Spectral Inversion of Modular Hamiltonians via Non-Commutative Kolmogorov Complexity: Unitary Information Retrieval in Type III1 Holographic Operator Algebras

Mathematical Applications of Science Fiction We present a rigorous derivation of the spectral inversion map for modular Hamiltonians within the framework of Type III von Neumann algebras, particularly those arising in the algebraic formulation of holographic field theories. By introducing a measure of Non-Commutative Kolmogorov Complexity defined on the predual space of the operator algebra, we establish a direct correspondence between the algorithmic information content of a quantum state and the spectral density of its associated modular operator. We prove that the reconstruction of bulk operators from boundary data is formally equivalent to a spectral inversion problem subject to an algorithmic entropy bound. To this end, we construct the Valerius–Kolmogorov Operator, a unitary mapping that retrieves information from the island regions of the entanglement wedge by inverting the modular flow. We provide detailed proofs of the existence, uniqueness, and stability of this inversion, showing that unitary information retrieval is possible precisely when the complexity growth of the modular flow is subexponential. This framework resolves the trans-temporal information leakage paradox in black hole evaporation models by interpreting Hawking radiation as a complexity-encoded modular scramble.