Scope of the finite Milnor-sphere calculation

The local construction and invariant calculation establish the explicit exotic sphere $M_{2,-1}$ and the formula $\lambda (M_{h,j})=(h-j{)}^2-1(\mathrm{m}\mathrm{o}\mathrm{d}7)$ for $h+j=1$, under the choice assumptions of The Milnor sphere $M_{2,-1}$ is homeomorphic but not diffeomorphic to $S^7$. Substituting $(h,j)=(1,0)$ gives $\lambda (M_{1,0})=1^2-1=0$, while $(h,j)=(2,-1)$ gives $\lambda (M_{2,-1})=3^2-1=8\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}7)$. The invariant is preserved by orientation-preserving diffeomorphisms and negated by orientation reversal (The Milnor lambda invariant is well defined modulo seven). Since neither $1$ nor $-1$ equals $0$ modulo seven, these two manifolds cannot be diffeomorphic in either orientation.

These are constructions and obstruction calculations for specified manifolds. This remark asserts no classification of all smooth homotopy seven-spheres, no group order, and no exhaustion of diffeomorphism types by this family. Such conclusions require additional proofs beyond the displayed local calculations.