Residual Freedom in Cold Dense Matter Order-Parameter Closure and Compact-Star Tests

We work in the cold regime (T ≃ 0) relevant for mature neutron stars, where the microphysics reduces to a barotropic equation of state P(ε). Starting from the energy density ε(n), we recall the T = 0 thermodynamic identities that generate P(n) and µ(n), and we show how a coherence variable χ(n) can modulate the EOS without breaking thermodynamic consistency. Following the structural-closure results and the residual-freedom classification, the present work explores one legitimate degree of freedom: how an admissible vacuum residue can be encoded, at T ≃ 0, into an effective order parameter χ(n) that controls both the cold EOS and (optionally) the macroscopic gravitational coupling, without modifying local microphysics. Finally, we propose a phenomenological TOV-like hydrostatic-equilibrium system with an effective gravitational coupling G(n) = G0 g(χ(n)), while explicitly isolating the terms that would depend on a microscopic completion (scalar–tensor, etc.).