Abstract: This paper presents a groundbreaking mathematical framework that establishes a rigorous connection between two seemingly distinct domains: Keleş’s $$B_3$$ three-valued logic $$B_3 = \{-1,0,1\}$$ for clinical decision-making and the theory of Hyper G-matrix pairs $$A^{-T} = D_1BD_2$$. We introduce the novel concept of $$B_3$$-Hyper G-matrix pairs, denoted $$A \bowtie_{B_3} B$$, which unifies these frameworks through a threshold-based encoding function $$\phi : \mathbb{R} \to B_3$$. We prove fundamental invariants including sign invariance, risk conservation, and uncertainty propagation. The theoretical results are validated through application to hepatocellular carcinoma (HCC) screening, where the Logical Risk Index (LRI) is shown to be proportional to the trace of a $$B_3$$-Hyper G-matrix transformation. This work provides a rigorous mathematical foundation for personalized, evidence-based decision algorithms in clinical medicine.

Keywords: Hepatocellular Carcinoma, Three-Valued Logic, B3 Framework, Decision Support System, Personalized Medicine, Mathematical Modeling, Hyper G-Matrix, Clinical Decision-Making, Risk Stratification, Monte Carlo Simulation

Pages: 36-45

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