:<br>- 45 = (K₃² − K₃ − 2τD)/2 where K₃ = 12 (3D kissing number)<br>- 13 = K₃ + 1 (same factor as Weinberg angle sin²θ_W = 3/13)<br>- −8/9 = −(K₃−4)/(K₃−3) from tetrahedral cluster geometry<br>- 2 = K₄/K₃ = 24/12 (Dirac g-factor)</p> <p>RESULT: 5.74×10⁻⁷ relative error (0.574 ppm) — 38× better than direct G measurements (22 ppm).</p> <p>CROSS-VALIDATION: sin²θ₁₃ = 1/45 (neutrino mixing angle) independently confirms the exponent, suggesting unified geometric origin for Planck scale and Standard Model.</p> <p>The formula emerges from the sedenionic dimensional cascade: physical reality projects from 16D sedenion algebra through octonions (8D) and quaternions ℍ (4D) to observable ℝ³. Kissing numbers K(16)=4320, K(8)=240, K(4)=24, K(3)=12 quantify information loss at each stage.</p> <p>Key identity: e^π − π = 19.999 ≈ 20 = K(8)/K(3) connects transcendental constants to lattice geometry (error 0.0045%).</p> <p>Package includes:<br>- UCT_Planck_Mass_v5_1.pdf (docx) - Complete derivation (4 pages, 3 figures)<br>- planck_mass_validation.py (ipynb) - Reproducible Python code (seed=42)<br>- Data with results - Bootstrap (N=10,000) and Monte Carlo (N=100,000) results<br><a title="Demo Colab Planck Mass Validation" href="https://colab.research.google.com/drive/1UtKmlkaYFHegx4S98Vv_B18FuylP3hlm?usp=sharing" target="_blank" rel="noopener">https://colab.research.google.com/drive/1UtKmlkaYFHegx4S98Vv_B18FuylP3hlm?usp=sharing</a></p> <p>All numerical verification uses CODATA 2022 constants.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18089296 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Geometric Derivation of Planck Mass: π⁴⁵ Formula with 0.574 ppm Precision Novgorodtsev, Aleksei Planck mass kissing numbers E8 lattice fine-structure constant gravitational constant sphere packing sedenionic cascade unified complexity theory <p>We derive M_Pl/m_e = π⁴⁵ × (1 + 2α + α/13 − (8/9)α²) from pure geometry.</p> <p>ZERO free parameters. ALL coefficients from sphere packing:<br>- 45 = (K₃² − K₃ − 2τD)/2 where K₃ = 12 (3D kissing number)<br>- 13 = K₃ + 1 (same factor as Weinberg angle sin²θ_W = 3/13)<br>- −8/9 = −(K₃−4)/(K₃−3) from tetrahedral cluster geometry<br>- 2 = K₄/K₃ = 24/12 (Dirac g-factor)</p> <p>RESULT: 5.74×10⁻⁷ relative error (0.574 ppm) — 38× better than direct G measurements (22 ppm).</p> <p>CROSS-VALIDATION: sin²θ₁₃ = 1/45 (neutrino mixing angle) independently confirms the exponent, suggesting unified geometric origin for Planck scale and Standard Model.</p> <p>The formula emerges from the sedenionic dimensional cascade: physical reality projects from 16D sedenion algebra through octonions (8D) and quaternions ℍ (4D) to observable ℝ³. Kissing numbers K(16)=4320, K(8)=240, K(4)=24, K(3)=12 quantify information loss at each stage.</p> <p>Key identity: e^π − π = 19.999 ≈ 20 = K(8)/K(3) connects transcendental constants to lattice geometry (error 0.0045%).</p> <p>Package includes:<br>- UCT_Planck_Mass_v5_1.pdf (docx) - Complete derivation (4 pages, 3 figures)<br>- planck_mass_validation.py (ipynb) - Reproducible Python code (seed=42)<br>- Data with results - Bootstrap (N=10,000) and Monte Carlo (N=100,000) results<br><a title="Demo Colab Planck Mass Validation" href="https://colab.research.google.com/drive/1UtKmlkaYFHegx4S98Vv_B18FuylP3hlm?usp=sharing" target="_blank" rel="noopener">https://colab.research.google.com/drive/1UtKmlkaYFHegx4S98Vv_B18FuylP3hlm?usp=sharing</a></p> <p>All numerical verification uses CODATA 2022 constants.</p> |
| title | Geometric Derivation of Planck Mass: π⁴⁵ Formula with 0.574 ppm Precision |
| topic | Planck mass kissing numbers E8 lattice fine-structure constant gravitational constant sphere packing sedenionic cascade unified complexity theory |
| url | https://doi.org/10.5281/zenodo.18089296 |