Bernoulli mean exceedance for two-valued weights with a small class

Candidate mean-exceedance results are assembled for normalized two-valued Bernoulli sums whose smaller positive weight class has at most seven coordinates, with either weight order. For a common bias 0<p<1/3, the strict mean-exceedance probability is at least p; the non-strict form includes 0<=p<=1/3. The seven-exceptional case groups the middle two conditional tails by monotonicity and reflects the other three pairs. Gaussian, binomial-tail and small-ratio estimates, with exact polynomial bounds, cover ordinary multiplicity at least 128. A plateau-aware capacity reduction and two complete exact calculations close 727,957 residual endpoints for 7<=n<=127. The earlier one- through six-exceptional cases are explicit inherited candidate inputs. The general arbitrary-multiplicity problem remains outside the result, and no strict-event claim at p=1/3 or uniform positive surplus is made.