The Fifth Link: Deriving the Gravitational Constant from the Tower Equation

Paper 2026v (Carpenter, 2026v) traced a five-link algebraic chain from the fine structure constant α to the gravitational constant G and reported that four links were closed. The fifth — the Planck-to-proton mass hierarchy — remained open: no derivation of the exponent η = ln(m_P/m_p) existed. This paper closes Link 5 by deriving η from the same tower equation that produces α. The result is: η = 14π + π²√α / 28 where 14 = 2 × 7 (equation degree × coefficient sum) and 28 = 4 × 7 (leading coefficient × coefficient sum), with 7 = 4 + 1 + 1 + 1 from the tower polynomial coefficients. The gravitational constant then follows from G = ℏc / (m_p² · e^{2η}). Predicted value: G = 6.67436 × 10⁻¹¹ m³ kg⁻¹ s⁻² CODATA 2022: G = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² Agreement: 0.37σ (8.4 parts per million) This is well within the 22 ppm measurement uncertainty of G — the worst-measured fundamental constant, and the only one never previously derived from theory. The complete chain — from π to G — now contains five closed links, zero free parameters, and one measured input (R∞, the Rydberg constant). The script that computes it is provided for independent verification.