Quantum Mechanics, General Relativity and Bekenstein-Hawking Entropy as Structural Limits of Fisher-Rao Information Geometry

Starting from Čencov's uniqueness theorem for the Fisher-Rao metric—the only Riemannian geometry on the space of probability distributions invariant under Markov morphisms—we derive a six-dimensional extended space E⁶ with conformal factor Ω(I)=(1+I/Λ²)⁻¹. The operator ellipticity condition $A(I)>0$ generates a geometric UV cutoff at x₀<1/3 without renormalization. The α=-1 connection selects Mη as the unique physically admissible spacetime via Hadamard well-posedness. Kaluza-Klein reduction yields Einstein's equations with MPl² = M₆⁴ · 2π Rφ ∫ Ω⁴ dτ. The Bekenstein-Hawking law S=Area/4GN follows from a geometric phase transition between volume-law (~ R³) and area-law (~ R²) minimal surfaces, with GN determined by the same integral. The statistical Hawking temperature TH=27/(256π) is fixed by the universal conformal potential Veff=-1/(4τ²) independent of normalization. The dark energy equation of state follows from an absolutely convergent hypergeometric series with ratio $1/3$, yielding an effective 4D observable value of w ≈ -0.802.