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/Ωₘ = √ (3π/2) — 0. 04% error 3. Optical depth: τ = Ωₘ/2√ (8π/3) — 0. 12% error 4. CMB/Structure: TCMB/f_σ8 = 2√ (8π/3) — 0. 04% error 5. Quantum (Unruh): λU/LH ≈ 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: Ωₘ = 1/ (1 + √ (3π/2) ) = 0. 31538 exactly. Code: https: //github. com/carlzimmerman/zimmerman-formula
Carl Zimmerman (Fri,) studied this question.