Scale-invariant exact anisotropic barotropes in bulk-boundary Lagrangian decomposition: from stiff matter to dark energy stars

We investigate static, spherically symmetric anisotropic matter configurations in the recently proposed Boehmer–Jensko bulk-boundary gravity (referred to as [Formula: see text]) theory, focusing on the quadratic model [Formula: see text] where [Formula: see text] refers to the bulk and [Formula: see text] the boundary. By imposing a linear barotropic equation of state [Formula: see text] together with a power-law ansatz for the metric potentials, we reduce the field equations to an algebraic constraint on the metric exponents and obtain a family of exact scale-invariant solutions parameterized by the stiffness parameter [Formula: see text]. Four representative cases are studied in detail: [Formula: see text] (incoherent radiation), [Formula: see text] (stiff matter), [Formula: see text] (dark-energy-like matter), and [Formula: see text] (exotic negative-pressure matter). For each branch we derive exact expressions for the metric functions, energy density, radial pressure, and tangential pressure, and analyze anisotropy, energy conditions, and parameter dependence. The radiation and stiff branches provide the most physically reasonable members of the family, while the [Formula: see text] branch furnishes an exact dark-energy-like configuration in the modified-gravity setting. At the same time, all four solutions share a scale-invariant structure, leading generically to a central singularity and preventing the existence of a finite radius at which [Formula: see text] vanishes. The solutions are therefore best interpreted as exact benchmark configurations or matching layers rather than complete stellar interiors. These results provide the first explicit scale-invariant anisotropic barotropes in the Boehmer–Jensko framework and clarify both the possibilities and limitations of the power-law ansatz.