Unified inverse rigidity of matrices: minimal rank restoration, order of vanishing of the determinant, restricted and weighted support, structured restoration and prescribed solution

This preprint develops a unified framework for inverse matrix rigidity: the minimum number of matrix entries that must be modified to restore a prescribed algebraic property. For a singular square matrix, the unrestricted restoration cost equals its nullity. Starting from this identity, the manuscript studies the order of vanishing of the determinant, classifies all minimum-cardinality restoration supports, derives the complete target-rank profile, and proves that prescribing a nonzero determinant does not increase the minimum cost, including under restricted and positively weighted support constraints. The paper also develops symmetric diagonal restoration with prescribed inertia, positive-semidefinite regularization, minimal grounding of graph Laplacians, spectral-robustness bounds, prescribed-solution constructions, and an integer theory in which the minimum-cost prescribed determinant is governed by divisibility properties of maximal minors. Particular attention is paid to the distinction between entrywise algebraic intervention and physically realizable intervention. The manuscript states explicitly which results are classical, which are reformulations, which are operational interpretations, and which algorithmic, numerical, and arithmetic problems remain unresolved. This manuscript extends and unifies the author's earlier preprints on inverse matrix rigidity and minimum-intervention zero-mode removal. Previously published results are identified and cited in the text. This record is a research preprint and has not undergone journal peer review.