  The protocluster SPT2349$-$56 at $z\,{=}\,4.3$ contains one of the most
actively star-forming cores known, yet constraints on the total stellar mass of
this system are highly uncertain. We have therefore carried out deep optical
and infrared observations of this system, probing rest-frame ultraviolet to
infrared wavelengths. Using the positions of the spectroscopically-confirmed
protocluster members, we identify counterparts and perform detailed source
deblending, allowing us to fit spectral energy distributions in order to
estimate stellar masses. We show that the galaxies in SPT2349$-$56 have stellar
masses proportional to their high star-formation rates, consistent with other
protocluster galaxies and field submillimetre galaxies (SMGs) around redshift
4. The galaxies in SPT2349$-$56 have on average lower molecular gas-to-stellar
mass fractions and depletion timescales than field SMGs, although with
considerable scatter. We construct the stellar-mass function for SPT2349$-$56
and compare it to the stellar-mass function of $z\,{=}\,1$ galaxy clusters,
finding consistent shapes between the two. We measure rest-frame galaxy
ultraviolet half-light radii from our HST-F160W imaging, finding that on
average the galaxies in our sample are similar in size to typical star-forming
galaxies at these redshifts. However, the brightest HST-detected galaxy in our
sample, found near the luminosity-weighted centre of the protocluster core,
remains unresolved at this wavelength. Hydrodynamical simulations predict that
the core galaxies will quickly merge into a brightest cluster galaxy, thus our
observations provide a direct view of the early formation mechanisms of this
class of object.
