Alternating binomial sums weighted by logarithms — expressions of the form involving (-1)^k times binomial coefficients times logarithmic terms — arise in the analysis of collecting processes. An inequality bounding such sums was conjectured based on numerical evidence and the structure of minimum-order statistics.
The proof exploits a probabilistic identity (arXiv:2603.11129). The alternating binomial logarithmic sum equals the variance of the logarithm of the maximum of n independent exponential random variables. Since variance is nonnegative, the inequality follows immediately.
The method is striking because the algebraic object — a complicated alternating sum — has no obvious sign. The terms oscillate, and direct estimation requires careful cancellation tracking. But the probabilistic representation converts the sum into a variance, which is nonnegative by definition. The inequality is not proved by bounding; it is proved by reinterpreting.
This is a specific instance of a broader technique: translating combinatorial identities into probabilistic statements where structural properties (nonnegativity, monotonicity, convexity) are automatic. The alternating sum looked like it needed estimation. It needed recognition.