After getting a relatively good night of sleep, it has occurred to me that most of what I said last night after the equation \[E\ket{k,E}=\sum_i Eb_i\ket{k,a_i}=\sum_i b_i H\ket{k,a_i}=\sum_i\sum_j b_i H_{ij}\ket{k,a_j}\] Was indeed trivial