Modular Hamiltonian Reconstruction of Scrambled States: A Non-Perturbative Isomorphism between Asymptotic Quantum Complexity and Unitary Information Retrieval in Type-III1 von Neumann Factors

Mathematical Applications of Science Fiction We present a rigorous derivation of a non-perturbative isomorphism between the asymptotic growth of quantum circuit complexity and the recoverability of information in scrambled quantum states, formulated within the framework of Type-III von Neumann algebras characteristic of local quantum field theories. By extending the Complexity equals Volume conjecture through Tomita–Takesaki modular theory, we construct a correspondence that links the geodesic length on the manifold of unitaries to the spectral properties of the modular Hamiltonian. We prove that the scrambling time corresponds to a phase transition in the Connes cocycle derivative, demonstrating that perfect information retrieval is dual to the saturation of the Lloyd bound. The work includes ten supporting lemmas and five main theorems, culminating in the Thorne–Reconstruction Theorem, which establishes that modular flow functions as a universal decoding mechanism for Hawking radiation in the infinite-N limit.