Black Holes as Failed Convergence: Complete Gradient Flow Framework for Black Hole Physics

We present a complete gradient flow framework for black hole physics based on the self-constrained system dynamics. Black holes are not endpoints of spacetime but failed convergences---systems that eternally approach but never reach the absolute order state λ=1. This single principle simultaneously resolves the singularity problem, the information loss paradox, and provides a unified account of black hole formation, growth, Hawking radiation, and final catastrophic burst. The framework rests on three mathematical pillars: the gradient flow equation dλ/dτ=-S'(λ), the theorem of unreachability of λ=1, and the theorem of global asymptotic stability of λ=1/2. Hawking radiation is derived as survival oscillations at the λ→1 boundary with frequency ν∝1/M. The final catastrophic burst emerges as the inevitable consequence of a positive-feedback loop. We provide comprehensive derivations of all scaling relations, detailed comparison with loop quantum gravity, string theory, fuzzball paradigm, and the holographic principle, and extensive discussion of observational tests including gravitational wave echoes, the KM3NeT 220 PeV neutrino event, and photon ring dynamics. A complete falsification matrix is provided.