Exact Fixed-Density Finite-Volume Realizations of Maximally Flat Newtonian Source Fields

This work proves that the maximally flat positive-mass Newtonian source configurations obtained in the thin-ring limit admit exact nearby finite-volume realizations using homogeneous source bodies of one prescribed positive material density. The continuation is established for arbitrary N from the nondegeneracy of the thin-source moment map, with local uniqueness, positivity, and source separation preserved. A corresponding enhanced-order result is obtained for prescribed-gradient configurations when one additional transversal physical control is available. The paper also gives explicit finite-thickness asymptotics and rigorous computer-assisted certification results: an N=1 square-annulus realization is certified for 0 < delta <= 0.39, an N=2 realization for the complete interval 0 < delta <= 0.17, and an N=3 finite-volume configuration is validated at a practical material-density point. The rigorous interval results are distinguished from broader numerical continuation evidence.