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 ags ¼ 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 hni ¼ 4 þ /3 ¼ 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 K3 Kähler manifold, we just need to extend the well known value v ¼ 24 of a crisp K3 to the case of a fuzzy K3. This leads then to vðfuzzyÞ ¼ 26 þ k ¼ ags. 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, namely K ¼ v ¼ D ¼ ags ¼ 26 þ k ¼ 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.