First CMB Acoustic Peak ℓ₁

τ-Formula

ℓ₁ = ℓ_A · (1 − φ₁) ≈ 220.6

Derivation

The first acoustic peak multipole is determined by the Friedmann pipeline with $τ$-native inputs:

where $_A = π d_A(z_rec) / r_s(z_rec)$ is the acoustic scale, $d_A$ is the angular diameter distance, $r_s$ is the sound horizon at recombination, and $_1$ is the phase shift from the neutrino and gravitational driving. All quantities are computed from $ι_τ$ with zero free parameters. The Planck measurement gives $_1 = 220.0 ± 0.5$. The deviation is $+0.28%$ ($+2840$ ppm). (Registry: V.T190, $τ$-effective, Wave 8A.)

The computation of $_1$ requires three $τ$-native inputs, each derived from $ι_τ$:

• Baryon density: $ω_b = 0.02209$, derived from $η_B = (121/270)\,ι_τ^19$ (Section (sec:ch62-baryon-density)).

Source

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