Minimum-Intervention Zero-Mode Removal via Inverse Matrix Rigidity

Zero modes play a central role in mechanical metamaterials, topological mechanics, and constraint networks. This preprint develops a complementary minimum-intervention interpretation based on inverse matrix rigidity. For a square singular matrix CCC, its nullity is interpreted as the exact minimum number of independently modifiable matrix entries required to restore full rank in the unconstrained entry-level model. Building on the author’s prior inverse-rigidity theorem, the paper introduces the entry-support admissibility gap V(C,Ω)V(C,\Omega)V(C,Ω), a three-regime classification of matrix–entry-support pairs, an exact matching–minor characterization of restricted rank restoration, physical-operation rigidity, a zero-mode overlap identity, and an exact unstructured zero-mode reappearance radius. It also establishes connections with Maxwell–Calladine counting and chiral Hamiltonians and develops a grounding–connectivity criterion for spring networks. The theory is illustrated through a path-graph spring network and the fully dimerized open Su–Schrieffer–Heeger chain. The accompanying Python GUI, verify_zero_mode_removal_gui.py, reproduces the finite-dimensional examples, verifies the matching–minor criterion and the reappearance radius, and permits predefined, manually entered, and randomly generated matrices of different sizes. Supplementary files include the graphical verification program, its README, dependency specification, and software license. The underlying inverse-rigidity theorem and the previously published general-purpose calculator are cited as separate prior contributions.