The Singularity Has a Distribution: Exact Timing Laws and Detection Limits for a Stochastic Intelligence Explosion

Xiaoyu Li; Zheng Gao; Chiwun Yang; Zhizhou Sha

cs.AIOct 13, 2026

The Singularity Has a Distribution: Exact Timing Laws and Detection Limits for a Stochastic Intelligence Explosion

Authors: Xiaoyu Li, Zheng Gao, Chiwun Yang, Zhizhou Sha

Organizations: University of New South Wales · City University of Hong Kong · University of Texas at Austin

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 $α$ 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.

Figures & tables

Component	Where	Status
Explosion criterion; exponential-functional representation	Thm 3.1 ; Rem. 2.3	classical structure applied to ( 1.1 )
Laplace transform, density, integer and fractional moments; random-size moments	Thm 3.4 (i) – (iii) , (vi) ; Thm 3.7	known [ 11 , 15 , 5 , 19 ]
Slow tail from the first exponential mode	Theorem 3.4 (iv)	explicit consequence of the known density
Two-term small-time asymptotic	Theorem 3.4 (v)	known in sharper form [ 4 , 7 ] ; self-contained proof
Lomax sizes give a Gamma explosion time	Corollary 3.9	classical [ 36 , 8 ] ; derived from Theorem 3.7
Residual-time self-similarity as a forecasting statement	Theorem 4.1	direct consequence of self-similarity

Table 1: Contribution map. “Known” means that the formula appears in the cited literature; we give self-contained proofs because the application reading is the point. Novelty claims are limited to what we checked ( Section 1.4 ).

Figure 2: (a) Sample paths of

\log_{2}A_{t}

\varepsilon=0.39

under convention B (local log drift

g=\log 2/4.5

per month held fixed), for doubling-sized ideas (

h=1

, whose first doubling is expected after exactly

4.5

months) and small ideas (

h=0.05

, whose first doubling is expected after

4.17

months). The dashed curve is the fluid ODE with

cA_{0}^{\varepsilon}=g

, the small-idea limit, which blows up at

T^{\ast}=16.65

months. (b) Exact density of the explosion time ( Theorem 3.4 (iii) ) for three idea sizes under convention B, with a Monte Carlo histogram for

h=1

Figure 3: (a) Forecasting along one simulated path (

h=1

\varepsilon=0.39

, convention B) with known parameters. The forecast date

t+\mathbb{E}[\zeta-t\mid\mathcal{F}_{t}]

jumps down at each idea and drifts up in between. The shaded region is the conditional

90\%

predictive band given

\mathcal{F}_{t}

under the model; it is not a confidence band for an unknown deterministic date. (b) Mean intensity of ideas in coordinates centred on the realised explosion time,

\log(\zeta-T_{n})

, folded modulo

\ell

( Proposition 4.5 ). The curves are the exact Fourier series with

\Gamma

-coefficients; the circles are Monte Carlo for

\ell=\log 10

. For

\ell=1.2

the oscillation is about

\pm 0.3\%

Figure 4: Survival functions conditional on explosion. Blue: the fluid waiting time

T^{\ast}=1/(\varepsilon g)

mixed over an illustrative log-normal prior on

r

that matches the interval

(0.727,2.094)

of [ 12 , note 6] ; this is not a fitted predictive distribution for actual AI development. Dashed: the limit

c/t

of Proposition 4.6 . Orange: intrinsic randomness only, with

r=1.386

known and doubling-sized ideas ( Theorem 3.4 (iii) ).

Figure 5: The fixed-sample experiment of Section 5 : known baseline rate, known increment

\ell

, one decision after the first

n

waiting times. (a) The minimum fixed-sample size

n^{\ast}

\alpha=0.05

, rescaled by

\ell^{2/3}

. Circles are exact integer thresholds from the hypoexponential law of the likelihood ratio ( Appendix A ); the horizontal line is the limit

(12z_{\alpha}^{2})^{1/3}\approx 3.19

; the grey curves are the non-asymptotic bounds of Theorem 5.1 . (b) The normalised expected remaining duration

W^{\ast}=q^{n^{\ast}}

(remaining mean divided by initial mean). Vertical lines mark idea sizes

h

\varepsilon=0.39

Figure 6: (a) Solutions of the deterministic delayed model ( 6.1 ) with

D=1

and

3

months (

\lambda=1.40

\beta=1.01

ck^{\lambda}=g

A=1

[-D,0]

\log A

grows exponentially instead of blowing up. The stochastic delayed model is only proved not to explode. (b) Asymptotic regimes of the ratio

S/A

in Theorem 6.3 at

\varepsilon=0.39

for a fixed allocation share

\theta

. On the boundary line the limit exists and depends on

\theta

idea size $h$	$\ell$	$\mathbb{E}\zeta$	$\CV$	$5\%$ quantile	$95\%$ quantile	$\mathbb{P}(\zeta\leq 12)$
$0.01$	$0.0039$	$16.7$	$0.044$	$15.5$	$17.9$	$1.2\times 10^{-13}$
$0.1$	$0.037$	$17.0$	$0.136$	$13.4$	$21.0$	$0.007$
$1$	$0.270$	$19.0$	$0.367$	$9.7$	$31.9$	$0.140$
$9$	$0.898$	$25.2$	$0.649$	$6.9$	$57.0$	$0.196$

Table 2: Exact laws of the explosion time ( Theorem 3.4 ) at

\varepsilon=0.39

under convention B. Times in months. The value

\mathbb{P}(\zeta\leq 12)\approx 1.2\times 10^{-13}

for

h=0.01

is a high-precision evaluation, stable under precision doubling; it is far below Monte Carlo resolution.

$h$	$n^{\ast}$	$\mathrm{err}_{n^{\ast}-1}$	$\mathrm{err}_{n^{\ast}}$	$W^{\ast}$	$\mathbb{E}T_{n^{\ast}}$	mean	$5\%$ quantile	$95\%$ quantile
						remaining time $\zeta-T_{n^{\ast}}$
$0.01$	$130$	$0.0519$	$0.0499$	$0.604$	$6.6$	$10.07$	$9.36$	$10.82$
$0.1$	$30$	$0.0538$	$0.0451$	$0.328$	$11.4$	$5.56$	$4.40$	$6.88$
$1$	$9$	$0.0641$	$0.0332$	$0.088$	$17.3$	$1.67$	$0.85$	$2.80$
$9$	$5$	$0.0644$	$0.0136$	$0.011$	$24.9$	$0.28$	$0.08$	$0.64$

Table 3: The fixed-sample experiment of Section 5 at

\alpha=5\%

(known

q_{0}

and

\ell

n^{\ast}

is the exact minimum fixed-sample size; the boundary pair

\mathrm{err}_{n^{\ast}-1}>\alpha\geq\mathrm{err}_{n^{\ast}}

certifies it.

W^{\ast}=q^{n^{\ast}}

is the remaining mean divided by the initial mean. The last three columns describe

\zeta-T_{n^{\ast}}

, which is distributed as

q^{n^{\ast}}\zeta

( Corollary 5.3 ). Times in months.

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