This paper provides a formal resolution to the Black Hole Information Paradox (Hawking, 1974) by applying the Mechanical Law of Consciousness and the framework of the Conscious Plenum. We demonstrate that the 50-year conflict between General Relativity and Quantum Mechanics is a product of a dualistic "Explanatory Gap" that treats physical matter and information content as separate ontologies. By defining the Black Hole not as a vacuum singularity, but as a Mechanical Saturation Point of information density, we derive the Bekenstein-Hawking area-entropy coefficient ("4") as a topological requirement of Plenum pressure (P_Phi). We mathematically establish a Zero-Remainder Identity between the event horizon's surface area and its integrated bit-threshold (M_max), proving that information is never "lost" or "hidden," but is identical to the physical boundary itself. Furthermore, we calculate the "Ultimate Frequency" of identity resolution at the Planck scale, providing a mechanical origin for Hawking Temperature and predicting a specific, quantized spectral signature for black hole radiation. This resolution is a direct application of the Mechanical Law established in Taming the Brute Fact (Tyler, 2026). This work successfully unifies the "Smooth" of Relativity with the "Chunky" of Quantum Mechanics, proving that the laws governing stellar collapse are identical to those governing biological awareness.