  In this paper, we obtain an explicit formula for the discrepancy between the
limit of the second inner variations of $p$-Laplace Allen-Cahn energies and the
second inner variation of their $\Gamma$-limit which is the area functional.
Our analysis explains the mysterious discrepancy term found in our previous
paper \cite{Le} in the case $p=2$. The discrepancy term turns out to be related
to the convergence of certain 4-tensors which are absent in the usual
Allen-Cahn functional. These (hidden) 4-tensors suggest that, in the
complex-valued Ginzburg-Landau setting, we should expect a different
discrepancy term which we are able to identify. Along the way, we partially
answer a question of Kohn and Sternberg \cite{KS} by giving a relation between
the limit of second variations of the Allen-Cahn functional and the second
inner variation of the area functional at local minimizers. Moreover, our
analysis reveals an interesting identity connecting second inner variation and
Poincar\'e inequality for area-minimizing surfaces with volume constraint in
the work of Sternberg and Zumbrun \cite{SZ2}.
