This study introduces the syntropy function ($S_N$) and expectancy function ($E_N$), derived from the novel function $\phi$, to provide a refined perspective on complexity, extending beyond conventional entropy analysis. $S_N$ is designed to detect localized coherent events, whereas $E_N$ encapsulates expected system behaviors, offering a comprehensive framework for understanding system dynamics. The manuscript explores essential theorems and properties, underscoring their theoretical and practical implications. Future research will further elucidate their roles, particularly in biological signals and dynamic systems, suggesting a deep interplay between order and chaos.

翻译：本研究通过新型函数φ推导出了协熵函数（$S_N$）和期望函数（$E_N$），为复杂性研究提供了超越传统熵分析的精细化视角。$S_N$旨在检测局域相干事件，而$E_N$则表征预期系统行为，二者共同构成了理解系统动力学的综合框架。本文探讨了相关核心定理与性质，着重阐释其理论与应用价值。未来研究将进一步揭示这些函数在生物信号及动态系统中的关键作用，提示有序与混沌之间存在深层互动关系。