The Correlation Shadow of Signed Zero-Relations: A Rank–Defect Theory for Planar General-Position Configurations

This research introduces a mathematical framework for studying the hidden structure of signed zero-relations in two-dimensional vector configurations. The framework constructs a correlation shadow from the complete family of signed vectors whose sum is zero, and then uses linear algebra, spectral methods, and Hamming geometry to investigate the relationship between the zero-relation space, the rank of the correlation shadow, structural defect, and the forced correlations among coordinates. The central goal is to move beyond simply counting zero-relations and instead study their internal structure, independence, and geometry. The framework also suggests potential applications in discrete and combinatorial geometry, optimization, binary decision systems, resource allocation, portfolio balancing, and mathematical economics.