Unified Computational Universe Terminal Object:\\Discrete Complexity Geometry, Information Geometry,\\Multi-Observer Causal Network, and Capability--Risk Structure Article Swipe

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Ma HaoBo , Zhang Wenlin · YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.5281/zenodo.17695810

This paper constructs on foundation of previous ``computational universe'' series works a unified computational universe terminal object with explicit categorical meaning. Previous works have axiomatized computational universe as four-tuple $ U_{comp} = (X,T,C,I), and established on it: discrete complexity geometry (complexity distance, volume growth, and discrete Ricci curvature), discrete information geometry (task information manifold (S_Q,g_Q) and embedding \Phi_Q), control manifold (M,G) induced by unified time scale and time--information--complexity joint variational principle, multi-observer consensus geometry and causal network, topological complexity and undecidability, as well as theory of universal catastrophic safety and capability--risk frontier. Goal of this paper is to unify these discrete and continuous, geometric and logical, single-observer and multi-observer, capability and risk structures into single categorical object U_{comp}^{term} ---called unified computational universe terminal object. Specifically, we perform following steps: enumerate \item Define computational universe category with unified time scale CompUniv_\kappa: objects are computational universes U_{comp} satisfying axioms, morphisms are ``safe simulation maps'' simultaneously preserving complexity geometry, information geometry, and catastrophe specifications. \item Construct 2-layer structure on this category: one layer is discrete configuration--event--causal diamond layer; one layer is continuous control--information geometry layer, with multi-observer network, knowledge graph families, and capability--risk frontier placed on top. \item Prove there exists object U_{comp}^{term} = \big( X,\, G_{comp},\, G_{info},\, E_{obs},\, S_{cat},\, F_{CR} \big), and for each U_{comp} \in CompUniv_\kappa ``contraction'' morphism F_{U} : U_{comp} \to U_{comp}^{term}, such that: itemize \item F_U at discrete level is embedding of configuration--event--diamond, at continuous level is embedding of control--information--observer states; \item F_U preserves complexity distance and unified time scale (at most linear rescaling); \item F_U makes all task information geometry and multi-observer consensus geometry become some class of ``submanifold--subnetwork'' on U_{comp}^{term}; \item F_U maps catastrophe specifications and capability--risk frontier to substructures of S_{cat} and F_{CR}. itemize \item Prove in natural 2-category sense (allowing natural transformations between morphisms), U_{comp}^{term} satisfies ``terminal object'' property: for any two such unified objects morphisms between them have unique (in natural isomorphism sense) factorization. \item Finally using previously established physical universe--computational universe categorical equivalence, construct correspondence between unified physical universe terminal object U_{phys}^{term} and U_{comp}^{term}$, and explain both are equivalent terminal objects under unified time scale, boundary diamonds, observer network, and capability--risk structure. enumerate This paper thus gives ultimate unified description of ``universe as computation'' under purely discrete, axiomatic framework: all concrete finite or local computational universes contract through safe--geometric--information compatible morphisms to same unified computational universe terminal object, which simultaneously plays terminal object role at categorical, geometric, and logical levels.

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article

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https://doi.org/10.5281/zenodo.17695810

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https://openalex.org/W7106500378

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https://doi.org/10.5281/zenodo.17695810

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Unified Computational Universe Terminal Object:\\Discrete Complexity Geometry, Information Geometry,\\Multi-Observer Causal Network, and Capability--Risk Structure

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2025

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2025-11-24

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Ma HaoBo, Zhang Wenlin

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https://doi.org/10.5281/zenodo.17695810

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Morphism, Mathematics, Embedding, Theoretical computer science, Universe, Computational complexity theory, Categorical variable, Manifold (fluid mechanics), Algorithm, Descriptive complexity theory, Object (grammar), Discrete time and continuous time, Computer science, Information geometry, Time complexity, Computational topology, Computational geometry, Discrete mathematics, Scale (ratio), Robustness (evolution), Tree (set theory), Decision tree model, Structural complexity theory, Information theory, Topology (electrical circuits), Homotopy

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