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%).
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