The Immirzi parameter, area spectrum, and entropy correction from ε = 1/π²

The Barbero-Immirzi parameter, treated as a free parameter in loop quantum gravity since 1996, is derived from the K4 tiling of S² within the Boundary Condition Cosmological Framework (BCCF). The result γ = (κ/π)(1 + κε²/3) = 0.27404, where κ = 4−π and ε = 1/π², matches the Domagala-Lewandowski/Meissner value to 0.009% with zero free parameters. The same structure yields the area spectrum A_j = 8κ√(j(j+1)) ℓ_P² and identifies the universal logarithmic entropy correction −3/2 = −β₁(K4)/χ(S²) as a topological invariant. Three predictions for the LQG programme are stated. This paper is a companion to the main BCCF framework (doi:10.5281/zenodo.18874865).