Homeomorphism type does not determine smooth structure in dimension seven

By The Milnor sphere $M_{2,-1}$ is homeomorphic but not diffeomorphic to $S^7$ the manifold $M_{2,-1}$ is homeomorphic to $S^7$.

The same theorem gives $\lambda (M_{2,-1})\equiv (2-(-1){)}^2-1=8\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}7)$, while $\lambda (S^7)=0$ because the standard sphere bounds the disk $D^8$ with $q=\sigma =0$, and the invariant is negated by orientation reversal so its zero value is fixed by reversal.

If the two closed smooth seven-manifolds were diffeomorphic with either orientation, the invariant of step 2.1 would agree, since an orientation-preserving diffeomorphism preserves $\lambda$ and an orientation-reversing one sends $\lambda (S^7)=0$ to $-\lambda (S^7)=0$; the values $1$ and $0$ differ modulo seven, so no such diffeomorphism exists.

Therefore $S^7$ and $M_{2,-1}$ are homeomorphic closed smooth seven-manifolds with non-diffeomorphic smooth structures, which refutes the statement.