Quantitative Density Stratification and Nonlinear Compactness Bounds for Static Barotropic Fluid Spheres: Finite-Compactness Defects Along a Sharp Density-Pressure Exponent Frontier

We derive a quantitative relation between local compactness and density stratification for regular, static, spherically symmetric isotropic perfect-fluid solutions of the Einstein equations, and use it to quantify the finite-compactness defect of a previously identified sharp density-pressure comparison family. In geometrized units, suppose that throughout the positive-pressure fluid region 0 < r < R, epsilon(r) > 0, p(r) > 0, 0 < 2m(r)/r < 1, and that the fluid obeys a C^1 barotropic equation of state satisfying 0 < v(epsilon) := dp/d(epsilon) <= v_max, where v_max > 0 is a finite uniform upper bound. Define x(r) := 2m(r)/r, s(r) := 4pi r^3 epsilon(r)/m(r) = 3 epsilon(r)/bar_epsilon(r), where bar_epsilon(r) = 3m(r)/(4pi r^3) is the average enclosed density. With lambda_vmax = min{3/(10v_max), 3}, we prove 3 - s(r) > lambda_vmax x(r), for 0 < r < R. Equivalently, 1 - epsilon(r)/bar_epsilon(r) > [lambda_vmax/3] [2m(r)/r]. For v_max >= 1/10, the coefficient lambda_* = 3/(10v_max) is the largest universal coefficient within the linear comparison family 3 - s > lambda x for the class 0 < v <= v_max. In particular, within the causal class 0 < v <= 1, the coefficient 1/10 in 1 - epsilon/bar_epsilon > x/10 is sharp among universal bounds linear in x. We then consider the mixed comparison family K_(u,w)(r) = [epsilon(r)/epsilon_c]^u [p(r)/p_c]^w / [1 - x(r)]. Writing q := p/epsilon, for every nonnegative exponent pair satisfying (1+q)(1+3q)[u/v_max + w/q] >= 4 for all q > 0, we obtain an exact decomposition of the logarithmic saturation defect D_(u,w) := -ln K_(u,w) into three nonnegative contributions associated with density stratification, exponent mismatch, and response slack. Retaining only the universally positive stratification contribution gives K_(u,w)(r) < exp[-G_vmax(x(r))] < 1, where G_vmax(x) = [lambda_vmax/6][-x - ln(1-x)]. Thus the sharp exponent frontier remains sharp as an exponent-space classification, while the coupled Tolman-Oppenheimer-Volkoff evolution enforces a universal compactness-dependent separation from saturation of the corresponding comparison inequalities at every nonzero radius. For an interior Pareto-frontier parameter 0 < xi < 1/sqrt(3), cancellation of the linear near-center defect requires q_c = xi, v_c = v_max. Under this tuning, liminf_(r->0) D_xi(r)/x(r)^2 >= 3(1+xi)^2/(40v_max), and this coefficient is attained by a constant-slope affine equation of state in a neighborhood of the center. The results distinguish sharpness of the exponent region from finite-compactness saturability of its inequalities. No optimality claim is made for the global nonlinear gap function G_vmax.