Spectral Signatures of Constraints: Density Field Examination of Millennium Mathematical Problems

This paper proposes density field spectral statistics as a universal examination layer for mathematical constraints, and conducts a systematic examination of six out of the seven Millennium Prize Problems. By mapping different types of mathematical structures into the eigenmode spectra of density fields, and employing Random Matrix Theory (RMT) statistics such as the nearest-neighbor spacing ratio and spectral rigidity for quantitative analysis, we find that mathematical constraints manifest in density fields as detectable spectral structures. The six-station examination reveals a unified constraint-spectral rigidity hierarchy: from super-rigidity (sub-GUE) produced by exact symmetry constraints, through selective rigidity and gap-chain constraints, to Poisson decorrelation caused by constraint fragmentation.