Mean exceedance for Bernoulli sums with three equal exceptional weights

This preprint presents a provisional candidate theorem for independent Bernoulli(p) variables with nonnegative weights consisting of three equal exceptional weights and n equal ordinary weights, where n >= 1 and 3a + nb = 1. It gives a written proof of the mean-exceedance bound P[a(B0 + B1 + B2) + b sum(Bi, i=3,...,n+2) >= p] >= p for 0 <= p <= 1/3; the event may be replaced by strict exceedance for 0 < p < 1/3. The exceptional weights may be larger or smaller than the ordinary weight. The argument combines two complementary reflected pairs among four conditional binomial tails with a one-third middle-tail estimate, a capacity reduction, and 103 exact finite inequalities. Together with the earlier one- and two-exceptional-weight results, it covers vectors whose positive entries take one value, or exactly two values with at least one positive-value class occurring at most three times. The supporting archive includes the article, source, finite checks, and verification records. The finite checks do not establish the infinite parameter range, and the general weight conjecture remains open.