Derivation of Newton's Gravitational Constant from First Principles

We derive Newton's gravitational constant from first principles. The result is G = ℏc / (9 m_t² e^{24π}), where ℏ is the reduced Planck constant, c is the speed of light, and m_t 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. M_Pl = 3 m_t e^{12π}. Combined with the textbook Planck-mass definition M_Pl² = 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 m_t, the same formula yields m_t = √[ 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 m_t = 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 m_t 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.