Admissible composites present the mod-two square algebra

Surjectivity follows from the definition of the image algebra. The reduction lemma spans each homogeneous quotient by admissible words. Suppose a nontrivial linear combination of distinct degree-$N$ admissible composites were zero as a natural operation. Choose $r=N+1$ and $L=N+1$. Every admissible sequence of degree $N$ has $e(I)\le N<r$, so the leading-monomial lemma applies to the same class $P$ on the same finite CW product for all terms. Pick the largest leading monomial among the composites occurring with coefficient one. No composite with a smaller leading monomial contains it, and its coefficient in its own composite is one. The evaluation of the combination on $P$ is therefore nonzero, a contradiction. Degree zero is the nonzero identity operation. Since all relations in the abstract quotient are homogeneous, injectivity in each degree proves injectivity of the graded algebra map.

Boundary of the theorem. It does not yet identify the square algebra with all stable cohomology operations. Such a classification would additionally use the Eilenberg–Mac Lane surjectivity calculation and the published universal-operation corollary. No such classification is imported into the admissible-basis theorem's proof.