The gravitational hierarchy problem—why gravity is ~10⁴5 times weaker than electromagnetism at the electron mass scale—has resisted parameter-free resolution despite decades of effort in supersymmetry, extra-dimension models, and warped geometry frameworks. All existing approaches introduce new degrees of freedom or symmetry principles without deriving the gravitational coupling constant alphaG = G mₑ² / (hbar c) from first principles. This paper derives alphaG solely from the Tier-1 axioms of Cognitional Mechanics (CM) and the algebraic structure of M₃ (C), the minimal noncommutative finite-dimensional C*-algebra. The principal result is alphaG = alpha^ (n (2n+1) ) * sqrt (n (n+1) / (2n+1) ) with n=3, yielding alphaG^ (CM) = 1. 75192 x 10^-45. All factors—the exponent 21, numerator 12, and denominator 7—emerge purely from the dimension parameter n; no empirical fitting is performed. The zeroth-order residual against CODATA 2022 is 6. 1 x 10^-5, within the 5. 5 x 10^-4 experimental scatter band of independent G measurements. The leading correction C₃G = - (delta² - delta/ (2*sqrt (n) ) ) * alpha², derived from Tier-1 constants delta² = n/2 and n alone, reduces the residual to 5. 5 x 10^-8, well below current experimental resolution. This establishes alphaG as a Tier-1 structural invariant of M₃ (C) and provides an algebraic resolution to the gravitational hierarchy problem. The CM-implied value Gᵢmplied = 6. 67471 x 10^-11 m³ kg^-1 s^-2 serves as a theoretical convergence point for future precision G metrology.
T.O. (Thu,) studied this question.