  High-precision exoplanet eclipse light curves, like those possible with JWST,
enable flux and temperature mapping of exoplanet atmospheres. These eclipse
maps will have unprecedented precision, providing an opportunity to constrain
current theoretical predictions of exoplanet atmospheres. However, eclipse
mapping has unavoidable mathematical limitations because many map patterns are
unobservable. This ``null space'' has implications for making comparisons
between predictions from general circulation models (GCMs) and the observed
planet maps, and, thus, affects our understanding of the physical processes
driving the observed maps. We describe the eclipse-mapping null space and show
how GCM forward models can be transformed to their observable modes for more
appropriate comparison with retrieved eclipse maps, demonstrated with
applications to synthetic data of an ultra-hot Jupiter and a cloudy warm
Jupiter under JWST-best-case- and extreme-precision observing scenarios. We
show that the effects of the null space can be mitigated and manipulated
through observational design, and JWST exposure times are short enough to not
increase the size of the null space. Furthermore, we show the mathematical
connection between the null space and the ``eigenmapping'' method,
demonstrating how eigenmaps can be used to understand the null space in a
model-independent way. We leverage this connection to incorporate null-space
uncertainties in retrieved maps, which increases the uncertainties to now
encompass the ground truth for synthetic data. The comparisons between observed
maps and forward models that are enabled by this work, and the improved
eclipse-mapping uncertainties, will be critical to our interpretation of
multidimensional aspects of exoplanets in the JWST era.
