Holonomic phase residuals control distance to the consistent phase manifold

At a point of the compact regular chart $K\subset Q_{\mathrm{r}\mathrm{e}\mathrm{g}}$, permute coordinates as $q=(x,y)$ so $D_{y}c$ is an invertible $m\times m$ block. Apply [F2] for each finite regularity order; uniqueness makes these graphs agree locally, so the graph $y=\psi (x)$ is smooth. Write $\varphi (x)=(x,\psi (x))$ and $T(x)=D\varphi (x)$. Since $J(\varphi (x))T(x)=0$ and both kernels have dimension $d-m$, $\mathrm{im}T=\ker J$.

After shrinking to a convex coordinate neighborhood, $B(x)=D_{y}c(x,\psi (x))$ remains invertible and $\mathrm{\parallel }B(x{)}^{-1}(D_{y}c(x,y)-B(x))\mathrm{\parallel }\le \frac{1}{2}$. Set $R_x(y)=c(x,y)-B(x)(y-\psi (x))$. Then $R_x(\psi (x))=0$ and the derivative of $B(x{)}^{-1}{R}_x$ has norm at most $\frac{1}{2}$. The mean-value inequality [F4] gives $\mathrm{\parallel }B(x{)}^{-1}{R}_x(y)\mathrm{\parallel }\le \frac{1}{2}\mathrm{\parallel }y-\psi (x)\mathrm{\parallel }$. Since $B^{-1}c=(y-\psi )+B^{-1}{R}_x(y)$, it follows that $\mathrm{\parallel }q-\varphi (x)\mathrm{\parallel }=\mathrm{\parallel }y-\psi (x)\mathrm{\parallel }\le 2\mathrm{\parallel }B(x{)}^{-1}\mathrm{\parallel }\mathrm{\parallel }c(q)\mathrm{\parallel }$. Compactness makes the constant uniform on a finite chart cover.

For $\bar{q}=\varphi (x)$, put $G=T^{T}MT$ and define $\bar{p}=MT{G}^{-1}{T}^{T}p$. Positive definiteness of $M$ and injectivity of $T$ make $G$ invertible; its smooth entries and positive determinant give a smooth inverse by $G^{-1}=\mathrm{adj}(G)\mathrm{/}\det G$. Then $J(\bar{q})M^{-1}\bar{p}=0$ and $T^T(p-\bar{p})=0$. Since $\ker T^T$ and $\mathrm{im}J(\bar{q}{)}^T$ both have dimension $m$ and the latter is contained in the former, $p-\bar{p}=J(\bar{q}{)}^T\eta$ for a unique $\eta$.

The identity $J(\bar{q})M^{-1}(p-\bar{p})=A(\bar{q})\eta$ and [F1] imply $\mathrm{\parallel }p-\bar{p}\mathrm{\parallel }\le C\mathrm{\parallel }A(\bar{q}{)}^{-1}\mathrm{\parallel }(\mathrm{\parallel }J(q)M^{-1}p\mathrm{\parallel }+\mathrm{\parallel }J(\bar{q})-J(q)\mathrm{\parallel }\mathrm{\parallel }{M}^{-1}p\mathrm{\parallel })$. The momentum set is bounded and $J$ is locally Lipschitz, so step 2.1 bounds the right side by $C(\mathrm{\parallel }J(q)M^{-1}p\mathrm{\parallel }+\mathrm{\parallel }c(q)\mathrm{\parallel })$. Thus $\mathrm{\Pi }(q,p)=(\bar{q},\bar{p})$ has the asserted distance bound and lies in $\mathcal{M}$. If $(q,p)\in \mathcal{M}$, then $\bar{q}=q$ and the formula for $\bar{p}$ returns $p$, so $\mathrm{\Pi }$ is the identity there. By [F3] the coordinate formula is smooth, and it is the identity on the consistent phase set in this neighborhood. Shrink the finitely many neighborhoods so their closures lie in an open regular set; continuity keeps $A^{-1}$ and the required derivatives uniformly bounded there. A finite subcover gives finitely many such local retractions and a uniform constant.