Power-law winding budget threshold

For a real exponent $\alpha$, let the “finite winding budget” proposition mean that the function $x\mapsto x^{-\alpha}$, using real exponentiation, is Lebesgue integrable on the open interval $(0,1)$. The theorem asserts the conjunction that, for every $\alpha\in\mathbb R$, this integrability proposition holds if and only if $\alpha<1$, and that, for every real $h$ satisfying $0\le h$, the function with exponent $1+h$, namely $x\mapsto x^{-(1+h)}$, is not integrable on $(0,1)$. The second universal quantifier includes the boundary case $h=0$; the integration domain excludes both endpoints.