Mean exceedance for Bernoulli sums with five equal exceptional weights

Let n ≥ 1, let a, b ≥ 0 satisfy 5a + nb = 1, and let U ∼ Bin(5,p) and V ∼ Bin(n,p) be independent. This preprint gives a candidate proof that P[aU + bV ≥ p] ≥ p for 0 ≤ p ≤ 1/3, with strict exceedance allowed for 0 < p < 1/3. Both weight orderings are included. A capacity reduction leaves six conditional binomial tails. Adjacent reflection fails at genuine capacity points; a weighted comparison of two tails, proved using rigorous normal approximation and rational Gaussian bounds, replaces it. A conditional-mixture estimate treats bounded weight ratios. The arguments cover every n ≥ 200 and leave an explicitly complete finite calculation of 605,963 endpoints. The result extends earlier small-multiplicity candidate results and does not resolve the general weighted conjecture. The principal result remains provisional: complete verification of the load-bearing analytic arguments and inherited candidate inputs, and appropriate critical examination, remain necessary.