Friedmann Equation

$$H^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda c^2}{3}$$

Status: Proven.

Derived directly from Einstein’s field equations (see Tensor Calculus and the Einstein Field Equation) applied to a homogeneous, isotropic universe, this is the master equation of modern cosmology — it relates the universe’s expansion rate $H$ to its energy density $\rho$ and the cosmological constant $\Lambda$.

Real, independent confirmation: see Standard Cosmology for the fuller status — the expansion/thermal-history framework this equation describes is proven to high precision (Hubble expansion, the CMB, Big Bang nucleosynthesis), though the literal nature of the density terms feeding into it (dark matter, dark energy) remains genuinely open — see Unknowns.

Used in: Paper 7, Paper 9, and Paper 12, where PBT proposes its own residual hierarchical energy density as the source of $\rho_{eff}$ and $\Lambda$ — a genuine attempt at a real Unknown, not a borrowed proven result. The equation itself is real and unmodified; PBT’s specific $H_0$ figure derived from it was found in the 2026-07-21 audit to sit between, and not distinctly match, the real Planck and SH0ES measurements (the Hubble tension).