arXiv:2609.00137 Defines Recursive Reproduction Number R_AI for AI Capability Loops

The model treats AI R&D as a discrete cycle whose output feeds the next cycle. Baseline productivity sets the initial slope while research difficulty grows with each increment. R_AI is the ratio of feedback gain to the difficulty gradient. Values above one produce compounding gains; values below one produce decay regardless of absolute model scale.

Empirical proxies include measured cycle time between model releases, the fraction of successor-model gains traceable to prior-model outputs, and the observed slowdown in benchmark improvement per additional FLOP. The paper shows shared code and weights across labs can push ecosystem-level R_AI above one even when every single organization remains below threshold.

Cycle duration acts as the binding constraint on amplification speed. Shorter cycles allow more iterations before difficulty saturates the loop. The framework therefore separates visible acceleration from true recursion and supplies observables that distinguish the two without requiring ex-ante knowledge of capability ceilings.

Operational consequence is that monitoring must target feedback strength and propagation efficiency rather than headline benchmark scores alone.