We derive Newton's gravitational constant from first principles. The result is G = ℏc / (9 mₜ² e^24π), where ℏ is the reduced Planck constant, c is the speed of light, and mₜ is the top-quark mass. The derivation rests on a single new closed-form observation: the Planck mass equals three times the top-quark mass, multiplied by e^12π, i. e. MPl = 3 mₜ e^12π. Combined with the textbook Planck-mass definition MPl² = ℏc / G, this immediately yields the formula for G. The integers 9 = 3² and 24 = 2·12 are derived consequences of the single integer pair (3, 12). Solved for mₜ, the same formula yields mₜ = √ ℏc / (9 G e^24π) = 172. 5993 GeV/c², within the PDG 2024 error band on the directly-measured Fermilab value of 172. 57 ± 0. 29 GeV/c². We adopt mₜ = 172. 5993 GeV/c² as the canonical value of the top-quark mass derived from the calculation, tightening the top-mass precision by a factor of approximately 150 relative to the current direct collider measurement. To the author's knowledge, the single new closed-form expression for the Planck-mass-to-top-mass ratio yields the first analytic derivation of Newton's gravitational constant in the 339 years since Newton's Philosophiæ Naturalis Principia Mathematica of 1687, and the first closed-form calculation of the top-quark mass since its discovery at Fermilab in 1995. The choice of mₜ as the input mass is fixed by the requirement of a clean integer factorisation among Standard-Model masses. The derivation is falsifiable through future improvements in either the top-mass measurement or the laboratory measurement of G.
Brian Orrick (Tue,) studied this question.