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tokens: a sum over all mass within λ C of the test point. We take for galaxies with NSA information and for those without, where V max is the maximum rotational velocity estimated by correcting the full-width half-max of the radio detection for turbulence and projection effects (Tully Fouque 1985 ). These contributions to the total potential derive primarily from the test galaxy’s dark matter. Note that in the case of cluster galaxies, the potential of the cluster itself is part of the external contribution. We calculate by summing the contributions of all unscreened mass within λ C . We then calculate the equilibrium H i -OC offset predicted for a given galaxy: if it is unscreened and 0 otherwise, where M ( r * ) is the dark matter plus gas mass between the H i and optical centroids. This follows from the requirement that the extra force on the stellar disc due to its offset from the halo centre compensate for its not feeling the fifth force, so that the stars, gas, and dark matter continue to move together (Jain VanderPlas 2011 ). We calculate M ( r * ) by assuming a constant density ρ 0 within r * (justified post-hoc: r * for the fifth-force models we are sensitive to is 10 −2 − 10 −1 kpc, much less than the halo scale radius r s ), and estimate it separately for each galaxy using the empirical relation between central baryonic and dynamical surface mass densities (Lelli 2014 ; Lelli et al. 2016 ; Milgrom 2016 ). This yields As spans a very small angle on the plane of the sky we compare separately its orthogonal RA ( r *, α ) and Dec. ( r *, δ ) components with those of the measured displacement for each galaxy. We feed these calculations into a Bayesian likelihood formalism. First, we generate N MC = 1000 Monte Carlo realizations of the predicted signal for each Alfalfa galaxy, sampling independently for each one the galaxy–halo connection (from 200 independent AM realizations), the distribution of mass in the smooth density field (from 10 particle-mesh BORG realizations), the contribution to Φ ex and from haloes too faint to be recorded in 2M+ + (calibrated with the darksky-400 N -body box), and the Gaussian observational uncertainties on the structural galaxy properties used to derive M ( r * ) and Φ in . The full probability distributions that we marginalize over are given in table 1 of Desmond et al. ( 2018b ). We estimate the probability that a given galaxy is unscreened as f ≡ N (|Φ ex | + |Φ in | |Φ c |)/ N MC . The likelihood function then has separate screened ( r * = 0) and unscreened (equation 4 ) components, with relative weights 1 − f and f , respectively. We model the unscreened component using a normalized histogram of the distributions of r *, α and r *, δ over all N MC realizations, obviating the need for assumptions on the form of the likelihood function such as Gaussianity. We convolve this likelihood with the Gaussian H i measurement uncertainty for each galaxy, θ i , and treat galaxies as uncorrelated and RA and Dec.