Geometric Derivation of Planck Mass: π⁴⁵ Formula with 0.574 ppm Precision

We derive M_Pl/m_e = π⁴⁵ × (1 + 2α + α/13 − (8/9)α²) from pure geometry. ZERO free parameters. ALL coefficients from sphere packing:- 45 = (K₃² − K₃ − 2τD)/2 where K₃ = 12 (3D kissing number)- 13 = K₃ + 1 (same factor as Weinberg angle sin²θ_W = 3/13)- −8/9 = −(K₃−4)/(K₃−3) from tetrahedral cluster geometry- 2 = K₄/K₃ = 24/12 (Dirac g-factor) RESULT: 5.74×10⁻⁷ relative error (0.574 ppm) — 38× better than direct G measurements (22 ppm). CROSS-VALIDATION: sin²θ₁₃ = 1/45 (neutrino mixing angle) independently confirms the exponent, suggesting unified geometric origin for Planck scale and Standard Model. The formula emerges from the sedenionic dimensional cascade: physical reality projects from 16D sedenion algebra 𝕊 through octonions 𝕆 (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. Key identity: e^π − π = 19.999 ≈ 20 = K(8)/K(3) connects transcendental constants to lattice geometry (error 0.0045%). Package includes:- UCT_Planck_Mass_v5_1.pdf (docx) - Complete derivation (4 pages, 3 figures)- planck_mass_validation.py (ipynb) - Reproducible Python code (seed=42)- Data with results - Bootstrap (N=10,000) and Monte Carlo (N=100,000) resultshttps://colab.research.google.com/drive/1UtKmlkaYFHegx4S98Vv_B18FuylP3hlm?usp=sharing All numerical verification uses CODATA 2022 constants.