One Knot, All Forces: The Gravitational Constant from the Poincare Sphere of the Trefoil

Papers 13-14 derived eighty Standard Model observables from the trefoil knot T(2,3). This paper argues that (+1)-Dehn surgery on the trefoil--which produces the Poincare homology sphere S^3/2I and the icosahedral completion A_5 used for nuclear binding in companion Paper 17--also organises the integers appearing in the gravitational constant. The first 2I-invariant Laplacian eigenvalue on S^3/2I is lambda_1 = 168, which factorises two complementary ways: 168 = 8 x 21 (matching the spinor and adjoint dimensions of Spin(7) = B_3) and 168 = 7 x 24 = |Irr(2T)| x |2T| via the McKay correspondence on the binary tetrahedral group. The integer 7 appears simultaneously as the spectral quotient (n_1+2)/2, as the…