Changes in anvil clouds with warming remain a leading source of uncertainty in estimating the Earth's climate sensitivity. Here, we develop a novel feedback analysis that decomposes changes in anvil clouds and creates testable hypotheses for refining their proposed uncertainty ranges with observations and theory. To carry out this storyline approach, we derive a simple but quantitative expression for the anvil area feedback, which is shown to depend on the present day, measurable cloud radiative effects and the fractional change in anvil area with warming. Satellite observations suggest an anvil cloud radiative effect of about \(\pm\)1 Wm^-2, which requires the fractional change in anvil area to be about  \(\mp\)50 % K^-1 to produce a feedback equal to the current best estimate of its lower bound. We use theory and observations to show that the change in anvil area is closer to about -4\% K^-1. This constrains the area feedback and leads to our new estimate of 0.02 \(\pm\) 0.07 Wm^-2 K^-1 which is many times weaker and more constrained than the overall anvil cloud feedback. In comparison, we show the anvil cloudy albedo feedback to be much less constrained, both theoretically and observationally, and thus to embody the uncertainty that poses an obstacle for bounding the Earth's climate sensitivity.