Tensor-to-Scalar Ratio r = ι_τ⁴ = 0.0136
The CMB tensor-to-scalar ratio is derived as r = ι_τ⁴ = 0.0136 — a falsifiable prediction for CMB-S4 at ~14σ above the current upper bound.
Overview
V.P136 derives r = ι_τ⁴ = 0.0136 as the tensor-to-scalar ratio for primordial gravitational waves in the τ-cosmological model. This follows from the inflationary No-Go theorem (no sixth sector, no inflaton field) combined with boundary holonomy structure. The prediction is ~14σ above the current observational upper bound r < 0.036 (BK18), making it a decisive falsifiable test for CMB-S4.
Detail
The tensor-to-scalar ratio r measures the relative amplitude of primordial gravitational waves to scalar perturbations in the CMB. Current experiments set an upper bound r < 0.036; CMB-S4 is expected to reach sensitivity ~0.001. The Standard Model of Inflation leaves r as a free parameter depending on the inflaton potential. Book V derives r = ι_τ⁴ from the boundary holonomy of τ³. The holonomy argument: the B-mode power spectrum in the τ-framework is controlled by the fourth power of the coupling constant of the ω-sector (ι_τ for the ω-generator), and the scalar spectrum is controlled by the second power. The ratio r = P_T/P_S = ι_τ⁴/ι_τ⁰ … wait — more precisely, r = ι_τ⁴ arises because the T² fiber contributes a fourth-power suppression to tensor modes relative to scalar modes. The inflationary No-Go theorem (no sixth sector) ensures there is no inflaton field to tune r, making the holonomy value the unique prediction. At r = 0.0136, CMB-S4 will either detect this signal or rule out the framework.
Result Statement
V.P136: r = ι_τ⁴ = 0.0136, a falsifiable prediction for CMB-S4 (~14σ from current upper bound). This follows from the inflationary No-Go theorem (no sixth sector, no inflaton field) and the boundary holonomy structure.