Geometry from Information: Paper 21 — The Euler Metric and Topological Phases of Gravity

We introduce the Euler metric d_χ(g,g') = |χ(M_g) - χ(M_{g'})| as topological complement to the GFI informational metric. Three results: (I) Entropy-topology theorem: S = A/4G_N iff χ(horizon) = 2. (II) Singularity resolution Schwarzschild → micro-dS is topologically free (d_χ = 0), since both have χ = 2. (III) Information paradox = topological obstruction: evaporation Schwarzschild → Minkowski requires d_χ = 1. Black holes cannot evaporate to flat space — endpoint is Planck-scale de Sitter remnant with χ = 2.