Fisher Information Metrics and Entropic Dynamics in Quantum Statistical Manifolds: A Motivic Approach

Mathematical Applications of Science Fiction We present a rigorous derivation of the Quantum FisherInformation Metric (QFIM) within the framework of derivedalgebraic geometry and non-commutative spectral theory.By treating the manifold of quantum states as a modulispace of stable sheaves over a non-commutative C∗-algebra,we establish a canonical isomorphism between the tangentbundle of the statistical manifold and the first derived coho-mology group of the entropic structure sheaf. We introducethe Malcor-Bures Spectral Sequence, which converges to theBures metric, and prove that the entropic dynamics gener-ated by the gradient flow of the von Neumann entropy isgeodesically complete with respect to the Wigner-Yanase-Dyson skew information metric. Furthermore, we analyze thestability of these dynamics using motivic L-functions associ-ated with the modular automorphism group, demonstratingthat the information geometry admits a unique hyper-Kählerstructure in the thermodynamic limit.