Gravitational Constant from ι_τ: α_G at 3 ppm
The gravitational fine structure constant is derived as α_G = α¹⁸√3(1 − (3/π)α) at 3 ppm from observation.
Overview
V.T154 derives the gravitational coupling constant (gravitational fine structure constant) α_G = G·m_p²/(ℏc) as α_G = α¹⁸√3(1 − (3/π)α), where α is the fine structure constant. The derivation uses the D-sector (gravity, α-generator) position in the τ-ABCD chart: gravity is the first sector (A=α), and α_G is the lowest-level coupling — it is the 18th power of α because gravity operates at the deepest sector level. Agreement with observation is at 3 ppm.
Detail
Newton’s gravitational constant G (equivalently, the gravitational fine structure constant α_G = G·m_p²/(ℏc) ≈ 5.9 × 10⁻³⁹) is one of the least understood constants of nature. Its value is 36 orders of magnitude smaller than the electromagnetic coupling α ≈ 1/137. The hierarchy problem of gravity asks why G is so small. Book V derives α_G from ι_τ and α through the τ-sector structure. The D-sector (gravity) is assigned to the α-generator, which occupies the first ρ-orbit position in the τ-ABCD chart. The sector depth (ρ-orbit level) determines the coupling power: the D-sector at depth 1 couples through α¹, but the full sector coupling to matter at depth 18 (the number of ρ-orbit traversals to close the sector loop) gives α¹⁸. The correction term (1 − (3/π)α) is the NLO contribution from the B-sector (EM, π-generator) boundary interaction. The formula α_G = α¹⁸√3(1 − (3/π)α) gives 3 ppm agreement. This provides a structural explanation for the gravitational hierarchy problem: G is small because it involves the 18th power of α, not because of fine-tuning.
Result Statement
V.T154: α_G = α¹⁸√3(1 − (3/π)α) at 3 ppm. Gravitational hierarchy explained by D-sector depth 18.