A sharp surplus bound for weighted Bernoulli sums with at most seven terms

For positive real weights summing to one, at most seven terms, common Bernoulli bias 0 < p <= 1/3, and every weight below p, a candidate exact computer-assisted argument gives non-strict mean-exceedance probability at least p + 3/256. The constant is a sharp uniform infimum for each fixed support from 4 through 7. An explicit restricted Manickam-Miklos-Singhi consequence follows.