The Systems Science Foundation of Contingency: Entropy Increase, Synergistic Evolution, and AI Governance from a Five-Dimensional Mathematical Perspective

Based on five-dimensional mathematics, this paper defines contingency as the significantdeviation between the estimated synergy coefficient ˆ κ and the actual value κ act (deviation degree ∆κ =|κ act −ˆ κ|/ˆ κ). Two types are distinguished: negative deviation (high estimate, low realization) and positivedeviation (low estimate, high realization). We prove that the long-term decline of κ act is equivalent toentropy increase, and the probability of contingency is positively correlated with κ act falling belowthe critical threshold. Using the Challenger disaster, rare-input AI misjudgment, and an old bridge’sunexpected stabilization as examples, we demonstrate the framework’s quantitative power. We thenextend it to AI safety governance via a κ guardian. Finally, we discuss the Big Bang as a primordialcontingent event. This work transforms contingency from philosophical speculation into a computabledeviation index, providing a unified epistemological and instrumental foundation for systems science,engineering reliability, and AI safety.