Chapter 13: Nonlinear τ-Einstein: Address Resolution, Not PDE Solving

General relativity in its orthodox formulation is a system of ten coupled nonlinear partial differential equations. Proving that solutions exist, establishing their uniqueness, controlling their regularity—these are among the deepest problems in mathematical physics. The Cauchy problem for the Einstein equations was solved by Choquet-Bruhat in 1952, but global existence and uniqueness remain open in many physically relevant cases. Cosmic censorship, the Penrose conjecture, the BKL analysis of generic singularities—all are unsolved precisely because the PDE approach demands control over solutions that the equations resist providing.

This chapter shows that none of these problems arise in the τ-framework. The τ-Einstein equation (the relevant chapter, V.D06) is not a PDE. It is a boundary-character identity in the holonomy algebra H_∂[ω], and its nonlinear regime is solved by address resolution: finding the unique boundary character that satisfies the cocycle-defect minimization principle. Existence, uniqueness, and selection are theorems (§), not conjectures. The profinite structure of Category τ provides a natural resolution scale that eliminates point masses and curvature singularities (§). Horizons emerge as present surfaces of the null structure on τ³ (§).