One axiom, one manifold, one metric. Read one way, it gives Schrödinger. Read another, it gives Einstein. Three peer-reviewed theorems (Chentsov, Brody–Hughston, Jacobson) plus the single axiom σ = 1/(1+σ) complete the bridge.
A single self-referential equation generates a statistical manifold with a unique Riemannian metric. Three peer-reviewed theorems published independently over four decades collectively show that such a manifold produces both quantum mechanics and general relativity as different limits of the same geometry. Chentsov proved the metric is unique. Brody and Hughston proved this metric is the quantum metric. Jacobson proved thermodynamics on local horizons yields the Einstein equations. The fourth link is the content supplied by σ = 1/(1+σ).