The Milnor lambda invariant is well defined modulo seven

Glue the two fillings to $N=W{\cup }_M(-W^{\mathrm{\prime }})$ and apply [L4]; reducing modulo $7$ gives $45\equiv 3$, so $3\sigma (N)+p_1^2[N]\equiv 0$, that is $p_1^2[N]\equiv 4\sigma (N)$ and $2p_1^2[N]-\sigma (N)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}7)$.

Substituting [L2] and [L3] into step 1.1 gives $2(q(W)-q(W^{\mathrm{\prime }}))-(\sigma (W)-\sigma (W^{\mathrm{\prime }}))\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}7)$, equivalently ${\lambda }_W(M)={\lambda }_{W^{\mathrm{\prime }}}(M)$; hence the class $\lambda (M)\in \mathbb{Z}\mathrm{/}7$ is independent of the supplied filling.

Let $f:M\to M^{\mathrm{\prime }}$ be an orientation-preserving diffeomorphism of closed oriented seven-manifolds. The same filling $W$, with its boundary identification changed by $f$, is a filling of $M^{\mathrm{\prime }}$. The relative fundamental class, tangent bundle, relative lift and boundary middle form on $W$ are unchanged, so it computes the same candidate. By step 2.1 every supplied filling of $M^{\mathrm{\prime }}$ gives this value. Hence $\lambda (M^{\mathrm{\prime }})=\lambda (M)$.

Reversing the orientation of the filling reverses the relative fundamental class and the boundary orientation, so both $q$ and $\sigma$ change sign while $p_1$ is orientation-independent; therefore $\lambda$ is negated.

Consequently $\lambda (M)=2q(W)-\sigma (W)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>7$ is a well-defined invariant of the oriented boundary, invariant under orientation-preserving boundary diffeomorphism and negated by orientation reversal, with no filling-existence assertion.