奇点具有分布：随机智能爆炸的精确时序定律与检测极限
The Singularity Has a Distribution: Exact Timing Laws and Detection Limits for a Stochastic Intelligence Explosion
AI总结：
该研究构建随机智能爆炸模型，推导爆炸时间的精确分布，提出想法到达率的固定样本检验，分析参数不确定性等对爆炸时序与可检测性的影响。
摘要：
自动化AI研究在软件能力与研究投入间形成正反馈循环，我们研究该循环的随机实现形式，其中独立的正乘法改进以与当前能力的幂次成正比的速率到达。在该模型中，随机性未改变确定性爆炸阈值，但将爆炸日期替换为显式分布：对于等大小改进，爆炸时间是独立指数变量的几何加权和，其q级数表示给出变换、矩和尾部行为，包括两项短时渐近；对于服从Lomax分布的改进大小，爆炸时间精确服从Gamma分布。自相似性表明，在参数已知时，剩余时间的相对不确定性不会随爆炸临近而消失。我们构建了固定样本检验，用于检验恒定与几何增长的想法到达率；若基线率与替代对数率增量ℓ已知，误差水平α下的最小样本量渐近为(12z_α²)^(1/3)ℓ^(-2/3)，其中z_α为标准正态分布的上α分位数。增长指数在临界值附近的不确定性产生带显式常数的1/t尾部，因此存在无限条件平均等待时间，尽管在指数为正时爆炸是确定的。额外扩展表明，固定部署延迟可阻止指定延迟反馈模型中的爆炸，幂律式的投入转移会改变增长阈值，在确定性监督竞赛中，固定分配份额不会改变渐近机制，除非指数达到临界相等。数值示例（非经验预测）展示了时序与可检测性如何依赖于改进的粒度。
英文摘要：
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 $α$ is asymptotic to $(12z_α^2)^{1/3}\ell^{-2/3}$, where $z_α$ is the upper-$α$ 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.
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