We extract the proton charge radius from the elastic form factor data using a
theoretical framework combining chiral effective field theory and dispersion
analysis. Complex analyticity in the momentum transfer correlates the behavior
of the spacelike form factor in different $Q^2$ regions and permits the use of
data up to $Q^2 \sim$ 0.5 GeV$^2$ in constraining the radius. The predictive
theory describes the data with the same accuracy as current descriptive models
(global fits). We obtain a radius of 0.844(7) fm, consistent with the
high-precision muonic hydrogen results.