Causal prediction of rare recovery responses and the optimal scale of observation time

Bilingual edition: the complete English manuscript appears first, followed by the complete Chinese translation. Both versions include the Technical Supplement, Appendices A–E. Mathematical content version: 2026-11-11.2; bilingual edition: 2026-11-11.3. We determine the information and physical waiting needed to predict a rare intervention response with error smaller than its occurrence probability in a one-dimensional growth process with renewed Poisson interval catastrophes. Deletions of initial sources in macroscopic intervals define a mixed response, with all intervention arms sharing the same stationary past and future forcing. A fixed number of growing submacroscopic gaps determines its rare-event probability scale. We identify the response probability and amplitude, characterize when terminal extinction bits lose predictive information, and construct a gate that recovers this information from the recovery trajectory. For absolute and squared loss, a deterministic-age Bayes-risk expansion holds uniformly with additive error smaller than the rare-event probability. Our main waiting theorem covers every honest causal stopping rule, allowing the complete realized catastrophe history. Its average-time benchmark is the sum of each ordered gap length weighted by the extinction probabilities of preceding gaps. Prediction error smaller than the rare-event probability requires mean waiting divided by this benchmark to diverge; every prescribed diverging multiplier is achievable using actual recovery records. With gap scales $n^{2/5}$ and $n^{4/5}$ and integrated catastrophe tail $A_x\sim C/x$, deterministic observation requires age larger than the $n^{4/5}$ scale, whereas a stopping rule achieves mean waiting $O(n^{2/5}\log n)$. The proofs preserve the rare-event scale through weighted response approximation, risk transfer over arbitrary information fields, and posterior lower bounds on rare history branches. 中文版题名:稀有恢复响应的因果预测与观察时间的最优尺度。 本文确定了在具有更新泊松区间灾变的一维生长过程中,以小于响应发生概率的误差预测稀有干预响应所需的信息与物理等待。在宏观区间内删除初始源,由此定义混合响应,所有干预臂共享同一平稳过去与未来驱动。固定数量、不断增长的亚宏观间隙决定其稀有事件概率尺度。我们识别响应概率与幅度,刻画终局灭绝位何时丧失预测信息,并构造一个从恢复轨迹中重获该信息的门控变量。对于绝对损失与平方损失,确定观察时龄的贝叶斯风险展开一致成立,其加性误差小于稀有事件概率。本文的主要等待定理涵盖每一种合法因果停止规则,并允许使用完整的已实现灾变历史。其平均时间基准,是各个按长度排序的间隙长度以先前间隙灭绝概率加权后的总和。若预测误差小于稀有事件概率,则平均等待除以这一基准必须发散;利用实际恢复记录,可以实现任意预先指定的发散乘子。当间隙尺度为 $n^{2/5}$ 和 $n^{4/5}$、积分灾变尾为 $A_x\sim C/x$ 时,确定时龄观察要求时龄超过 $n^{4/5}$ 尺度,而一种停止规则可实现平均等待 $O(n^{2/5}\log n)$。证明通过加权响应近似、任意信息域上的风险转移,以及稀有历史分支上的后验下界,始终保持稀有事件尺度上的精度。 The source archive includes editable English and Chinese Markdown, TeX, deterministic diagnostic code and recorded output, and offline rebuilding scripts. Original prose: CC BY 4.0. Original code: MIT. AI-use and competing-interest disclosures are included in both language versions. No empirical dataset is used. Related public version-1 preprints: https://doi.org/10.5281/zenodo.23911961 and https://doi.org/10.5281/zenodo.23912425. Complete PDF: 76 pages. English: pages 1–41. Chinese: pages 42–76. 完整双语PDF共76页,英文第1–41页,中文第42–76页。