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The next step on the path toward another Earth is to find atmospheres similar to those of Earth and Venus - high-molecular-weight (secondary) atmospheres - on rocky exoplanets. Many rocky exoplanets are born with thick (> 10 kbar) H$_2$-dominated atmospheres but subsequently lose their H$_2$; this process has no known Solar System analog. We study the consequences of early loss of a thick H$_2$ atmosphere for subsequent occurrence of a high-molecular-weight atmosphere using a simple model of atmosphere evolution (including atmosphere loss to space, magma ocean crystallization, and volcanic outgassing). We also calculate atmosphere survival for rocky worlds that start with no H$_2$. Our results imply that most rocky exoplanets orbiting closer to their star than the Habitable Zone that were formed with thick H$_2$-dominated atmospheres lack high-molecular-weight atmospheres today. During early magma ocean crystallization, high-molecular-weight species usually do not form long-lived high-molecular-weight atmospheres; instead they are lost to space alongside H$_2$. This early volatile depletion also makes it more difficult for later volcanic outgassing to revive the atmosphere. However, atmospheres should persist on worlds that start with abundant volatiles (for example, waterworlds). Our results imply that in order to find high-molecular-weight atmospheres on warm exoplanets orbiting M-stars, we should target worlds that formed H$_2$-poor, that have anomalously large radii, or which orbit less active stars.