  In this paper, we prove two improved versions of the Finiteness Principle for
nonnegative $ C^2(\mathbb{R}^2) $ interpolation, previously proven by
Fefferman, Israel, and Luli. The first version sharpens the finiteness constant
to $ 64 $, and the second version carries better computational practicality.
Along the way, we also provide detailed construction of nonnegative $ C^2 $
interpolants in one-dimension, and prove the nonexistence of a bounded linear $
C^2 $-extension operator that preserves nonnegativity.
