N14 — Silk damping ratio $_D / _1 = _D / _B$
N14: Silk damping ratio $_D / _1 = _D / _B$. Planck legacy, CMB-S4.
Falsification Details
ell-D/ell-1 = kappa-D/kappa-B. Silk damping!prediction pred:n14 The ratio $_D / _1 = 5.647$ is a boundary holonomy invariant (V.D254). $_D = 1244.0$ at $+9$~ppm.
Planck legacy, CMB-S4
confirmed at $+9$~ppm.
Context
N14: Prediction
ell-D/ell-1 = kappa-D/kappa-B.
The ratio $_D / _1 = 5.647$ is a boundary holonomy invariant (V.D254). $_D = 1244.0$ at $+9$ ppm. Experiment: Planck legacy, CMB-S4. Timeline: confirmed at $+9$ ppm.
Derivation Context
The Silk damping multipole—the scale beyond which photon diffusion erases primordial anisotropies—has the $τ$-native expression
_D \;=\; _1 × κ_Dκ_B \;=\; _1 × 1 - ι_τι_τ^2 \;≈\; 220 × 5.655 \;=\; 1244.0,
at $+9$ ppm from the Eisenstein–Hu (1998) fitting formula $_D ≈ 1244$. (Registry: V.D254, Wave 14B.)
The structural content. The ratio $κ_D/κ_B = (1 - ι_τ)/ι_τ^2$ is the holonomy-to-baryon coupling ratio—the same ratio that gives $ω_m/ω_b - 1$ in the matter density formula. The damping multipole exceeds the first peak multipole by exactly the matter-to-baryon coupling ratio: photon diffusion reaches the scale where holonomy mass balances baryon mass. That a single algebraic ratio governs both the matter hierarchy and the damping scale is a quantitative prediction of the $τ$ architecture, not a coincidence.