The
k
k
-connective bounding group
θ
n
(
k
)
{\theta ^n}(k)
and the
k
k
-connective inertial group
I
n
(
k
)
{I^n}(k)
are defined as subgroups of
θ
n
{\theta ^n}
, the group of smooth
n
n
-spheres,
n
≧
7
n \geqq 7
. It is shown
I
n
(
k
)
{I^n}(k)
is contained in
θ
n
(
k
)
{\theta ^n}(k)
. Consequently, the image of the Milnor-Novikov pairing
τ
n
,
k
{\tau _{n,k}}
is contained in
θ
n
+
k
(
k
)
{\theta ^{n + k}}(k)
when
n
≧
k
+
2
n \geqq k + 2
. It follows that
τ
7
,
3
=
0
{\tau _{7,3}} = 0
.