The Cascade Depth Theorem

We formalize the tool-building cascade: a recursive decision process in which an agent, instead of solving problem instances directly, invests in tools that solve problem classes, thereby revealing higher-level problems amenable to the same strategy. The central result is the Cascade Depth Theorem: under a multiplicative leverage model with per-level multipliers \(m_k > 1\), the total output at cascade depth \(k\) is

\[O(k) = \prod_{i=1}^{k} m_i \cdot O_0,\]

and the optimal depth \(k^*\) satisfies a Bellman equation with a threshold policy. We prove:

1. Threshold structure. There exists \(k^*(C, T, \lambda, \mathbf{m})\) such that the optimal policy invests in level \(k+1\) if and only if the marginal-multiplier-to-cost ratio exceeds a horizon-dependent threshold.

2. Phase transition. There exists a critical cost \(C^\) below which \(k^ \geq 3\) and the invest-always policy dominates, producing super-linear output growth in AI capability.

3. Monotonicity. \(k^*\) is non-decreasing in horizon \(T\), non-decreasing in problem arrival rate \(\lambda\), and non-decreasing in \(1/C\) (inverse AI cost).

4. Self-improvement equivalence. The optimal cascade depth \(k^\) equals the self-improvement ceiling \(K^(N)\) under appropriate identification of compute \(N\) with inverse AI cost \(1/C\).

The model predicts a bimodal distribution of AI productivity gains across firms and individuals, determined by cascade depth rather than task-level speed. A 4-level cascade with per-level multipliers of \(m = 3\) yields \(3^4 = 81\times\) total improvement; a 6-level cascade with heterogeneous multipliers easily exceeds \(300\times\). This multiplicative structure explains the observed two-orders-of-magnitude variance in reported AI productivity gains without requiring any assumption about task-level speed differences.