The cosmological constant and the cosmic microwave background temperature are derived from a single algebraic axiom — σ = 1/(1+σ) — with zero free parameters. The same framework predicts cosmic energy fractions, resolves the Hubble tension, and identifies a 9% transition sector testable by Euclid, LSST, or DESI DR3 within five years.
The Pentagon Physics programme derived the cosmological constant from a single algebraic axiom: σ = 1/(1+σ), whose positive solution is σ = 1/φ. The same framework now yields the cosmic microwave background temperature to within one part in three thousand of the FIRAS measurement, without fitted parameters. Geometry, rather than spatial assumptions, generates the fundamental constants — the axiom selects the √5 algebraic field, which determines the pentagon and thus π itself.
Three interconnected predictions emerge: cosmic energy comprises 61.8% dark energy, 23.6% dark matter, 9% transition sector, and 5.57% baryonic matter. The Hubble parameter is predicted at approximately 0.634, resolving current tensions by revision downward. The unobserved 9% transition sector should be detected or ruled out by Euclid, LSST, or DESI DR3 within five years.