arXiv：cs.AI· Xiaoyu Li, Zheng Gao, Chiwun Yang, Zhizhou Sha·· 4 小时前
奇点有分布：随机智能爆炸的精确时间规律与检测极限
The Singularity Has a Distribution: Exact Timing Laws and Detection Limits for a Stochastic Intelligence Explosion
AI 导读
研究将AI研究自动化视为软件能力与研究投入之间的正反馈循环，发现随机性不改变确定性爆炸阈值，但把爆炸时刻替换为显式分布：等量改进下爆炸时间为独立指数分布的几何加权和，Lomax分布改进下则精确服从Gamma分布。在恒定与几何增长想法到达率之间的固定样本检验中，已知基线率与对数率增量ℓ时，误差水平α下的最小样本量渐近于(12z_α^2)^{1/3}ℓ^{-2/3}。
正文
Abstract:Automating AI research creates a positive feedback loop between software capability and research effort. We study a stochastic realisation of this loop in which independent, positive, multiplicative improvements arrive at a rate proportional to a power of current capability. Within this model, randomness leaves the deterministic explosion threshold unchanged but replaces the blow-up date by an explicit distribution. For equal-sized improvements the explosion time is a geometrically weighted sum of independent exponentials; its $q$-series representation gives its transforms, moments and tail behaviour, including a two-term small-time asymptotic. For Lomax-distributed improvement sizes the explosion time is exactly Gamma distributed. Self-similarity implies that, with parameters known, the relative uncertainty about the remaining time does not vanish as explosion approaches. We formulate a fixed-sample test between constant and geometrically increasing idea-arrival rates. If the baseline rate and the alternative log-rate increment $\ell$ are known, the minimum sample size at error level $\alpha$ is asymptotic to $(12z_\alpha^2)^{1/3}\ell^{-2/3}$, where $z_\alpha$ is the upper-$\alpha$ standard normal quantile. Uncertainty about the growth exponent near its critical value produces a $1/t$ tail with an explicit constant, and hence an infinite conditional mean waiting time, although explosion is certain given a positive exponent. Separate extensions show that a fixed deployment delay prevents explosion in the specified delayed-feedback models, that power-law diversion of effort shifts the growth threshold, and that in a deterministic oversight race a fixed allocation share does not change the asymptotic regime, except at a critical equality of exponents. Numerical illustrations, which are not empirical forecasts, show how timing and detectability depend on the granularity of improvements.
| Comments: | 24 pages, 6 figures, 3 tables. Code and data: this https URL |
| Subjects: | Artificial Intelligence (cs.AI); Computers and Society (cs.CY); Probability (math.PR) |
| MSC classes: | 60J27 (Primary), 60G51, 33D15, 62F03, 34K25, 91B62 (Secondary) |
| Cite as: | arXiv:2610.18445 [cs.AI] |
| (or arXiv:2610.18445v1 [cs.AI] for this version) | |
| https://doi.org/10.48550/arXiv.2610.18445 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Zheng Gao [view email]
[v1]
Tue, 13 Oct 2026 14:16:55 UTC (3,748 KB)
来源：arXiv：cs.AI · arxiv.org