The cosmological constant from the semiclassical S^4 partition function

Preprint — not peer reviewed. Feedback welcome.

We present a formula with no fitted parameters for the cosmological constant:

Λ = (1991/720) × (m_p/M_P) × exp(−24π²) = 2.85×10⁻¹²²,

where m_p = 0.938 GeV is the proton mass and M_P = 1.221×10¹⁹ GeV is the Planck mass. The observed value is 2.86×10⁻¹²²; the discrepancy is 0.3%. Each factor is determined entirely by known physics. The exponential exp(−24π²) ≈ 10⁻¹⁰³ is the semiclassical weight of the Euclidean de Sitter instanton—the Einstein–Hilbert action evaluated on the round four-sphere S⁴ at Planck curvature. The ratio m_p/M_P ≈ 10⁻¹⁹ is the QCD gauge hierarchy from dimensional transmutation. The coefficient 1991/720 = 2.765 is the a-type trace (conformal) anomaly of the Standard Model—a rational number determined by the number and spin of known particles, independent of their masses. Together, the three factors account for all 122 orders of magnitude.

The 122-decade hierarchy decomposes as 103+19: a gravitational suppression times a gauge hierarchy, weighted by the matter degrees of freedom. Both the gravitational action ratio 24π²/8π² = 3 and the requirement of Yang–Mills confinement are specific to four spacetime dimensions; the formula is intrinsically four-dimensional. The formula identifies the cosmological constant with the Helmholtz free energy of the cosmological horizon, computed from a partition function that factorises into gravitational and matter sectors. Combined with CMB-measured Ω_m h² = 0.1424, it predicts H₀ = 67.4 ± 0.8 km s⁻¹ Mpc⁻¹. The identification of the coefficient with the conformal anomaly makes the formula a precision test of the Standard Model particle content: any new light species shifts Λ by a calculable amount through its contribution to Δa.