SAT solvers are empirically known to perform poorly when reasoning about multiplication. Yet for over a decade we have lacked a theoretical explanation for this phenomenon. CDCL SAT solvers implicitly search for resolution proofs, and no lower bound on proof size has ruled out the existence of short proofs that solvers simply fail to find. We give the first lower bound of this kind by showing that general resolution proofs of the associativity of $n$-bit multiplication require size $2^{Ω((n/\log n)^{1/4})}$. This lower bound holds for a broad class of multiplier encodings based on partial product summation, including the standard array and Wallace-tree multipliers used to bit-blast multiplication in SMT solvers. This result resolves an open problem of Beame and Liew. The proof constructs a reduction from a perfect-matching principle on bounded-degree bipartite expander graphs to multiplier associativity. Itsykson, Slabodkin, and Sokolov proved that this principle is hard for resolution. The lower bound for multiplier associativity follows. The same reduction, when combined with Håstad's recent lower bound for the perfect-matching principle of the odd grid, yields an exponential lower bound for multiplier associativity in the stronger bounded-depth Frege proof systems.

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