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 E^6 with conformal factor (I) = (1+I/^2) ^-1. The operator ellipticity condition A (I) >0 generates a geometric UV cutoff at x₀<1/3 without renormalization. The =-1 connection selects M_ as the unique physically admissible spacetime via Hadamard well-posedness. Kaluza-Klein reduction yields Einstein's equations with M₋^2 = M₆^4 2 R_ ^4 d. The Bekenstein-Hawking law S=Area/4G₍ follows from a geometric phase transition between volume-law (R^3) and area-law (R^2) minimal surfaces, with G₍ determined by the same integral. The statistical Hawking temperature T₇=27/ (256) is fixed by the universal conformal potential V₄₅₅=-1/ (4^2) 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.
Franklin Ramirez Azofeifa (Thu,) studied this question.