Twenty Nine Comparisons

A comparison sort is a decision tree: each internal node is one comparison with two outcomes, and each leaf is one possible output permutation. For $n = 12$ distinct elements, what is the smallest number of comparisons that some input is guaranteed to force, for any comparison sort?

Useful values: $12! = 479001600$, $2^{28} = 268435456$, $2^{29} = 536870912$, $\log_2 12 \approx 3.585$.