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Abstract We present 0.″22-resolution Atacama Large Millimeter/submillimeter Array (ALMA) observations of CO(2−1) emission from the circumnuclear gas disk in the red nugget relic galaxy PGC 11179. The disk shows regular rotation, with projected velocities near the center of 400 km s⁻¹. We assume the CO emission originates from a dynamically cold, thin disk and fit gas-dynamical models directly to the ALMA data. In addition, we explore systematic uncertainties by testing the impacts of various model assumptions on our results. The supermassive black hole (BH) mass (M_BH) is measured to beM_BH= (1.91 ± 0.04 [1σstatistical] ${}_{-0.51}^{+0.11}$[systematic]) × 10⁹M_⊙, and theH-band stellar mass-to-light ratioM/L_H= 1.620 ± 0.004 [1σstatistical] ${}_{-0.107}^{+0.211}$[systematic]M_⊙/L_⊙. ThisM_BHis consistent with the BH mass−stellar velocity dispersion relation but over-massive compared to the BH mass−bulge luminosity relation by a factor of 3.7. PGC 11179 is part of a sample of local compact early-type galaxies that are plausible relics ofz∼ 2 red nuggets, and its behavior relative to the scaling relations echoes that of three relic galaxy BHs previously measured with stellar dynamics. These over-massive BHs could suggest that BHs gain most of their mass before their host galaxies do. However, our results could also be explained by greater intrinsic scatter at the high-mass end of the scaling relations, or by systematic differences in gas- and stellar-dynamical methods. AdditionalM_BHmeasurements in the sample, including independent cross-checks between molecular gas- and stellar-dynamical methods, will advance our understanding of the co-evolution of BHs and their host galaxies. 

Time-delay cosmography with gravitationally lensed quasars plays an important role in anchoring the absolute distance scale and hence measuring the Hubble constant, H 0 , independent of traditional distance ladder methodology. A current potential limitation of time-delay distance measurements is the mass-sheet transformation (MST), which leaves the lensed imaging unchanged but changes the distance measurements and the derived value of H 0 . In this work we show that the standard method of addressing the MST in time-delay cosmography, through a combination of high-resolution imaging and the measurement of the stellar velocity dispersion of the lensing galaxy, depends on the assumption that the ratio, D s / D ds , of angular diameter distances to the background quasar and between the lensing galaxy and the quasar can be constrained. This is typically achieved through the assumption of a particular cosmological model. Previous work (TDCOSMO IV) addressed the mass-sheet degeneracy and derived H 0 under the assumption of the ΛCDM model. In this paper we show that the mass-sheet degeneracy can be broken without relying on a specific cosmological model by combining lensing with relative distance indicators such as supernovae Type Ia and baryon acoustic oscillations, which constrain the shape of the expansion history and hence D s / D ds . With this approach, we demonstrate that the mass-sheet degeneracy can be constrained in a cosmological model-independent way. Hence model-independent distance measurements in time-delay cosmography under MSTs can be obtained.