The Complex Numbers and Unitarity of the Self-Referential Universe 2; The Geometric-Algebraic Duality of Self-Referential Systems: A Unified Framework of Information Manifolds and Sheaf Categories
Year: 2026
Type: preprint
Abstract: Previous work has deepened the mathematical foundations of self-referential systems from the two directions of information geometry and category theory. Information geometry reconstructs self-referential systems as geodesic motion on a manifoldof quantum states, introducing quantum Fisher information and the Bures metric tocharacterize their dynamics and critical behavior; category theory reconstructs selfreferential systems as self-referential monads on tensor categories, revealing theiralgebraic and entanglement essence through sheaf categories and monad structures.This paper proves that these two descriptions are profoundly equivalent: the Fisherinformation flow corresponds to the derivative of the self-referential monad multiplication, and the Bures metric corresponds to the Riemannian metric on the sheafcategory. We construct an explicit functor from the sheaf category to the information manifold and prove that this functor constitutes a categorical equivalence;under this equivalence, the multiplication natural transformation of the monad induces the generator of the Fisher information flow, while the inner product structureof the morphism space in the sheaf category precisely yields the Bures metric. Thisunification provides a dual perspective for understanding the information processing and algebraic structure of self-referential systems, lays a unified mathematicallanguage for physical predictions such as the black hole information paradox andgravitational wave echoes, and reveals deep connections among quantum gravity,information theory, and category theory.
Authors: zhou, changzheng, zhou, ziqing
Language: English
Topic: Complex Systems and Dynamics
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7134932153