Mathematical Redefinition of Nocturnal Darkness Based on Finite Quantum Measurement Threshold Boundaries and Geometrical Occupancy Limits

This framework presents a rigorous spatiotemporal measurement model defining localized nocturnal darkness. By introducing generalized distributions via mathematical framework configurations, conventional continuous space approximations are replaced with discrete sources. Incorporating discontinuous step-thresholds of the quantum measurement operator, the exact effective total irradiance \(E_{\text{eff}}\) is derived, establishing that nocturnal darkness is a deterministic consequence of discrete geometric dilution and discontinuous measurement thresholds evaluated at the localized non-linear measurement interface.