Chapter 14: The τ-Schwarzschild Readout: Torus Vacuum

In 1916, Karl Schwarzschild schwarzschild1916gravitationsfeld found the first exact solution of Einstein’s field equations einstein1915field: the vacuum metric surrounding a spherically symmetric, non-rotating mass. The solution contains R_S = 2GM/c² as its single parameter—the Schwarzschild radius—and predicts an event horizon at r = R_S, a coordinate singularity at the horizon, and a curvature singularity at r = 0. A century of work has clarified the coordinate singularity (removable by Kruskal–Szekeres coordinates) but not the curvature singularity (still present, still physically absurd).

This chapter derives the τ-native Schwarzschild relation as a readout of the stabilized torus vacuum. The starting point is not a PDE on a manifold but the torus vacuum geometry of τ³: the fibered product τ¹ ×_f T², when the D-sector saturates, settles into a stabilized torus with shape ratio r/R = ι_τ (§). The gravitational constant G_τ emerges as the coherence conversion invariant relating the radius index to the mass index (§). The orthodox Schwarzschild metric is recovered as a sphere-proxy corollary: the chart readout Φ_p projects the torus-shaped boundary character onto a spherically symmetric metric (§). No singularity forms (§), and the stabilized torus relaxes through two channels: geometric (shape normalization) and topological (handle settling) (§). The chapter concludes with the full Schwarzschild relation R_S = 2 G_τ M and the No-Shrink forward reference (§).

Lean reference. The structures and theorems of this chapter are formalized in TauLib.BookV.Gravity.GravitationalConstant (V.D01, V.D02, V.T01, V.P01) and TauLib.BookV.Gravity.Schwarzschild (V.D07, V.D08, V.D09, V.T03, V.R02).