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REVIEW 4 major objections 6 minor 46 references

Will the Technological Singularity Come Soon? Modeling the Dynamics of Artificial Intelligence Development via Multi-Logistic Growth Process

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Stacking S-shaped growth curves onto 70 years of AI milestones puts the current deep-learning wave's peak at 2024 and its fade at 2035–2040, implying the technological singularity is not near.

desk verdict A decent curve-fitting exercise with an honest historical decomposition, but the headline singularity forecast is just a labeled logistic parameter, not a robust prediction. read the letter →

arxiv 2502.19425 v1 pith:ULPTB5TQ submitted 2025-02-11 physics.soc-ph cs.CY

Guangyin Jin , Xiaohan Ni , Kun Wei , Jie Zhao , Haoming Zhang , Leiming Jia This is my paper · ORCID
classification physics.soc-phcs.CY
keywords technologicalsingularityartificialintelligencemulti-logisticgrowthlogisticprocesstechnologyforecastingdeeplearninglargelanguagemodelsinnovationwaves
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to test a contested question with data: is the technological singularity imminent? It hypothesizes that AI development is not one exponential takeoff but a sequence of overlapping S-shaped growth waves, and it fits a multi-logistic model to the cumulative number of famous AI systems from 1950 to 2023. The fitted third wave peaks around 2024 and reaches saturation around 2035–2040 unless a fundamental new breakthrough starts another wave. Independent counts of AI papers, GPU transistors, and internet users corroborate the same timing. The authors conclude that a near-term singularity is unlikely, while acknowledging the current wave is still transformative.

What carries the argument

The multi-logistic growth function $$L_n(t) = a_0 + \sum_{i=1}^{N} \frac{a_i}{1+\exp\left(-\frac{t-m_i}{w_i}\right)}$$ sums one logistic curve per technology wave, each with ceiling $a_i$, midpoint $m_i$, and speed width $w_i$. The midpoint $m_i$ is the fastest-growth point of that wave, and the interval rules $\sigma_i=\pi w_i/\sqrt{3}$ divide each wave into emergence, growth, maturity, and saturation stages. This object carries the argument because the fitted third-wave midpoint $m_3\approx 2024$ and its $2\sigma$ upper edge near 2040 convert the historical dataset into a dated forecast. The first derivative of the sum is treated as the speed or heat of AI development, and the Levenberg-Marquardt method is used to estimate the parameters.

What would settle it

If, after 2024, the annual increase in notable AI systems or AI papers does not begin falling along the fitted curve, or if a new wave of foundational AI systems appears before roughly 2035, the central claim is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that the dynamics of AI development are multi-logistic: the cumulative number of well-known AI systems from 1950–2023 is well described by a sum of logistic curves, with three waves for overall AI and two for industry. The estimated midpoint of the current deep-learning wave is $m_3 \approx 2024.46$, and its $2\sigma$ saturation interval runs to about 2040, so the model predicts that the current wave will plateau in 2035–2040 absent a new fundamental innovation. The first derivative of the fitted sum, read as the speed of AI development, is near its maximum now, which explains the intensity of LLM progress without invoking runaway growth. The consistency of the fits across the system-count, paper-count, GPU transistor, and internet-user datasets is taken as evidence that the forecast is robust.

Load-bearing premise

The forecast stands on the assumption that the current deep-learning wave is one S-shaped curve whose shape is already set by data through 2023, and that the number of waves (three overall, two for industry) is known in advance.

Editorial extensions

If this is right

  • The current LLM boom is the steepest part of the third S-curve, so the rate of new notable AI systems should start decelerating from here.
  • Without a fundamental theoretical breakthrough, cumulative AI output will approach saturation in the 2030s, with the growth phase ending around 2035–2040.
  • The second wave's long duration supplied the deep-learning foundations, and industry now drives the third wave, a shift that frames where the next wave would likely originate.
  • The exponential-growth view of AI, and with it the near-term technological singularity, is rejected by the data; any singularity scenario is pushed beyond the current wave's horizon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to fit the same stacked S-curves to capability-weighted AI measures, such as benchmark scores or economic output, to check whether the 2024 peak and 2035–2040 saturation hold beyond simple system counts.
  • Because the paper fixes the number of waves in advance and does not model how a fourth wave would be born, the 2035–2040 fade is best read as the no-breakthrough baseline rather than as a hard ceiling.
  • If the midpoint is really 2024, year-over-year growth in new notable AI systems should be visibly declining by the late 2020s; this is a concrete calendar check of the model.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper proposes a multi-logistic growth model for the cumulative number of famous AI systems, hypothesizing that AI development is a superposition of logistic components. Using the AI Historical Statistics, AI Arxiv Papers, GPU Transistor, and Internet User datasets, the authors fit the model with N=3 waves for total/academia and N=2 for industry/collaboration, compare against logistic, exponential, LPPL, and polynomial fits, and read off from the fitted third-wave midpoint (m3≈2024) that the current AI wave peaks around 2024 and will decline around 2035-2040 absent fundamental innovations. The paper concludes that the technological singularity is unlikely in the foreseeable future.

Significance. If the central forecast were supported, the paper would provide a quantitative, data-driven counterpoint to exponential-growth singularity forecasts and an interpretable decomposition of AI history into logistic waves. The in-sample fits are excellent (R² ≈ 0.999 on the AI Historical Statistics dataset), and the comparison against ordinary logistic and exponential growth is a useful demonstration that a single S-curve cannot capture the multiple waves. The cross-validation on Arxiv paper counts with a single logistic component is a reasonable check of the current-wave timing. However, because the headline decline prediction is extracted from parameters that are weakly identified from pre-inflection data, and because the model has no mechanism to forecast the fourth wave that the authors themselves acknowledge, the paper does not establish the 'objective law' it claims. The contribution is a plausible descriptive model plus an unsupported extrapolation.

major comments (4)
  1. [4.4, Table 3, Eq. (9)] The claims that the current wave peaks around 2024 and declines in 2035-2040 are direct readings of the fitted parameters m3=2024.46 and w3=4.33 in Table 3, not independent forecasts. Because the data end in 2023, before m3, the third logistic component is observed only on its pre-inflection, accelerating branch. In this regime, a3 and w3 are weakly identified: many (a3,w3,m3) triples reproduce the observed early trajectory with very different saturation behavior. The confidence intervals from Eq. (9) are linear-regression intervals around the fitted curve and do not propagate the joint uncertainty in the nonlinear parameters. The paper should provide profile-likelihood or bootstrap intervals for m3 and w3 and should treat the 2035-2040 decline window as a consequence of the assumed logistic form, not as a data-derived bound.
  2. [4.4, Section 5] The conclusion that 'the technological singularity appears unlikely in the foreseeable future' does not follow from the fitted decline of the third wave. The multi-logistic model is a sum of independent logistic components and contains no arrival process for future components. Section 4.4 itself states that the timing of a fourth wave 'remains uncertain' and that historical waves overlap; a fourth wave beginning around 2030-2035 is consistent with the model's historical pattern. The fitted decline can therefore support only the conditional statement 'if no fundamental technological innovation emerges,' not a judgment about the probable occurrence of a singularity, which would depend on future waves that the model does not predict.
  3. [4.4, Table 2] The number of waves N is fixed a posteriori from the 'three peaks and two troughs' narrative (N=3 for total/academia, N=2 for industry/collaboration), and no comparison is made to alternative values of N on the same data. Because the chosen N determines which model component produces the headline m3 and w3, the model comparisons in Table 2 do not select among multi-logistic models with different numbers of waves. The claim in Section 5 that the multi-logistic process is 'an objective law of AI technology development' therefore overstates the evidence, which supports only a descriptive fit to the historical waves.
  4. [4.5, Table 5] The cross-validation experiment on the AI Arxiv Papers dataset fits a single logistic function to cumulative counts from 2008 to 2023 and then uses the fitted function to assert a decline around 2035-2040. Only 16 data points are used, no data are held out, and the decline time is again an extrapolation of the fitted functional form beyond the observation period. This is a consistency check of the current-wave midpoint, not a validation of the forecast; the term 'cross-validation' is therefore misleading.
minor comments (6)
  1. [Section 4] The research questions are misnumbered: 'RQ1' appears twice; the second should be RQ2, and the subsequent questions should be renumbered accordingly.
  2. [Section 3] The description of the four growth stages contains duplicated words ('which which') and a confusing statement that the development speed 'begins to decline exponentially' in both the maturity and saturation stages; the text should be edited for clarity.
  3. [Section 4.1] There are several typos: 'verity' should be 'verify', and 'are are' appears in the definition of N and Nv. Additionally, Table 1 labels four datasets while the text describes categories within datasets; the wording should be aligned.
  4. [Introduction] Figure 2 is described before it is introduced, and the claim that the logistic model has a higher R² than exponential growth for LLM papers is not quantified in the text; the authors should provide the numerical comparison.
  5. [Section 4.4] The sentence 'From the results in Table 4, we obtain the 2σ time range of total AI development in the third wave is [2009, 2040]' is grammatically incomplete; 'is' should be 'as' or the sentence should be restructured.
  6. [Eq. (9)] Equation (9) is the standard formula for a linear-regression confidence interval; its use for a nonlinear model with estimated parameters should be justified or replaced with a nonlinear method (e.g., delta method or bootstrap).

Circularity Check

2 steps flagged · score 6.0 of 10

The headline predictions reduce to fitted logistic parameters: the '2024 fastest point' is the fitted midpoint m3, and the '2035-2040 decline' is the 2-sigma interval computed from the fitted width w3. The paper's self-citations are not load-bearing.

  1. fitted input called prediction [Section 3 Eq. (1); Section 4.4 and Table 3]
    "M represents the midpoint of the logistic function, the point where the rate of development is fastest. ... From the results in Table 3 and Figure 7(b), the projected midpoint of the ongoing third wave of AI is approximately 2024, suggesting we are currently experiencing the peak velocity of this technological surge."

    Table 3 lists m3 = 2024.46 for the total dataset as a fitted parameter of Eq. (3). The paper's own definition in Section 3 says M is 'the point where the rate of development is fastest.' Therefore the headline result 'around 2024 marks the fastest point' is not an independent forecast; it is the fitted value of m3 relabeled as a 'projected midpoint.' No quantity beyond the parameter estimate is involved.

  2. fitted input called prediction [Section 4.4, Table 4, and Eq. (2)]
    "Nevertheless, as indicated by the predictive trend illustrated in Figure 7 and the temporal analysis detailed in Table 4, absent any substantial new technological advancements, the current AI wave is anticipated to plateau between 2035 and 2040."

    The 'temporal analysis' in Table 4 computes T3(2σ3) = 2009-2040 using σ3 = π × w3 / sqrt(3) from Eq. (2), with the fitted width w3 = 4.33. Thus the predicted 'plateau between 2035 and 2040' is just the M ± 2σ interval of the fitted third logistic term, a direct function of the fitted parameters m3 and w3. Because the data end in 2023, before the fitted midpoint 2024.46, the declining branch after the midpoint is not observed; the 2035-2040 window is dictated by the assumed logistic shape and the fitted width, not by data showing saturation.

full rationale

The paper is not circular in its model-building or baseline comparisons: fitting a multi-logistic curve to the AI Historical Statistics data and showing that it outperforms exponential, logistic, and polynomial alternatives is a legitimate empirical exercise. The self-citations [13,14] are incidental references in a limitations paragraph and do not carry the argument. The circularity is concentrated in the two headline quantitative conclusions. The 'fastest point around 2024' is exactly the fitted midpoint m3 of the third logistic term, and the paper's own definition of M is 'the point where the rate of development is fastest,' so this is a fitted parameter presented as a prediction. Similarly, the 'decline around 2035-2040' is the sigma-based interval computed from the fitted width w3 via Eq. (2) and displayed in Table 4; it is a relabeling of fitted parameters rather than an out-of-sample forecast. The cross-validation on Arxiv papers does not break this circularity, because it repeats the same logistic fitting procedure and again reports the fitted midpoint and sigma interval as the predicted peak and decline. The paper also notes that the timing of a fourth wave 'remains uncertain' and that historical waves overlap, which is an honest caveat; however, the abstract's unconditional-sounding claim that the technological singularity appears unlikely in the foreseeable future depends on the fitted decline of the current wave. That dependence is a consequence of the model's construction, not an independent empirical finding. Overall, the circularity is partial: the historical fit and model comparisons are meaningful, but the central forecast reduces to the fitted parameters of the assumed logistic form, warranting a score of 6.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model has 10 free parameters for the total series (a0, plus a_i, w_i, m_i for each of three waves). All predictions are deterministic functions of these fitted values. The number of waves is an unsupported modeling choice. No new entities are postulated.

free parameters (4)
  • a3, w3, m3 (third wave ceiling, speed, midpoint) = a3=1664.85, w3=4.33, m3=2024.46
    These parameters are fitted to the AI Historical Statistics 'Total' data and directly determine the predicted peak and decline time.
  • a1, w1, m1, a2, w2, m2 (first and second wave parameters) = See Table 3
    Fitted parameters for earlier waves, used to reconstruct the multi-logistic curve.
  • a0 (initial value) = Not reported in table
    Baseline value in Eq. 3, fitted to data.
  • Number of waves = 3 for total/academia, 2 for industry
    Chosen by visual inspection of the historical data, not learned from the model. This choice affects the entire fit and prediction.
assumptions (4)
  • domain assumption AI development can be represented as the sum of independent logistic growth curves.
    Stated as a hypothesis in Section 3. No physical mechanism is given for why superposition should hold.
  • domain assumption Cumulative number of famous AI systems is a valid proxy for AI development level.
    The dataset definition is subjective and may be biased; this proxy is not independently validated.
  • ad hoc to paper The number of waves is known a priori and fixed.
    The paper sets 3 waves for total/academia and 2 for industry based on visible peaks, without a formal model selection test.
  • standard math Standard nonlinear least squares and linear-regression confidence intervals apply.
    Assumes measurement errors are independent and normally distributed, which is not tested.

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Cite this review

Pith. "Pith review of Will the Technological Singularity Come Soon? Modeling the Dynamics of Artificial Intelligence Development via Multi-Logistic Growth Process." pith.science (2026). https://pith.science/paper/ULPTB5TQ

@misc{pith2026250219425,
  author       = {Pith},
  title        = {Pith review of: Will the Technological Singularity Come Soon? Modeling the Dynamics of Artificial Intelligence Development via Multi-Logistic Growth Process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ULPTB5TQ}},
  note         = {Machine review of arXiv:2502.19425}
}
read the original abstract

We are currently in an era of escalating technological complexity and profound societal transformations, where artificial intelligence (AI) technologies exemplified by large language models (LLMs) have reignited discussions on the 'Technological Singularity'. 'Technological Singularity' is a philosophical concept referring to an irreversible and profound transformation that occurs when AI capabilities surpass those of humans comprehensively. However, quantitative modeling and analysis of the historical evolution and future trends of AI technologies remain scarce, failing to substantiate the singularity hypothesis adequately. This paper hypothesizes that the development of AI technologies could be characterized by the superposition of multiple logistic growth processes. To explore this hypothesis, we propose a multi-logistic growth process model and validate it using two real-world datasets: AI Historical Statistics and Arxiv AI Papers. Our analysis of the AI Historical Statistics dataset assesses the effectiveness of the multi-logistic model and evaluates the current and future trends in AI technology development. Additionally, cross-validation experiments on the Arxiv AI Paper, GPU Transistor and Internet User dataset enhance the robustness of our conclusions derived from the AI Historical Statistics dataset. The experimental results reveal that around 2024 marks the fastest point of the current AI wave, and the deep learning-based AI technologies are projected to decline around 2035-2040 if no fundamental technological innovation emerges. Consequently, the technological singularity appears unlikely to arrive in the foreseeable future.

Figures

Figures reproduced from arXiv: 2502.19425 by the authors.

Figure 1
Figure 1. AI technology has experienced three peaks and two troughs in history [3]. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The trends and fitting curves of the cumulative numbers of arXiv papers that contain the keyword [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The logistic growth and regrowth process of technological development. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The visualization of segmented fitting for different AI waves of overall and academia. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: The visualization of segmented fitting for different AI waves of industry and industry-academia [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: The prediction of the number of famous AI systems with 95% confidence intervals. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: The visualization of the attributes for different AI waves. [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: The data and prediction results on the annual cumulative number of AI-related papers. [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: The data and prediction results on the annual cumulative number of internet users and GPU [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.