Silk Damping Scale ℓ_D

τ-Formula

ℓ_D = ℓ₁ · κ_D / κ_B = 1244.0

Derivation

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.

Source

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