  I explore the form of the effective interaction in harmonic-oscillator-based
effective theory (HOBET) in next-to-next-to-next-to-leading order (N3LO). As
the included space in a HOBET (as in the shell model) is defined by the
oscillator energy, both long-distance (low-momentum) and short-distance
(high-momentum) degrees of freedom reside in the high-energy excluded space. A
HOBET effective interaction is developed in which a short-range
contact-gradient expansion is combined with an exact summation of the relative
kinetic energy. By this means the very strong coupling of the included (P) and
excluded (Q) spaces by the kinetic energy is removed. One finds that the
interplay of QT and QV is governed by a single parameter kappa, the ratio of an
observable, the binding energy |E|, to a parameter in the effective theory, the
oscillator energy. Once the functional dependence on kappa is identified, the
remaining order-by-order subtraction of the short-range physics residing in Q
becomes systematic and rapidly converging. Numerical calculations are used to
demonstrate how well the resulting expansion reproduces the running of Heff
from high scales to a typical shell-model scale of 8 hbar omega. At N3LO
various global properties of Heff are reproduced to a typical accuracy of
0.01%, or about 1 keV.
  The state-dependence of the effective interaction has been a troubling
problem in nuclear physics, and is embodied in the energy dependence of
Heff(|E|) in the Bloch-Horowitz formalism. It is shown that almost all of this
state dependence is also extracted in the procedures followed here, isolated in
the analytic dependence of Heff on kappa. Thus there exists a simple, Hermitian
Heff that can be use in spectral calculations.
