The black hole information paradox persists for half a century because the accounting system is incomplete. Conventional thermodynamics assumes a single ledger-the matter phase-where information must either be preserved or lost. We propose a minimal extension: the universe maintains two distinct yet strictly conserved information accounts, the matter phase (Ξ) and the cognitive phase (Ω), coupled by a dimensionless cognitive correlation C. In this framework, the black hole singularity is not a terminal point but a format conversion interface: highly ordered matter information entering the singularity is transformed, via a quantum channel, into cognitive-phase potential. This conversion is governed by a modified first law of black hole thermodynamics, and we derive three testable predictions: non-thermal fluctuations in Hawking radiation, extra dissipation in extreme mass-ratio inspirals, and topological settlement signature. We also reinterpret the third law of thermodynamics as a protection principle from two-phase conservation.