Mean exceedance for eight equal exceptional Bernoulli weights

A candidate proof is given for a normalized Bernoulli sum with eight equal weights and any number of other equal weights, with either weight order. For independent Y ~ Bin(8,p) and X ~ Bin(n,p), n >= 1, a,b >= 0 and 8a+nb = 1, the strict mean-exceedance probability P[aY+bX > p] is at least p for 0 < p < 1/3; the non-strict event includes 0 <= p <= 1/3. The proof reduces real weights to rational capacity endpoints, reflects three pairs of conditional binomial tails and retains the middle three tails. An additional coefficient inequality handles the range in which the upper Gaussian shifts reverse order. Explicit analytic bounds cover ordinary multiplicity at least 100. Complete exact calculations cover the remaining 480,600 endpoints and 1,584 binomial faces. Direct small-case treatment avoids earlier candidate family theorems as mathematical inputs. The result remains provisional and concerns this fixed eight-member weight class; no arbitrary-multiplicity theorem or uniform positive surplus for original weights is claimed.