Higher-Order Spectral Geometry and the Resolution of the Black Hole Information Problem

The black hole information problem arises from the apparent conflict between Hawking radiation, which suggests information loss, and the fundamental unitarity of quantum mechanics. This work introduces the Tri-World System, a closed geometric–spectral–dynamical framework consisting of three universes D→ΣE→ΦF, connected by definable maps and closed at stable fixed points. Within this framework we construct the higher-order spectral manifold Q=Σ (2) (D), and identify the phase transition surface S=Λ∈Q∣det⁡ (∂Λ/∂Γ) =0, where Jacobian degeneracy, spectral clustering, and curvature divergence occur. We prove the Information Preservation Theorem, showing that crossing S induces a condensed spectral phase but does not destroy information. Instead, the Tri-World Isomorphism reintegrates condensed spectral data back into curvature geometry, ensuring global conservation: I (Γ) =I (λ) =I (S) =I (Λ). The theory is shown to be model-theoretically complete, guaranteeing that information cannot be logically lost. Compatibility with semiclassical gravity, holography, and AdS/CFT is demonstrated, and examples from Schwarzschild and Kerr geometries illustrate how S corresponds to physically meaningful loci. The framework provides a mathematically complete geometric resolution of the black hole information problem.