Operatorization: A Framework for Transforming Fixed Solutions into Executable Knowledge

Abstract

Human knowledge has accumulated through the discovery of theorems, laws, algorithms, logical principles, and formal proofs. While these artifacts have enabled significant scientific and technological progress, they are typically stored, communicated, and utilized as static knowledge objects. Modern artificial intelligence systems primarily learn from data and textual representations of such knowledge, rather than directly leveraging the operational structures embedded within them [11,15,16]. This paper proposes a conceptual framework termed Operatorization, a process that transforms fixed solutions and static knowledge artifacts into machine-executable operators. Instead of treating a theorem, law, or formal result solely as an object of interpretation, the proposed framework seeks to identify and extract its reusable behavioral structures, constraints, invariants, and operational semantics, thereby enabling its representation as an executable computational entity. The framework introduces a general mapping from knowledge objects to executable operators and illustrates the process through three representative case studies: the Tuy's Cut Operator [1,2], the Brauer Height-Zero Operator [3,4], and the DEO-2 Evolution Operator (Dynamic Evolution Operator derived from Second-Order Differential Evolution Equations and Chernoff Approximation Theory) [5-7]. These examples demonstrate how established mathematical structures may be reformulated as reusable computational components suitable for reasoning systems, decision-support frameworks, simulation environments, and future hybrid AI architectures. To demonstrate practical executability, a lightweight reference implementation containing representative operators and a reusable operator schema accompanies the proposed framework. The paper hypothesizes that Operatorization may provide a useful intermediate layer, enabling artificial intelligence to utilize not only information but also selected forms of executable behavior derived from scientific and mathematical knowledge. A lightweight reference implementation is provided as a proof-of-concept to support reproducibility, independent validation, and future research.