Counts in an Expanding Volume: The Departure of the Count Slope from the Euclidean Value

This paper addresses a specific question in the accounting of the cosmic optical and infrared backgrounds: how and why does the logarithmic slope of cumulative galaxy counts fall below the Euclidean value of 0.6 per magnitude at faint limits? 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: cosmo count ratio 25 to 24 = 1.96; euclidean ratio per mag = 3.98; cosmo slope 24 25 dex per mag = 0.292; cosmo slope 25 26 dex per mag = 0.25. 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.