The Forgotten Theorem
Date:
The Forgotten Theorem
This is a companion to the fictional story “The Divergence”, exploring the mathematical idea that inspired it.
In 2027, a mathematician named Arnaud Mehran published a little-known blog post titled “Nonlinear Systems and the Collapse of Shared Cognitive Space.” The post went unnoticed—until the rise of Mira-X in 2036 made its predictions disturbingly real.
Here is the essence of Mehran’s argument.
Modeling Human Productivity Under AI Amplification
Let:
- $x$ = baseline human capability (e.g., IQ, education, expertise)
- $P(t)$ = productivity at time $t$
- $\gamma$ = strength of AI’s effect
- $\alpha$ = how much human capability amplifies AI leverage
- $\beta > 1$ = nonlinearity or feedback strength (recursive productivity effects)
The productivity evolution is governed by:
\[\frac{dP}{dt} = \gamma \cdot x^\alpha \cdot P^\beta\]This describes a positive feedback loop: the more capable and productive someone is, the faster their productivity grows.
Finite-Time Blowup
Solving the differential equation:
\[\frac{dP}{P^\beta} = \gamma \cdot x^\alpha \cdot dt\]Integrating gives:
\[P(t) = \left[(1 - \beta)(\gamma \cdot x^\alpha \cdot t + C)\right]^{1 / (1 - \beta)}\]For $\beta > 1$, this solution blows up in finite time. That is:
\[P(t) \to \infty \text{ as } t \to t^* \text{ for some finite } t^*\]Where:
\[t^* = \frac{P(0)^{1 - \beta}}{\gamma \cdot x^\alpha \cdot (\beta - 1)}\]This means: small differences in capability can lead to exponentially large differences in outcomes in finite time, not just over decades or centuries.
Societal Implications
Let $P_i(t)$ be productivity for individual $i$, and define inequality:
\[\sigma_P(t) = \text{std deviation across } \{P_i(t)\}\]Then societal stability can be modeled as:
\[S(t) = \frac{1}{1 + \sigma_P(t)}\]If $\sigma_P(t) \to \infty$, then $S(t) \to 0$. Society destabilizes.
Mehran concluded:
“Societal divergence becomes unmanageable when cognition becomes recursive.”
Postscript
At the time, the proof felt theoretical—just another curve on another blog.
Now, it feels like prophecy.