The Critical Count Slope: Why a Faint-End Slope of 0.4 Divides Finite Backgrounds from Divergent Ones

This paper addresses a specific question in the accounting of the cosmic optical and infrared backgrounds: why does the convergence of the extragalactic background hinge on whether the faint-end logarithmic count slope lies below or above a specific value, and how close is the measured slope to that boundary? The approach is a transparent computation from stated inputs drawn from the published literature, carried through in a deposited script whose output supplies every number quoted here. The principal computed values are: alpha critical = 0.4; total alpha 0.21 = 7.67; total alpha 0.30 = 8.8; total alpha 0.35 = 10.3; total alpha 0.45 to35 = 20.6; ratio 0.35 to 0.21 = 1.34; residual beyond 30 alpha 0.21 = 0.0959. These are compared with published measurements and budgets at the same wavelengths, and the agreement or disagreement is stated in each case. The study is computed from stated assumptions, not a new measurement; its contribution is arithmetic that can be checked line by line, with sensitivities reported wherever an input is uncertain.