Flux-limited selection for identical standard candles ($r_{\max }=\sqrt{L/(4\pi F_{\min })}$)

A noise-free flux threshold truncates the distance distribution of objects with exactly identical relevant-band luminosity. It cannot preferentially select intrinsically brighter members when no luminosity scatter exists. A luminosity distribution introduces the accessible-volume weighting $L^{3/2}$ in homogeneous Euclidean space. Measurement scatter across the flux threshold and nonuniform completeness introduce additional selection effects even for identical intrinsic luminosities.