Friedmann Equations from the 0Σ Cosmological Model: Cavalieri's Theorem, FRW Scale Factor, and the Baryon Dilution Factor

The 0Σ cosmological model, formulated in 1999 as a singularity sphere whose decompression initiates the Big Bang, is identified in [1] as the total space of the Trit principal bundle. This paper derives the Friedmann equations from the 0Σ sphere geometry using Cavalieri’s principle: two solids with equal cross-sectional areas at every height have equal volumes. Applied to the 0Σ sphere (a 9-sphere S⁹) and the FRW universe (3D spatial slices expanding in conformal time), Cavalieri’s principle gives the FRW scale factor a(η) ∝ sin(η) — the radiation-dominated closed FRW solution — as a geometric consequence of the spherical topology of the 0Σ. The 0Σ sphere radius R0Σ equals the Hubble radius at the Trit phase transition: R0Σ = c/H(MTrit) ≈ 7.3 Planck lengths. The Cavalieri mode count on (S²)³ gives the number of Trit degrees of freedom: g_*Trit = (D × Dobs × |T|/e)⁴ = (120/e)⁴ = 3.79×10⁶, where D = 10, Dobs = 4, |T| = 3, and e is Euler’s number. This closes the open calculation of the baryogenesis paper [6]: the complete baryon asymmetry formula η_B = (28/79) × sin(arccos(1/√6)) × αTrit/π × e⁴/(120)⁴ = 5.81×10⁻¹⁰, against the observed 6.1×10⁻¹⁰ (−4.7 percent). All quantities are derived from {-1,0,+1} without free parameters.