Part 12: A Program That Generalizes All Inequalities and Incorporates Them into Functions, While Simultaneously Optimizing Them via the Function C

We develop a function-based framework for quantitative remainders in inequalities arising from weak majorization and informationtheory. Under a positive sum-gap assumption, the majorization comparison controls three cyclic remainders formed from absolute values offunction evaluations. Applied to the entropy coordinates of two probability vectors, the method gives conditional linear estimates, equalitystable quadratic remainders, and a curvature correction for Gibbs’inequality. We also investigate convex and concave transformations ofthe variational Kullback–Leibler inequality, Pinsker’s inequality, andthe conditional-expectation inequality of Tao in its sharp form due toAhlswede. Exact double-integral identities explain the transformed estimates and yield explicit strong-convexity or strong-concavity terms,together with equality and sharpness statements. The overarchingprogram is formulated as a template: the existence of a nontrivialpositive correction for a given inequality requires additional hypotheses and is not a universal consequence of its original ordering.Keywords: mathematical inequalities; functional remainders; weak majorization; convex and concave functions; Kullback–Leibler divergence; Gibbs’inequality; Pinsker’s inequality; conditional mutual information; sharp constants.2020 Mathematics Subject Classification: Primary 26D15, 94A17; Secondary 26D20, 60E15.