Mixed Density-Pressure Compactness Bounds for Static Barotropic Fluid Spheres: A Sharp Exponent Frontier and a Sound-Speed-Independent Pressure Endpoint

We determine the exact universal exponent region within a two-exponent family of pointwise density-pressure compactness inequalities for regular, static, spherically symmetric isotropic perfect-fluid solutions of the Einstein equations. In geometrized units, suppose that the fluid is barotropic and satisfies 0 < v := dp/d(epsilon) <= v_max, where v_max > 0 is a finite uniform upper bound. Define x(r) := 2m(r)/r. For nonnegative exponents u, w, consider K_(u,w)(r) = [epsilon(r)/epsilon_c]^u [p(r)/p_c]^w / [1 - x(r)]. Writing U = u/v_max, W = w, we identify the exact universal exponent region within this mixed power-law family: A = {(U,W) in [0,infinity)^2 : (1+q)(1+3q)[U + W/q] >= 4 for all q > 0}. Every pair in A makes K_(u,w) strictly decreasing away from the regular center, whereas every pair outside A is excluded by suitable regular near-center solutions that violate both the corresponding monotonicity statement and compactness inequality. Thus A is the exact universal exponent region within the stated mixed power-law comparison family. The lower Pareto frontier of A is parametrized by 0 <= xi <= 1/sqrt(3), with U_xi = 4(1 - 3xi^2) / [(1+xi)^2(1+3xi)^2], W_xi = 8xi^2(3xi+2) / [(1+xi)^2(1+3xi)^2]. Thus, for u_xi = v_max U_xi, w_xi = W_xi, one has 2m(r)/r < 1 - [epsilon(r)/epsilon_c]^(u_xi) [p(r)/p_c]^(w_xi). The density endpoint xi = 0 reproduces the previously derived density-compactness inequality 2m(r)/r < 1 - [epsilon(r)/epsilon_c]^(4v_max). The opposite endpoint gives the sound-speed-independent pressure bound 2m(r)/r < 1 - [p(r)/p_c]^(4 - 2sqrt(3)). At the point xi = 1/3, where the rescaled density exponent U = u/v_max equals the pressure exponent W = w, U_(1/3) = W_(1/3) = 3/8, one obtains 2m(r)/r < 1 - [epsilon(r)/epsilon_c]^(3v_max/8) [p(r)/p_c]^(3/8). Under the causal sound-speed condition 0 < dp/d(epsilon) <= 1, this yields 2m(r)/r < 1 - [epsilon(r)p(r)/(epsilon_c p_c)]^(3/8). Finally, the pure-pressure endpoint requires no prescribed barotropic equation of state or sound-speed bound: under the broader regular static isotropic perfect-fluid hypotheses of the corresponding theorem, it remains valid assuming nonincreasing average density. The exponent 4 - 2sqrt(3) is minimal within the corresponding pure-pressure power-law comparison family under that broader hypothesis class. These are interior matter-stratification bounds and are not optimal global compactness bounds. Version 2 note: This revision clarifies the assumptions and scope of the mixed density-pressure theorem and the broader pure-pressure endpoint, explicitly states the positive-density, positive-pressure, no-trapping, and regular TOV hypotheses where relevant, defines q = p/epsilon in the abstract, and improves several cross-references and logical formulations. The admissible exponent region, Pareto frontier, endpoint bounds, sharpness results, and main mathematical conclusions are unchanged.