A Mathematical Model of Multiplicative Knowledge Growth

This paper develops a mathematical framework for human knowledge acquisition in which learning is multiplicative rather than additive. The model introduces a knowledge-dependent insight probability — formalising the expert advantage — and derives closed-form solutions for three growth regimes (accelerating, plateau, linear) governed by the sign of a critical parameter λ = rαE[I] − δ. Extensions include a Cox–Ingersoll–Ross stochastic differential equation capturing individual variance, an optimal control analysis showing front-loaded schedules achieve 55–96% higher cumulative knowledge under equal effort, and an information-theoretic characterisation linking the knowledge-leverage coefficient α to the mutual information between new and existing knowledge. The framework yields falsifiable predictions and quantitative guidance for educational design.