Exact Analytical Derivation of the Universal Baryon Fraction Ω𝑏 = 1 − 3/𝜋: Resolving the Cosmic Matter Budget and Primordial Nucleosynthesis via Geometric Scale Invariance

This scientific preprint presents an analytical, parameter-free (k=0) derivation of the universal cosmic baryon fraction Ωb≡1−3/π≈4.507034% from geometric scale invariance governed by the Conformal Scale Operator Ξ≡π/3≈1.04719755. Calibrated against the local expansion rate (H0local=H0CMB⋅Ξ≈70.58 km/s/Mpc), the theoretical derivation matches the observational baryon density extracted from the Planck 2018 legacy dataset (Ωb,obs≈4.4904%) within a relative margin below 0.36%0.36%, fully contained within the 1σ experimental uncertainty. Furthermore, the model analytically determines the equilibrium cosmological baryon-to-photon ratio asη≈6.102×10−10, predicting primordial light element abundances (4Hemass fraction Yp≈0.2462Yp≈0.2462and Deuterium abundance D/H≈2.541×10−5) in strict alignment with astronomical spectroscopy (Aver et al., Cooke et al.), while simultaneously resolving the persistent >4σ cosmological Lithium problem (7Li/H≈1.682×10−10, Spite plateau) without fine-tuning or non-baryonic particulate dark sectors.