A Pointwise Density-Compactness Bound for Static Barotropic Fluid Spheres: Sound-Speed Control and Sharpness within a Power-Law Comparison Family

We derive a pointwise relation between local compactness and density contrast 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 < dp/d(epsilon) <= v_max. Then epsilon(r)[1 - 2m(r)/r]^(-1/(4v_max)) is strictly decreasing away from the regular center. Consequently, 2m(r)/r < 1 - [epsilon(r)/epsilon_c]^(4v_max), for 0 < r < R. Under the causal barotropic sound-speed condition 0 < dp/d(epsilon) <= 1, this gives 2m(r)/r < 1 - [epsilon(r)/epsilon_c]^4. If a finite stellar surface has nonzero limiting density epsilon_s, the same comparison yields 2M/R <= 1 - [epsilon_s/epsilon_c]^(4v_max). We further show that 1/(4v_max) is the largest exponent whose monotonicity can be guaranteed from the stated assumptions within the comparison family epsilon(r)[1 - 2m(r)/r]^(-a). For every larger exponent, regular near-center solutions of self-bound linear equations of state can be chosen for which the corresponding comparison function increases. The result is an interior density-contrast estimate and is not an optimal global compactness bound. Version 2 note: This revision clarifies the stated assumptions in the abstract and surface corollary, and makes the treatment of v_max consistent throughout by identifying it as a finite uniform upper bound on dp/d(epsilon), with equality to the supremum when it is chosen as the least uniform upper bound for the solution. It also improves the wording of the surface-limit argument and the physical-units discussion for clarity. The main theorem, proof, causal specialization, surface bound, and sharpness result are unchanged.