  A combinatorial object is said to be quasirandom if it exhibits certain
properties that are typically seen in a truly random object of the same kind.
It is known that a permutation is quasirandom if and only if the pattern
density of each of the twenty-four 4-point permutations is close to 1/24, which
is its expected value in a random permutation. In other words, the set of all
twenty-four 4-point permutations is quasirandom-forcing. Moreover, it is known
that there exist sets of eight 4-point permutations that are also
quasirandom-forcing. Breaking the barrier of linear dependency of perturbation
gradients, we show that every quasirandom-forcing set of 4-point permutations
must have cardinality at least five.
