构建可解释的考试准备度指数，量化学习者备考状态
Exam Readiness Index (ERI): A Theoretical Framework for a Composite, Explainable Index
- 基于六项学习信号合成可解释的准备度评分
- 评分在0-100间，具备稳定性与唯一最优解保证
- 适合教育科技与个性化学习系统开发者参考
本文提出一种考试准备度指数（ERI）的理论框架：一个在[0,100]区间内的复合、可解释评分R，用于综合评估学习者对高风险考试的准备情况。ERI整合了六个信号——掌握度（M）、覆盖度（C）、保持率（R）、进度（P）、波动性（V）和耐力（E），均来自练习与模拟测试的交互数据流。论文形式化定义了分量映射与复合函数的公理，证明了单调性、Lipschitz稳定性及蓝图权重调整下的有界漂移，并在凸设计约束下证明了最优线性组合的存在性与唯一性。通过蓝图加权集中性分析，刻画置信区间，并证明其与先修知识可容许课程（知识空间/学习空间）兼容。研究聚焦理论推导，实证分析留待未来工作。
原文摘要 · Abstract (English)
We present a theoretical framework for an Exam Readiness Index (ERI): a composite, blueprint-aware score R in [0,100] that summarizes a learner's readiness for a high-stakes exam while remaining interpretable and actionable. The ERI aggregates six signals -- Mastery (M), Coverage (C), Retention (R), Pace (P), Volatility (V), and Endurance (E) -- each derived from a stream of practice and mock-test interactions. We formalize axioms for component maps and the composite, prove monotonicity, Lipschitz stability, and bounded drift under blueprint re-weighting, and show existence and uniqueness of the optimal linear composite under convex design constraints. We further characterize confidence bands via blueprint-weighted concentration and prove compatibility with prerequisite-admissible curricula (knowledge spaces / learning spaces). The paper focuses on theory; empirical study is left to future work.
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