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SDSS J1004+4112 is a well studied gravitational lens with a recently measuredtime delay between its first and fourth arriving quasar images. Using this newconstraint, we present updated free-form lens reconstructions using the lensinversion method {\tt GRALE}, which only uses multiple image and time delaydata as inputs. In addition, we obtain hybrid lens reconstructions by includinga model of the brightest cluster galaxy (BCG) as a Sersic lens. For bothreconstructions, we use two sets of images as input: one with all identifiedimages, and the other a revised set leaving out images that have beenpotentially misidentified. We also develop a source position optimization MCMCroutine, performed on completed {\tt GRALE} runs, that allows each model tobetter match observed image positions and time delays. All the reconstructionsproduce similar mass distributions, with the hybrid models finding a steeperprofile in the center. Similarly, all the mass distributions are fit by theNavarro-Frenk-White (NFW) profile, finding results consistent with previousparametric reconstructions and those derived from Chandra X-ray observations.We identify a $\sim 5 \times 10^{11} M_{\odot}$ substructure apparentlyunaffiliated with any cluster member galaxy and present in all our models, andstudy its reality. Using our free-form and hybrid models we predict a centralquasar image time delay of $\sim 2980 \pm 270$ and $\sim 3280 \pm 215$ days,respectively. A potential future measurement of this time delay will, whilebeing an observational challenge, further constrain the steepness of thecentral density profile.

Free-Form and Hybrid Lens Models for SDSS J1004+4112: Substructure and Central Image Time Delay Constraints