Endpoint obstruction to an algebraic inverse approximation theorem ($E_n(\sqrt{1-x^2})=O(n^{-1}),\omega (\sqrt{1-x^2},n^{-1})\gtrsim n^{-1/2}$)

The first Jackson theorem for periodic approximation applied to $|\sin \theta |$ gives algebraic best error $O(n^{-1})$ for $\sqrt{1-x^2}$. Its ordinary interval modulus of continuity at $1/n$ is at least $\sqrt{2/n-1/n^2}$. Substituting that error rate into the unmodified inverse theorem for trigonometric approximation would instead bound the modulus by $O(\log (n+1)/n)$, a contradiction. Algebraic inverse estimates must incorporate the endpoint compression of cosine substitution for polynomial approximation.