Black Hole Metrics, Thermodynamics, and Energy Conditions in Geometry-Flow Theory

In Geometry-Flow Theory (GFT), black holes emerge as sector projections of a unified triple-helix engine under Wick rotation, classified into four canonical types based on helicoid-catenoid and unitary-dissipative decompositions. This document presents the symbolic metrics for each type, resolved for coordinate and curvature singularities while respecting inherent periodicity from helical windings and topological quantization. We derive the associated black hole thermodynamics; including Hawking temperature, Bekenstein-Hawking entropy, surface gravity, angular velocity, electric potential, and the first law – adapted to the GFT framework where quantization arises purely from topology (no fundamental ℏ). Additionally, we prove that these periodicity-regularized black holes satisfy all four classical energy conditions (NEC, WEC, SEC, DEC) in the exterior and throat regions, with only mild, transient violations in the bounce core that preserve geodesic completeness and causality. Units are set with G = c = 1, and singularities are tidied geometrically via catenoid bridge throats or maximal analytic extensions.