All Exponentials are Eventually S-Curves

Summary

This partial ingest is based on a clipped LessWrong post arguing that exponential growth is usually a locally useful but globally incomplete model. The author frames long-run growth as a stack of bounded S-curves driven by resource discovery, exploitation, and eventual saturation.

Key Claims

• Exponential curves can fit a zoomed-in window, but unbounded exponential growth is structurally wrong for systems with finite resources.
• Positive feedback eventually weakens when the relevant limiting input is exhausted or saturated.
• Historical growth can look exponential because many S-curves overlap or stack across successive technologies and resources.
• A useful growth model should include the causally relevant inputs and their rate of exhaustion rather than only extrapolating recent acceleration.

Methods / Formalism

• The post is qualitative rather than formal, but the modeling contrast is between x(t)=x_0 e^{rt} and bounded S-shaped curves such as logistic growth.
• The reusable wiki formal anchor is the logistic growth model:

$$\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),$$

where K is a carrying-capacity or saturation parameter.

The linked annex Logistic Sigmoid and Saturating Growth keeps the reusable logistic equation and local-exponential comparison.

Evidence / Experiments

• The post uses broad examples such as technology booms, market efficiency, cultural diffusion, population, energy extraction, and GDP.
• It does not provide empirical curve fitting; its value is as a caution about extrapolating exponentials without modeling constraints.

Connections

• Core source for S-Curves and Saturating Growth.
• Complements Sigmoid Functions by connecting the mathematical S-shape to resource-limited growth reasoning.
• Logistic Sigmoid and Saturating Growth stores the formal anchor for the S-curve model.
• Relevant to Deep Learning Fundamentals when model-capability or compute-scaling narratives assume indefinite exponential improvement.
• Fits Probability and Statistics as a warning about model misspecification and extrapolation outside the observed regime.

Open Questions

• Which limiting resources matter most for AI capability growth: compute, data, energy, algorithmic ideas, deployment friction, or evaluation saturation?
• How should the wiki distinguish descriptive scaling laws from causal growth models?
• When do stacked S-curves approximate an exponential well enough for short-run forecasting?

Citation

LessWrong contributor. (2026). All Exponentials are Eventually S-Curves. LessWrong.