A Uniformly Sharp Lane–Emden Envelope for Response-Bounded Relativistic Fluid Spheres

This manuscript establishes a universal nonlinear density–compactness comparison for regular relativistic perfect-fluid spheres under a positive upper bound on the barotropic pressure response. Consider static, spherically symmetric, isotropic perfect-fluid solutions of the Einstein–Tolman–Oppenheimer–Volkoff equations. In geometrized units G = c = 1, assume positive pressure and energy density, 0 < 2m/r < 1, and a C¹ barotropic equation of state satisfying 0 < dp/dε ≤ V < ∞ throughout the positive-pressure interior. Define the normalized mass-to-radius variable y = m/(Vr) and the density-ratio variable s = 3ε/ε̄, where ε̄ denotes the average enclosed energy density. Let S(y) denote the first increasing-mass-to-radius branch of the regular Newtonian isothermal Lane–Emden trajectory, ending at its classical first maximum η* = 2.5175513244…. The principal theorem establishes the strict interior comparison 0 < y(r) < η*, s(r) < S(y(r)), at every positive interior radius, without assuming monotonicity of the local compactness m/r. The proof derives an exact normalized dynamical system directly from the original TOV equations. Its density-ratio evolution has the same structural form as the classical isothermal Lane–Emden system, but with a strictly larger coefficient multiplying normalized compactness. This ordering excludes ordinary first contact with the classical reference branch. A separate turning-point argument, using backward Taylor expansions and the square-root behavior of the reference graph, excludes escape through the endpoint where the classical graph develops a vertical tangent. The nonlinear comparison implies the response-dependent compactness ceiling 2m(r)/r < 2η*V = 5.0351026488… V, together with the strict density-stratification inequality 1 − ε(r)/ε̄(r) > 1 − S(y(r))/3 > [2m(r)/r]/(10V). The nonlinear envelope therefore provides a strict improvement over the corresponding linear density-stratification inequality. It also yields an inverse density-conditioned compactness restriction wherever the density ratio lies within the range of the first Lane–Emden branch. The manuscript establishes uniform optimality across the union of all positive response-bound classes. The coefficient 2η* is the least universal compactness coefficient, and S(y₀) is the least universal pointwise upper envelope at each prescribed normalized compactness y₀ ∈ (0, η*). These supremum statements are established through regular relativistic isothermal solutions with p = Vε, whose normalized trajectories converge to the Newtonian Lane–Emden solution as V approaches zero. For pointwise optimality, the construction produces admissible configurations at exactly the prescribed value y = y₀ before taking the limit. Separately, the coefficient 1/(10V) in the linear density-stratification inequality is shown to be optimal for every fixed V > 0 within the corresponding universal linear comparison family. This fixed-response sharpness is established through regular near-center solutions with affine equations of state and is distinct from the uniform optimality of the nonlinear envelope. The classical Lane–Emden trajectory, its turning-point constant, relativistic stellar state-space formulations, and reference-orbit comparison methods are established mathematical objects. Earlier dynamical-systems approaches, including the work of Nilsson and Uggla and subsequent developments by Heinzle, Röhr, and Uggla, provide relevant mathematical context. The contribution considered here is the specific universal nonlinear comparison for positive-response-bounded relativistic barotropes, together with its stated uniform and fixed-response optimality results. The manuscript introduces no modified gravitational dynamics, no new field equations, and no new Lane–Emden constant. It does not claim fixed-positive-V optimality of the nonlinear envelope, an equality-attaining relativistic configuration, a finite-surface stellar extremum, or a dynamical-stability criterion. The results concern regular positive-pressure interiors and their precise mathematical comparison with the first ascending branch of the classical Newtonian isothermal Lane–Emden trajectory.