Determine the optimal decomposition strategy and depth for the Waring macro regime in A_1

Determine the optimal recursive decomposition, at each stage, for representing m^k using the macro set M = {m^k : m ≥ 1} in A_1 that minimizes definitional depth, and characterize the minimal depth as a function of m.

Background

In the Waring macro regime for A_1, the authors analyze how to represent mk using smaller k-th powers. They provide an upper bound on depth via a halving strategy but do not know the optimal decomposition at each stage.

Identifying the optimal strategy would clarify the relation between depth and m in this regime and sharpen the connection between the theoretical model and empirical observations from MathLib.

References

We don't know the optimal decomposition at each stage, but can bound the depth: a halving strategy (expressing mk using k-th powers of integers \approx m/2, then recursing) gives depth O(log m) and a linear relationship in column 1; slower reductions give greater depth and a sublinear relationship, down to logarithmic if depth \sim m.

— Compression is all you need: Modeling Mathematics  (2603.20396 - Aksenov et al., 20 Mar 2026) in Subsubsection “A_n, polynomial-density macros / Waring” within Section 3 (Discriminating between regimes)