G and the Hierarchy Dilemma: A Bootstrap Prediction for the Gravitational Constant

The Bootstrap framework — in which fundamental constants arise from the icosahedral geometry of a lattice on the closed manifold S³ — has successfully derived the fine structure constant α, the proton-to-electron mass ratio μ, the hydrogen binding energy, and the coupling strengths of the electromagnetic, weak, and strong forces. Here we present a further prediction: the gravitational constant G is not a fundamental constant but a local, configuration-dependent quantity — the amplitude of a collective baryonic mode sampled at a particular position, orientation, and epoch. Unlike α (a geometric property of the icosahedron, the same everywhere), G depends on the distribution of matter across S³ and therefore varies with location. This makes G a statistical quantity, analogous to temperature: real and measurable, but emergent. Published data are consistent with this prediction: • The fine structure constant α is known to 12 significant figures with complete laboratory agreement. G is known to only 5 significant figures, with persistent disagreement exceeding five standard deviations (Quinn et al. 2014). • G measurements oscillate with a 5.9-year period correlated with Earth's length-of-day variation (Anderson et al. 2015). • Big Bang nucleosynthesis constraints are naturally satisfied: at the BBN epoch, the nearly uniform matter distribution predicts a spatially constant G, matching G_BBN/G₀ = 0.99 ± 0.05 (Alvey et al. 2020). The framework also reframes the hierarchy problem: the ratio α/α_G ≈ 10³⁶ compares a geometric constant with a statistical measurement — incommensurable quantities rather than a number requiring explanation. Eighth paper in the Bootstrap series. The previous seven derived constants. This one identifies a quantity that cannot be derived — and shows that the inability to derive it is itself a prediction of the framework. Part of the Bootstrap Universe series: https://doi.org/10.5281/zenodo.17906573