Tensor-to-Scalar Ratio r

τ-Formula

r = ι_τ^(2·dim(T²)) = ι_τ⁴ ≈ 0.0136

Derivation

dimensional suppression!theorem The tensor-to-scalar ratio $r = P_t/P_s$ is determined by the fibration structure $τ^3 = τ^1 ×_f T^2$:

r \;=\; ι_τ^\,2 · (T^2) \;=\; ι_τ^2 × 2 \;=\; ι_τ^4 \;≈\; 0.01357.

• Tensor perturbations (gravitational waves) are D-sector frame-holonomy fluctuations propagating on the base $τ^1$. They are insensitive to the fiber structure.
• Scalar perturbations (curvature/density fluctuations) are boundary-character fluctuations of the full arena $τ^3$, coupling to both fiber circles of $T^2$.
• Each fiber dimension contributes a breathing-fraction suppression $ι_τ$ to the scalar amplitude relative to the tensor amplitude.
• The power spectrum is quadratic in amplitude ($P |δ|^2$).

with the first factor equal to the number of lemniscate lobes and the second factor arising from $P |δ|^2$. (Registry: V.P136, $τ$-effective, Wave 13.)

Source

This prediction is derived in the Physics Ledger (Chapter 62 — inflation-cmb-bbn), Books IV–V of Panta Rhei.