Derivation of the Euler characteristic and the curvature of Cantorian-fractal spacetime using Nash Euclidean embedding and the universal Menger sponge

The present work gives an analytical derivation of the curvature K   of fractal spacetime at the point of total unification of all fundamental forces which is marked by an inverse coupling constant equal α¯gs=26.18033989. To do this we need to first find the exact dimensionality of spacetime. This turned out to be n = 4 for the topological dimension and ∼〈n〉=4+ϕ3=4.236067977∼〈n〉=4+ϕ3=4.236067977 for the intrinsic Hausdorff dimension. Second we need to find the Euler characteristic of our fractal spacetime manifold. Since E-infinity Cantorian spacetime is accurately modelled by a fuzzy K  3 Kähler manifold, we just need to extend the well known value χ=24χ=24 of a crisp K3 to the case of a fuzzy K  3. This leads then to χ(fuzzy)=26+k=α¯gs. The final quite surprising result is that at the point of unification of our resolution dependent fractal-Cantorian spacetime manifold we encounter a Coincidencia Egregreium, namelyK=χ=D=α¯gs=26+k=26.18033989.Finally we look for some indirect experimental evidence for the correctness of our result using the COBE measurement in conjunction with Nash embedding of the universal Menger sponge.