Scalar Perturbation Amplitude A_s

τ-Formula

A_s = (121/225) · ι_τ¹⁴ ≈ 2.14 × 10⁻⁹

Derivation

A_s \;=\; 121225\,ι_τ^18\, (1 - ι_τ^3/3) \;=\; α_τ\,ι_τ^14\, (1 - ι_τ^3/3) \;≈\; 2.096 × 10^-9,

where $α_τ = (11/15)^2 = 121/225$ is the $τ$-native fine-structure amplitude. The NLO factor $(1 - ι_τ^3/3)$ is structural: it represents $τ^3$ volume averaging over $(τ^3) = 3$ dimensions, not dynamical spectral running. The Planck 2018 value $A_s = (2.100 ± 0.030) × 10^-9$ gives a deviation of $-1979$ ppm ($-0.2%$). (Registry: V.D253, $τ$-effective, Wave 14A.)

From $r = ι_τ^4$ and $n_s = 1 - 2/57$, the slow-roll parameter $ε = r/16 = 8.48 × 10^-4$, which gives dynamical spectral running $dn_s/d k = O(ε) ∼ 10^-3$. The NLO correction $(1 - ι_τ^3/3)$ requires $O(ι_τ^3) ∼ 0.040$—a $156×$ gap. This confirms that the NLO correction is structural (volume averaging), not dynamical (spectral running). (Registry: V.T198.)

Source

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