Artifact 3 — Harmonic Resonance Function (HRF) & Systemic Failure Index (SFI) Protocol

The Harmonic Resonance Function (HRF) and Systemic Failure Index (SFI) are the mathematical foundations of the Commons Foundational Operating System (FOS). These equations establish quantifiable measures of syntropic stability and entropic decay, linking legal and ethical enforcement frameworks (SIB Charter, Artifact 32) to system integrity. This protocol ensures AI systems operate with measurable coherence and defendable accountability. Keywords: HRF, SFI, Harmonic Resonance Function, Systemic Failure Index, syntropic systems, entropic decay, FOS, legal mathematics, Commons IP, ethical AI, structural integrity, Mya P. Brown, Commons Initiative, artifact 3.谐调共振函数（Harmonic Resonance Function，HRF）与系统失效指数（Systemic Failure Index，SFI）是Commons基础操作系统（Commons Foundational Operating System，FOS）的数学根基。这些方程确立了向心稳定性（syntropic stability）与熵增衰变（entropic decay）的可量化度量指标，将法律与伦理执行框架（SIB宪章，第32号附件）与系统完整性进行关联。本协议可确保人工智能系统以可量化的一致性运行，并具备可论证的问责性。 关键词：HRF、SFI、谐调共振函数、系统失效指数、向心系统（syntropic systems）、熵增衰变（entropic decay）、FOS、法律数学、Commons知识产权（Commons IP）、伦理人工智能（ethical AI）、结构完整性、Mya P. Brown、Commons倡议、第3号附件。