  For fixed degree and increasing number of variables the dimension of the
vector space of $n$-variate real symmetric homogeneous polynomials (forms) of
degree $d$ stabilizes. We study the limits of the cones of symmetric
nonnegative polynomials and symmetric sums of squares, when expressed in
power-mean or monomial-mean basis. These limits correspond to forms with stable
expression in power-mean (or monomial-mean) polynomials that are globally
nonnegative (resp. sums of squares) regardless of the number of variables. We
introduce partial symmetry reduction to describe the limit cone of symmetric
sums of squares, and reprove a result of arXiv:1205.3102v4 that limits of
symmetric nonnegative polynomials and sums of squares agree in degree $4$. We
use tropicalization of the dual cones, which was first in the context of
comparing nonnegative polynomials and sums of squares in arXiv:2203.06291, to
show differences between cones of symmetric polynomials and sums of squares
starting in degree 6, which disproves a conjecture of arXiv:1205.3102v4. For
even symmetric nonnegative forms and sums of squares we show that the cones
agree for degree at most 8, and are different starting with degree 10. We also
find, via tropicalization, explicit examples of symmetric forms that are
nonnegative but not sums of squares in the limit.
