Universal Quantum Kasner Fixed Point at the Schwarzschild Singularity

The Schwarzschild spacelike singularity admits an anisotropic local blow-up to the vacuum Kasner geometry with exponents (−1/3, 2/3, 2/3). We show that this geometric limit extends to quantum correlations: after the corresponding spacetime and field rescaling, the Unruh and Hartle–Hawking two-point functions converge to the same Kasner kernel. Adiabatic/Hadamard renormalization of this fixed-point kernel yields a parameter-free terminal vacuum polarization and the complete leading renormalized stress tensor. For a massless minimally coupled scalar, ⟨Tâb̂⟩ren = σ−4 diag(−Cρ, C∥, C⊥, C⊥), with (Cρ, C∥, C⊥) = (5.9211, 13.018, 16.577) × 10−5. Converted to Schwarzschild variables, the prediction agrees at the percent level or better with an independent four-dimensional numerical calculation. For these two stationary states the leading stress is therefore state independent; a smooth added one-particle excitation grows only as r−3, compared with the fixed-point vacuum stress r−6.