I. J. Good's 1965 "intelligence explosion" idea: a system smart enough to improve its own intelligence enters a feedback loop, each gain making the next gain easier. This model integrates a capability index C forward in time under that feedback, capped by a safety/alignment ceiling.
dC/dt = R · C^p · (1 − C / C_max)
p ≤ 1 sub-linear → growth flattens out on its own
p ≈ 1 exponential → steady doubling
p > 1 hyperbolic → C can spike to the ceiling in finite time
- Starting compute — the seed system's initial capability index C₀.
- Self-improvement rate R — how strongly each unit of capability accelerates the next.
- Recursive exponent p — the shape of the feedback: at p>1 the growth is "hyperbolic" (a real mathematical singularity at finite time, before the ceiling clips it).
- Safety / alignment ceiling — a capability cap representing external oversight and control; lower it to see an S-curve instead of a runaway spike.
Real-world relevance: this hyperbolic-vs-ceiling framing is exactly the shape of the "fast takeoff vs slow takeoff" debate in AI-safety research (Bostrom, Yudkowsky, and others) — nobody knows the true value of p, which is precisely why the debate matters.