Extend Meir’s strong composition theorem to standard composition

Establish an analogue of Meir’s strong composition theorem for the standard block composition f ◇ g of Boolean functions, rather than for the strong composition variant.

Background

Meir’s strong composition theorem provides a significant partial result toward KRW-type lower bounds but for a stronger notion of composition. Bridging the gap between strong and standard composition is essential for applying these techniques to the usual block composition model.

The authors note that despite this progress, an analogous theorem for standard composition is not known.

References

Currently, the closest answer to Conjecture \ref{conj:1.2} is Meir's strong composition theorem , but we don't know how to prove it for the case of standard composition.

— A nearly-$4\log n$ depth lower bound for formulas with restriction on top  (2404.15613 - Wu, 2024) in Section 1, Introduction

However, analogous composition theorems for randomized query complexity and approximate degree remain open, even up to polylogarithmic factors .

— A General Composition Theorem for Approximate Degree  (2609.30139 - Pednekar et al., 24 Sep 2026) in Section 1, Introduction