From a Sharp Thin-Shell Obstruction to a Smooth Positive-Density Initial-Data Embedding of a Virialized Halo in Lambda-FLRW Cosmology
Within classical general relativity, we compare a sharp timelike junction and a smooth finite-width spacelike initial-data embedding of a positive-excess virialized halo in a homogeneous Lambda-FLRW environment. On the ordinary branch, timelike Israel matching at the virial boundary produces a negative surface layer: a distributional representation of the environmental compensation omitted by the sharp construction. We replace this zero-width source by a finite-width, background-relative underdensity whose local rest-frame energy density remains positive. The resulting conformally flat, constant-mean-curvature ADM initial slice satisfies the Hamiltonian and momentum constraints, with the residuals converging under resolution refinement. At finite radius, its geometric and matter variables return to the local-effective FLRW data within the declared tolerances: the constructed slice contains no residual distributional layer and satisfies the reconstructed weak, null, and dominant energy conditions. For radii beyond the numerical endpoint, the exterior is defined by the exact analytic local-effective FLRW solution. The GCT framework supplies the global motivation clock interpretation, while the present calculations are local-effective classical-GR calculations with the bound-sector constants c0 and G0 held fixed: no radial GCT lapse or constant interpolation is constructed.