  We quantify evolution in the cluster scale stellar mass - halo mass (SMHM)
relation's parameters using 2323 clusters and brightest central galaxies (BCGs)
over the redshift range $0.03 \le z \le 0.60$. The precision on inferred SMHM
parameters is improved by including the magnitude gap ($\rm m_{gap}$) between
the BCG and fourth brightest cluster member (M14) as a third parameter in the
SMHM relation. At fixed halo mass, accounting for $\rm m_{gap}$, through a
stretch parameter, reduces the SMHM relation's intrinsic scatter. To explore
this redshift range, we use clusters, BCGs, and cluster members identified
using the Sloan Digital Sky Survey C4 and redMaPPer cluster catalogs and the
Dark Energy Survey redMaPPer catalog. Through this joint analysis, we detect no
systematic differences in BCG stellar mass, $\rm m_{gap}$, and cluster mass
(inferred from richness) between the datsets. We utilize the Pareto function to
quantify each parameter's evolution. We confirm prior findings of negative
evolution in the SMHM relation's slope (3.5$\sigma$) and detect negative
evolution in the stretch parameter (4.0$\sigma$) and positive evolution in the
offset parameter (5.8$\sigma$). This observed evolution, combined with the
absence of BCG growth, when stellar mass is measured within 50kpc, suggests
that this evolution results from changes in the cluster's $\rm m_{gap}$. For
this to occur, late-term growth must be in the intra-cluster light surrounding
the BCG. We also compare the observed results to Illustris TNG 300-1
cosmological hydrodynamic simulations and find modest qualitative agreement.
However, the simulations lack the evolutionary features detected in the real
data.
