Eternal AI: Recursive Self-Improvement Curve

Recursive Self-Improvement

Feedback strength k 0.35

Growth exponent p 1.00

Compute ceiling R 300

Simulation speed 1.0×

Cognition State

Regime: —

Capability C(t)

10.0

Growth rate dC/dt

0.00

Ceiling reached

3.3%

Elapsed steps t

How it works

An "eternal" AI in this model is one whose own output — improved algorithms, better data curation, more efficient architectures — feeds back into its own capability, in an unbounded loop. This is the toy dynamical-systems model behind the "intelligence explosion" idea from AI-safety literature (Good 1965, Bostrom 2014):

dC/dt = k · C^p · (1 − C/R)

C  = capability (nodes in the cognition network)
k  = feedback strength (how much each unit of capability
     accelerates the next improvement)
p  = growth exponent: p<1 sublinear, p=1 exponential,
     p>1 superlinear ("hyperbolic") — reaches a finite-time
     singularity if uncapped
R  = compute ceiling — a resource/hardware bound that forces
     the logistic-style (1 − C/R) saturation term

k (feedback strength) — the leverage each improvement gives to the next one. Higher k pulls the curve toward the ceiling faster.
p (growth exponent) — the shape of the takeoff. p≈1 is ordinary exponential growth; p>1 means capability itself increases its own growth rate ("recursive" self-improvement, blows up without a resource cap); p<1 self-limits even before the ceiling.
R (compute ceiling) — models the hard constraint every real system has (hardware, energy, data). No feedback loop grows forever; the (1 − C/R) term is what turns an explosion into an S-curve.
The 3D structure is a branching cognition graph — every third capability unit spawns a new branch node, so the network visibly grows outward as C(t) rises, and stalls visually the moment growth saturates near R.

Try p = 1.6 with a high k to see an explosive early takeoff that still saturates at R; try p = 0.7 to see a system that plateaus long before reaching its ceiling — a "slow takeoff" in Bostrom's terminology.

Cognition nodes Branch links

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