One Equation Against the Sky — A CMB Test of a Predicted Matter Density

There is an equation. f² − 2πf + π = 0. From a geometric closure condition it fixes f₁ = 5.735434. Combined with the primordial baryon density, it fixes the total physical matter density before the CMB is calculated. We froze that number and asked a standard Boltzmann code what sky it predicts. The first six TT acoustic peak heights emerge within 3% of the Planck 2018 central values. No peak height was fitted. The third peak provides an additional diagnostic: it lies near k₃τc(z*) ≈ 1, where tight coupling breaks down at last scattering. The six peaks were a test, not a target.