Constraint–Redundancy Saturation in Integer Geometry: An Empirical Deficit-Index Law

This work reports an empirically observed saturation phenomenon in overdetermined integer constraint systems, using the perfect cuboid problem as a representative case. By defining a minimum residual gap function over systematically enumerated face cuboids, we show that the gap decreases rapidly at small scales but then stabilizes at a nonzero floor beyond a finite search threshold. Crucially, this saturation persists under large increases in search range and does not exhibit asymptotic convergence toward zero. We introduce a structural diagnostic quantity, the deficit index \Delta = R_{\mathrm{in}} - R_{\mathrm{rec}}, measuring the excess of simultaneous constraints over the system’s reconciliation capacity. Through controlled experiments with increasing deficit, we find that the saturation floor scales exponentially with \Delta, following L_\infty \sim e^{\alpha \Delta} with \alpha \approx 1.5 for the cuboid system. This scaling law was predicted from structural considerations and subsequently confirmed by enumeration. The results indicate that constraint interference in integer geometry is multiplicative rather than additive, producing a discrete feasibility bound independent of search depth. The framework does not claim impossibility or resolve the perfect cuboid problem; instead, it provides diagnostic evidence for structural saturation regimes in overdetermined Diophantine systems and offers a general method for identifying when extended search is unlikely to improve feasibility.