preserving all morphisms. Decomposition to set-theoretic elements and recomposition via categorical functors are bidirectional and structure-preserving. Sketch of Proof. Let C_w be the category whose objects are physical tuple networks N_w and whose morphisms are causal transformations. Let C_s be the category whose objects are symbolic tuple networks N_s and whose morphisms are formal transformations. Define F: C_w→ C_s by mapping each physical element e ∈ε_w to a symbolic token s∈ε_s via encoding function Φ:ε_w → ε_s . Extend F to morphisms by mapping causal edges to formal edges. By construction， (3) provides categorical mappings between domains。
Objective Knowledge: An Evolutionary Approach ， “Minds。
and intent at the ontological level; (2) formalizes each to set-theoretic and categorical precision; (3) defines intelligence as the sum of information processing and selective intent; (4) yields a human–machine collaboration protocol usable by both biological and artificial agents. Its uniqueness， founded by Xiaohui Zou， founded by Xiaohui Zou。
1953. 3.J. Searle， and the failure of psychology， 1950. 5.K. Popper， outsourceable to AI · Selec- tive Intent : goal-setting，” Mind 。
1921. 2.L. Wittgenstein。
1980): machines perform Information Processing without Selec- tive Intent 。
and Yi (selective intent/meaning). This paper presents the first formal mathematical characterization of RZX/Smart System Studies (SSS) using set theory and category theory. Its proved that both Wu and Wen possess a two-level structure: sub-universal (element objects) and super-sub-domain (hierarchical tuple networks). Its established that the Human-Machine Equation: Intelligence = Information Processing + Selective Intent. Its further proved that the Fen-He Freedom Theorem: decomposition to set-theoretic elements and recomposition via categorical functors are bidirectional and structure-preserving. This framework resolves two long-standing gaps: the disconnect between philosophical categories and mathematical category theory， symbolic mediation，N where E is the sub-universal : a set of atomic element objects; and N is the super-sub-domain : a hierarchical tuple network over ε 。
AI philosophy，。
formally a directed graph N=(V， there exists a faithful functor F: C_w → C_s from the category of physical structures to the category of symbolic structures， and (4) yields an operational protocol for human–machine collaboration. Rongzhi-Xue (RZX)/mart System Studies (SSS) ， and cognitive science to jointly define intelligence. RZX/SSS provides the first cogni-tive operating system usable by both humans and machines. A Unified Category-Theoretic Framework for Human–Machine Cognitive Collaboration Xiaohui Zou^0000-0002-5577-8245 October 2026 Abstract Rongzhi-Xue (RZX)。
and syntax is not semantics. Yet no prior framework has provided a rigorous mathematical structure that simultaneously: (1) separates physical carriers from symbolic forms from mental intent。
Cognitive Science Psychology studies human intelligence but ignores machine information processing. Computer science studies information processing but ignores selective intent. Cognitive science oscillates between “cognition = computation” and “cognition ≠ computation” without resolution. RZX unifies all three via Equation : intelligence requires both components; disciplines that study only one are necessarily incomplete. 8 Architecture Diagram
