Exploration 04 · a model with stated assumptions

Improve the generator

Not "can progress become exponential?" but "can the system generating progress improve the generator of progress — and what stops it?"

Generated landscape

01Intelligence
02Discovery
03Technology
04Resources
05Compute, energy, physical capacity
06More capable intelligence

The last arrow is the one that matters: the loop does not only produce more — it improves its own rate of producing. That is where the mathematics stops being exponential.

Recursion β0.50 How strongly progress improves the process that makes progress. 0 is ordinary exponential growth. Bottleneck b0% Share of every cycle that never speeds up — experiments, fabrication, validation. Amdahl's law. Physical ceiling Knone Finite energy, materials or resources. Capability cannot exceed it.

Finite-time singularity

With nothing in the way, the equations escape to infinity at t ≈ 1.39. Not a prediction: a sign the model has left the world where its assumptions hold.

Fig. The generator

dC/dt = r_eff·C·(1 − C/K) · dr/dt = β·r·C·(1 − C/K) · r_eff = 1 / ((1 − b)/r + b/r₀). Capability C grows at a rate r that itself improves with C. Integrated numerically (adaptive RK4, log-scale state). A toy with stated assumptions — not a model of any real institution, field or civilization.

Three regimes

Runaway: With feedback and nothing in the way, capability reaches infinity at a finite time — for β = 0.5, r₀ = 1, at t = ln(r₀/β)/(r₀ − β) ≈ 1.386. The singularity belongs to the equations; real systems leave the model's assumptions first.
Bottlenecked: If a share b of every cycle cannot be accelerated, the effective rate can never exceed r₀/b however good the generator gets. Growth is at most exponential. This is Amdahl's law, and it is exactly what EXP-001 is registered to measure for one capability class.
Saturating: A finite supply of energy, materials or resources caps capability itself: growth bends into a logistic curve.

Assumptions

Capability C and generator speed r are single numbers; real progress is not. Feedback is proportional to capability; it could be weaker, delayed or negative. Ceilings are hard; real ones move. The model exists to show which assumptions produce which shapes — not to say which assumptions are true.

Integrated with adaptive fourth-order Runge–Kutta in log C. Unit tests compare the numerical blow-up time with the closed form and check each regime: tests/math.test.ts.