use a discrete finite $J$ version of the method and take the limit $J \to \infty$ at the final stage of computation.

In this method $h^i_j$ is replaced by matrix elements of an effective Hamiltonian $H= H_0 + H_+ + H_-$. The action of H on gauge theory states/operators is given in (11) of [@golm1] in which the operators contain impurities of fixed spacing $l$, $\co^{J_0,J_1\ldots J_k}_l \sim \rm{Tr}(\phi Z^l \psi Z^{J_0-l})
\rm{Tr}Z^{J_1}\ldots \rm{Tr} Z^{J_k}$. After a discrete Fourier transform with respect to $l$, which is equivalent to that in (\[ops\]) for large J, one obtains the following equations: \[heff\] H\_0\_m\^[J\_0,J\_1,,J\_k]{}&=&ł’ \_m\^[J\_0,J\_1,,J\_k]{}\
H\_+\_m\^[J\_0,J\_1,,J\_k]{}&=&ł’g\_2\_[’\_[k+1]{}]{}V\_[\_k]{}\^[\_[k+1]{}]{}\_[m’]{}\^[J\_0-J\_[k+1]{},J\_1, J\_[k+1]{}]{}\
H\_-\_m\^[J\_0,J\_1,,J\_k]{}&=&ł’g\_2\_[i,’\_[k-1]{}]{}V\_[\_k]{}\^[\_[k-1]{}]{}\_[m’]{}\^[J\_0+J\_i,J\_1, J\_i\\,J\_k]{}where we introduce a collective index notation in the matrix elements $V_{\a_k}^{a_{(k \pm 1)}}$, namely $\a_k=\{m,s_0,\dots,s_k\}$. Here $s_i=J_i/J$ satisfy $\sum_i s_i =1$.

The right sides of the equations for $H$ define contributions to matrix elements $h^i_j$. Symbolically, the structure is $H \co_j = h^i_j \co_i$. Note that the interaction terms are purely of order $g_2$ and describe the splitting/joining of a $(k+1)$-trace operator into superpositions of $(k+1\pm 1)$-trace operators. The Hamiltonian $\tilde{H} = S H S^{-1}$ transformed to string basis agrees [@sv] with the Hamiltonian of string bit formalism which contains order $g_2$ splitting/joining interaction 