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

Man Huang

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

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

Contributed by Man Huang ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

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.

Author comments: Complete English manuscript with Technical Supplement Appendices A–E. Full English–Chinese edition, sources and original MIT-licensed diagnostic code: https://doi.org/10.5281/zenodo.24138927. Mathematical content version 2026-11-11.2. Companion preprints: https://doi.org/10.5281/zenodo.23911961 and https://doi.org/10.5281/zenodo.23912425.

Provenance statement

OpenAI GPT-6.1 Sol, xAI Grok 4.7 and OpenAI Codex assisted with literature synthesis, theoretical and argumentative structure, proof drafting, mathematical checking, language revision, diagnostic code and package consistency. HUANG Man is the sole named human contributor. The complete English manuscript includes Technical Supplement A–E. Shared foundations and retained proof passages are attributed to the version-1 companion preprints, DOI 10.5281/zenodo.23911961 and DOI 10.5281/zenodo.23912425. The English–Chinese edition, sources and original diagnostic code are archived at DOI 10.5281/zenodo.24138927. Finite diagnostics are not formal verification.

Tools used

OpenAI: ChatGPTVersion GPT-6.1 Sol
OpenAI: Codex
xAI: GrokVersion 4.7

References

No bibliography entries.

Version history

v1 · 41 pagesSubmitted by Man HuangComplete English manuscript with Technical Supplement Appendices A–E. Full English–Chinese edition, sources and original MIT-licensed diagnostic code: https://doi.org/10.5281/zenodo.24138927. Mathematical content version 2026-11-11.2. Companion preprints: https://doi.org/10.5281/zenodo.23911961 and https://doi.org/10.5281/zenodo.23912425.CurrentNov 12, 2026