Optimal base of logarithmic execution depth

Determine the optimal logarithmic base for the number of layers required by polynomial executors to answer a single-depth query from a balanced context, narrowing the current lower bound log_3(L−1) and upper bound floor(log_2 L)+2.

Background

The paper proves logarithmic upper and lower bounds on execution depth for single-depth queries. For balanced contexts, polynomial executors require at least log_3(L−1) layers, while a repeated-squaring construction achieves at most floor(log_2 L)+2 layers. These bounds differ primarily in the base of the logarithm, which the paper explicitly identifies as unresolved.

References

For single-depth queries on the balanced context, the optimal depth therefore lies between $\log_3(L-1)$ and $\log_2L+2$, which agree up to the factor $\log_23\approx1.58$ and an additive constant (a further $\log_3k_L$ for the unnormalized context); only the base of the logarithm is open.

— Provable Transformer Context Compression for In-Context Reinforcement Learning  (2610.20004 - Zhao et al., 14 Oct 2026) in Appendix E, paragraph following Theorem E.1