Reflected Newton Transforms and Concavity of Peak-Polynomial Coefficients

We study a polynomial transform whose rising-binomial coefficients are formed from reflected backward differences. In Newton coordinates its matrix is a folded minimum kernel. This identifies its kernel and its entire image on nonnegative data: the cone of symmetric concave sequences, with an explicit integral inverse. The same representation gives strong coefficientwise log-concavity for arbitrary multivariate nonnegative Newton data, both in the rising basis and in forward binomial bases. We also characterize the zero boundary needed for an adjoining-root recurrence and prove that the two-step displacement in the positivity criterion is sharp. As applications, peak-polynomial coefficients in Bencs's rising basis are positive integers, symmetric and ordinarily concave, proving his log-concavity conjecture. We determine their exact maximum plateau and all log-concavity equality cases. The forward-basis result answers a question of Diaz-Lopez, Harris, Insko and Omar at every integer center at or beyond the last peak.