The Finitism Black Hole: Structural Completion at the Horizon

The Schwarzschild metric diverges at the event horizon and predicts a curvature singularity at the core, signaling a breakdown of continuum general relativity. Standard approaches treat these infinities as mathematical artifacts to be regularized or resolved by unknown quantum gravity effects. Computational finitism proposes a different resolution: infinities are representational overflows that emerge when continuous mathematics is applied to a finite causal substrate. By replacing the divergent metric component with a smooth saturation function derived from finite state capacity per Planck volume, we show that the horizon is not a mathematical surface but a smeared computational phase boundary of thickness . The black hole core is not a singularity but a saturated computational cluster running at maximum update frequency. This structural completion yields three sharp, parameter-free observational signatures: gravitational wave echoes with quantized delay and geometric decay, a high-energy cutoff in Hawking radiation, and sub-Poissonian photon statistics. All predictions are anchored to , the finite alphabet limit. The horizon singularity does not require quantum gravity to fix it; it dissolves when the map stops pretending infinity is physical. We also make falsifiable predictions.