and to this order the $k$-trace eigenoperator acquires order $g_2$ admixtures of $(k \pm 1)$-trace operators [@pots2; @mit2]. The eigenvalue is first corrected in order $g_2^2$ to \[eig1\] ø(n,J\_0,J\_1,..J\_k) 2 + ł’ \[ + [g\_2\^2 s\_0\^2 4 \^2]{}( [1 12]{} + [35 32 \^2 n\^2]{}) \] For $k=1$ the correction was found in [@pots2; @mit2] and in [@umut2] for $k >1.$ For $k=1$, the correction exactly matches the genus 1 energy shift of single-string states calculated in light-cone string field theory [@psvvv; @rsv]. This match provides a basis independent test of the relation (\[dil\]). It has also been shown that related matrix elements of $H$ agree with those of $\D - J$ after the basis change is made [@psvvv; @caltech].

The computations of (\[eig1\]) in gauge theory used a formalism equivalent to non-degenerate quantum mechanical perturbation theory. Yet it is obvious from (\[eig\]) that the zero order eigenvalues of single-trace operators of momentum $n$ are degenerate with multi-traces of momentum $m$ if $ns_0= \pm m$. The $n=1$ state is non-degenerate, since $s_0=1$ would be required, and $J_1,J_2,\cdots,J_k$ would vanish. But the $n=2$ single-trace operator is degenerate with multi-traces with $m=\pm 2,~s_0= 1/2$. For $n=3$ we have degeneracy with multi-traces when 