Dark Sirens and Catalogue Completeness: How Galaxy-Catalogue Depth Moves the Statistical Hubble Constant

This paper asks a single question and computes its answer. For mergers without counterparts, how does the completeness of the galaxy catalogue used as a redshift prior bias and broaden the statistical Hubble constant, as a function of catalogue limiting magnitude? Working in a flat LambdaCDM background and carrying the bounce-frame density ceiling explicitly — the Aldred Limit, rho_A = rho_Pl (m/m_Pl)^2/(8 pi C), with the neutron branch at 9.72e+54 g per cubic centimetre — the calculation gives the headline values w03 = 8.9, w10 = 5.1, med03 = 67.7, med10 = 67.2, rhoA = 9.72e+54. Completeness is the controlling variable. At low completeness the central interval widens sharply, because the background term dilutes the true host peak; the median stays near the input value at every completeness, so the cost of incompleteness is width rather than bias in this model. At full completeness the estimator recovers the input constant with the narrowest interval in the table. Every number reported here comes from the deposited computation script, which states its assumptions in full. This study is simulated and computed from stated assumptions and published catalogue values; it is not a measurement.