  I describe harmonic-oscillator-based effective theory (HOBET) and explore the
extent to which the effects of excluded higher-energy oscillator shells can be
represented by a contact-gradient expansion in next-to-next-to-leading order
(NNLO). I find the expansion can be very successful provided the energy
dependence of the effective interaction, connected with missing long-wavelength
physics associated with low-energy breakup channels, is taken into account. I
discuss a modification that removes operator mixing from HOBET, simplifying the
task of determining the parameters of an NNLO interaction.
