Geometric Derivation of Planck Mass Achieves 0.574 ppm Precision Using π⁴⁵ Formula
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AlekseN
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Summary
Researchers present a geometric derivation of the Planck mass using pure geometry with zero free parameters, achieving 0.574 ppm precision. The formula M_Pl/m_e = π⁴⁵ × (1 + 2α + α/13 - (8/9)α²) emerges from sphere packing and sedenionic dimensional cascade theory, connecting fundamental constants to lattice geometry. The derivation shows 38× better precision than direct gravitational constant measurements and suggests unified geometric origins for Planck scale and Standard Model parameters.
Key quotes
· 5 pulledWe derive M_Pl/m_e = π⁴⁵ × (1 + 2α + α/13 - (8/9)α²) from pure geometry.
ZERO free parameters. ALL coefficients from sphere packing.
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.
Key identity: e^π - π = 19.999 ≈ 20 = K(8)/K(3) connects transcendental constants to lattice geometry (error 0.0045%).
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 kissin
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