Mean exceedance for Bernoulli sums with four equal exceptional weights

Let n be a positive integer and let a,b be nonnegative with 4a+nb=1. For independent Bernoulli(p) variables, this preprint gives a candidate proof that the probability of a weighted sum with four weights equal to a and n weights equal to b being at least p is at least p for 0 <= p <= 1/3. The strict exceedance event is also allowed for 0 < p < 1/3. Both orderings of the two weights are covered. The capacity reduction permits fractional raw thresholds and an inactive zero-exception row. Five conditional binomial tails compensate for one another: complementary reflections reduce the target to a joint first-and-central-tail inequality, and an adjacent reflection proves it at every ratio at least four. Classical binomial estimates cover the remaining ratios for n >= 148; an explicitly complete exact calculation settles 26,322 residual capacity cases. For unequal positive weights with n <= 3, the theorem uses the earlier one-, two- and three-exceptional candidates after exchanging classes; for n >= 4, it uses classical inputs and local analytic and exact finite arguments. Together with those earlier candidates, it gives the routine consequence for two positive weight values when one occurs at most four times. No priority claim or general two-valued theorem is made. The principal results remain provisional candidates; complete load-bearing verification is still missing before they can be treated as established.