Near-identity Sobolev loop factorization ($X=X_{-}{X}_{+}$, $X_{+}(0)=I$)

For matrix-valued circle loops in $H^s$, $s>1/2$, use a submultiplicative algebra norm with contractive Fourier projections. If $\Vert X-I\Vert <1$, solve $Y_{+}=I+y_{+}$ by $y_{+}=-P_{>0}((X-I)(I+y_{+}))$, and solve $V_{-}=I+v_{-}$ by $v_{-}=-P_{<0}((I+v_{-})(X-I))$. Both are contractions. Then $X{Y}_{+}$ has only nonpositive modes, $V_{-}X$ has only nonnegative modes, and their compatible products equal a constant matrix. Invertibility of the Hardy compression $I+P_{\ge 0}(X-I)$ makes that constant invertible. This yields the stated normalized factors; uniqueness follows because the two Fourier subalgebras intersect in constants.