The Friedmann Geometric Factor: Five Cosmological Relationships

We report that five independent cosmological quantities share relationships involving the geometric factor 2√(8π/3) = 5.7888, arising from the Friedmann equation through ρc = 3H²/(8πG): 1. MOND acceleration: a₀ = cH₀/2√(8π/3) — 0.8% error 2. Dark energy ratio: Ω_Λ/Ω_m = √(3π/2) — 0.04% error 3. Optical depth: τ = Ω_m/2√(8π/3) — 0.12% error 4. CMB/Structure: T_CMB/f_σ8 = 2√(8π/3) — 0.04% error 5. Quantum (Unruh): λ_U/L_H ≈ 2√(8π/3) — ~2% error All values from Planck 2018, SPARC, COBE/FIRAS. Four sub-0.15% matches with the SAME geometric factor yields P ~ 10⁻¹² against coincidence. Key finding: The factor 8π/3 from Einstein's field equations propagates through galaxy dynamics, dark energy, reionization, structure growth, and quantum effects. Testable prediction: Ω_m = 1/(1 + √(3π/2)) = 0.31538 exactly. Code: https://github.com/carlzimmerman/zimmerman-formula