  One of the most directly observable features of a transiting multi-planet
system is their size-ordering when ranked in orbital separation. Kepler has
revealed a rich diversity of outcomes, from perfectly ordered systems, like
Kepler-80, to ostensibly disordered systems, like Kepler-20. Under the
hypothesis that systems are born via preferred formation pathways, one might
reasonably expect non-random size-orderings reflecting these processes.
However, subsequent dynamical evolution, often chaotic and turbulent in nature,
may erode this information and so here we ask - do systems remember how they
formed? To address this, we devise a model to define the entropy of a planetary
system's size-ordering, by first comparing differences between neighboring
planets and then extending to accommodate differences across the chain. We
derive closed-form solutions for many of the micro state occupancies and
provide public code with look-up tables to compute entropy for up to ten-planet
systems. All three proposed entropy definitions exhibit the expected property
that their credible interval increases with respect to a proxy for time. We
find that the observed Kepler multis display a highly significant deficit in
entropy compared to a randomly generated population. Incorporating a filter for
systems deemed likely to be dynamically packed, we show that this result is
robust against the possibility of missing planets too. Put together, our work
establishes that Kepler systems do indeed remember something of their younger
years and highlights the value of information theory for exoplanetary science.
