Part 9: A Program That Generalizes All Inequalities and Incorporates Them into Functions, While Simultaneously Optimizing Them via the Function C

We develop conditional nonlinear refinements of Sobolev and Gagliardo–Nirenberg interpolation inequalities on bounded Lipschitz domains. Positive norm-gap terms are controlled either by quantita tive growth and geometric convexity of the nonlinearity or by uniform comparability of the underlying norms. In the critical regime we study finite-family logarithmic factors defined through Sobolev-norm ratios, especially the minimum-ratio functional Amin and a hybrid weighted-mean family Aλ,p. We establish ordering and parameter dependence results and a uniform error bound for approximating Amin by negative-order power means. Fourier examples rule out logarithmic exponents below 1/2 for the unrestricted two-dimensional critical Sobolev class. All estimates are stated with explicit hypothe ses and follow from classical interpolation and critical logarithmic embeddings; no new unrestricted Sobolev embedding is claimed.