The KAM limit parametrisation is an embedding of the torus: injectivity modulo $2\pi$, rank-$n$ differential and Lipschitz inverse

Contraction and injectivity. By [F1] each partial derivative obeys $\mathrm{\mid }{\mathrm{\partial }}_{{\vartheta }_i}{w}_j\mathrm{\mid }\le 2C{E}_0$ on the domain. For fixed real $\vartheta ,{\vartheta }^{\mathrm{\prime }}$ and each component $w_j$, reality follows from [F6], and the curve $t\mapsto w_j(\vartheta +t({\vartheta }^{\mathrm{\prime }}-\vartheta ),\nu )$ has derivative bounded by $2C{E}_0\mathrm{\mid }\vartheta -{\vartheta }^{\mathrm{\prime }}{\mathrm{\mid }}_1$ by [F2]; the one-variable mean-value inequality gives $\mathrm{\mid }{w}_j(\vartheta )-w_j({\vartheta }^{\mathrm{\prime }})\mathrm{\mid }\le 2C{E}_0\mathrm{\mid }\vartheta -{\vartheta }^{\mathrm{\prime }}{\mathrm{\mid }}_1$. Summing over $j$ gives $\mathrm{\mid }w(\vartheta ,\nu )-w({\vartheta }^{\mathrm{\prime }},\nu ){\mathrm{\mid }}_1\le 2nC{E}_0\mathrm{\mid }\vartheta -{\vartheta }^{\mathrm{\prime }}{\mathrm{\mid }}_1\le \frac{1}{2}\mathrm{\mid }\vartheta -{\vartheta }^{\mathrm{\prime }}{\mathrm{\mid }}_1$ by $4nC{E}_0\le 1$; then $\mathrm{\mid }{\mathrm{\Theta }}_{\nu }(\vartheta )-{\mathrm{\Theta }}_{\nu }({\vartheta }^{\mathrm{\prime }}){\mathrm{\mid }}_1\ge \mathrm{\mid }\vartheta -{\vartheta }^{\mathrm{\prime }}{\mathrm{\mid }}_1-\mathrm{\mid }w(\vartheta ,\nu )-w({\vartheta }^{\mathrm{\prime }},\nu ){\mathrm{\mid }}_1\ge \frac{1}{2}\mathrm{\mid }\vartheta -{\vartheta }^{\mathrm{\prime }}{\mathrm{\mid }}_1$, so ${\mathrm{\Theta }}_{\nu }$ is injective, and ${\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu )={\mathrm{\Psi }}_{\mathrm{\infty }}(0,{\vartheta }^{\mathrm{\prime }},\nu )$ implies ${\mathrm{\Theta }}_{\nu }(\vartheta )={\mathrm{\Theta }}_{\nu }({\vartheta }^{\mathrm{\prime }})$, hence $\vartheta ={\vartheta }^{\mathrm{\prime }}$. This is clause 1.

Weighted phase Jacobians with the retracted parameters. Write $\mathrm{\Lambda }:={\mathrm{\Omega }}_{\alpha ,\tau }$ and $D_j=B_{r_j\mathrm{/}2}\times S_{s_j-8{\sigma }_j}\times O_{h_{j+1}}(\mathrm{\Lambda })$ as in the composition supplier. For $x_j=(z,\nu )\in D_j$, recursively put $x_i=(z_i,{\lambda }_i):=T_i(x_{i+1})$ for $i=j-1,\dots ,0$, so $x_i\in D_i$ and ${\lambda }_i={\varphi }_i({\lambda }_{i+1})$. Since these parameter coordinates are independent of the input phase $z$, the phase chain rule gives $D_z{\mathrm{\Psi }}_j=M_0\cdots M_{j-1}$ with $M_i:=D_z{\tilde{\mathrm{\Phi }}}_i(z_{i+1},{\lambda }_i)$. The domain inclusions and retraction bounds of the composition supplier put each evaluation in the domain of [F7]'s one-step estimate. As $W_i{W}_{i+1}^{-1}=\mathrm{diag}({\eta }_i{I}_n,\frac{1}{2}{I}_n)$, [F7] gives $\mathrm{\parallel }{W}_i{M}_i{W}_{i+1}^{-1}\mathrm{\parallel }\le (1+C{E}_i)\mathrm{/}2\le 9\mathrm{/}16$, using $C{E}_i\le C{E}_0\le 1\mathrm{/}(4n)\le 1\mathrm{/}8$. Inserting the adjacent weights into the product yields $\mathrm{\parallel }{W}_0{D}_z{\mathrm{\Psi }}_j{W}_j^{-1}\mathrm{\parallel }\le (9\mathrm{/}16{)}^j\le 1$, including the identity product at $j=0$.

Descent and injectivity on the torus. By [F1] each ${\mathrm{\Psi }}_j$ is $2\pi$-equivariant and $\mathcal{L}$ is invariant under $\vartheta \mapsto \vartheta +2\pi k$; passing to the uniform limit on $\mathcal{L}$ gives ${\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta +2\pi k,\nu )={\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu )+(0,2\pi k)$, so $\vartheta \mapsto {\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu )$ maps equivalent points to points whose difference lies in $\{0\}\times 2\pi {\mathbb{Z}}^n$ and therefore induces a well-defined map ${\widehat{\mathrm{\Psi }}}_{\nu }$ on ${\mathbb{T}}^n$; it is continuous by the quotient property of [F3]; if ${\widehat{\mathrm{\Psi }}}_{\nu }(\pi \vartheta )={\widehat{\mathrm{\Psi }}}_{\nu }(\pi {\vartheta }^{\mathrm{\prime }})$, then ${\mathrm{\Theta }}_{\nu }(\vartheta )-{\mathrm{\Theta }}_{\nu }({\vartheta }^{\mathrm{\prime }})=2\pi k$ for some $k\in {\mathbb{Z}}^n$, and putting ${\vartheta }^{\mathrm{\prime }\mathrm{\prime }}:=\vartheta -2\pi k$ the equivariance gives ${\mathrm{\Theta }}_{\nu }({\vartheta }^{\mathrm{\prime }\mathrm{\prime }})={\mathrm{\Theta }}_{\nu }({\vartheta }^{\mathrm{\prime }})$, so step 1.1 forces ${\vartheta }^{\mathrm{\prime }\mathrm{\prime }}={\vartheta }^{\mathrm{\prime }}$, i.e. $\pi \vartheta =\pi {\vartheta }^{\mathrm{\prime }}$; hence ${\widehat{\mathrm{\Psi }}}_{\nu }$ is injective. This is clause 2.

Immersion. Let $v\in {\mathbb{C}}^n$ and suppose $D_{\vartheta }{\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu )v=0$ at a real $\vartheta$; the second component of this derivative is $(I_n+{\mathrm{\partial }}_{\vartheta }w(\vartheta ,\nu ))v$, so $\mathrm{\mid }v{\mathrm{\mid }}_1=\mathrm{\mid }-{\mathrm{\partial }}_{\vartheta }w(\vartheta ,\nu )v{\mathrm{\mid }}_1\le 2nC{E}_0\mathrm{\mid }v{\mathrm{\mid }}_1\le \frac{1}{2}\mathrm{\mid }v{\mathrm{\mid }}_1$ by the entrywise bound of step 1.1, and [F4] gives $v=0$; hence the differential has rank $n$. This is clause 3.

The finite normal-form residual. Fix a real label $\nu$ and a real phase point in $D_j$. The finite phase maps and parameter images are real by the one-step symmetry already proved in Conjugation symmetry of the KAM construction: real perturbations produce real tori; they are canonical by the composition supplier. Put $H:=H_0$ and freeze its original parameter at ${\chi }_j(\nu )$. The exact identity $H({\mathrm{\Psi }}_j(z,\nu );{\chi }_j(\nu ))=e_j(\nu )+\nu \cdot I+P_j(z,\nu )$ and the pointwise transport calculation [F5] give $X_H({\mathrm{\Psi }}_j;{\chi }_j(\nu ))-D_z{\mathrm{\Psi }}_j(0,\nu )=D_z{\mathrm{\Psi }}_j{X}_{P_j}$. Here $(0,\nu )$ denotes the constant phase vector, while the parameter is fixed. By step 1.2 and [F8], the maximum norm of this residual after multiplication by $W_0$ is at most $2{\epsilon }_j\mathrm{/}(r_j{\sigma }_j)=2\alpha E_j{\sigma }_j^{\mu -1}$. Since $E_j\le 4^{-j}{E}_0$, ${\sigma }_j={\sigma }_0{2}^{-j}$ and $\mu >1$, this bound tends to zero.

Lipschitz inverse. Let $x=\pi \vartheta$, $y=\pi \vartheta \mathrm{\prime }$. Choose an integer vector $k$ minimizing $\mathrm{\mid }{\mathrm{\Theta }}_{\nu }(\vartheta )-{\mathrm{\Theta }}_{\nu }(\vartheta \mathrm{\prime })-2\pi k{\mathrm{\mid }}_{\mathrm{\infty }}$; such a minimizer exists by choosing a closest integer in each coordinate. Step 1.1 applied to $\vartheta -2\pi k$ and $\vartheta \mathrm{\prime }$, with equivariance from step 2.1, gives $\mathrm{\mid }\vartheta -\vartheta \mathrm{\prime }-2\pi k{\mathrm{\mid }}_1\le 2\mathrm{\mid }{\mathrm{\Theta }}_{\nu }(\vartheta )-{\mathrm{\Theta }}_{\nu }(\vartheta \mathrm{\prime })-2\pi k{\mathrm{\mid }}_1\le 2n{d}_{{\mathbb{T}}^n}(\pi {\mathrm{\Theta }}_{\nu }(\vartheta ),\pi {\mathrm{\Theta }}_{\nu }(\vartheta \mathrm{\prime }))$. The domain quotient distance is at most the left-hand side, and the phase distance is at least its angular quotient distance, so $d_{{\mathbb{T}}^n}(x,y)\le 2n{d}_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}({\widehat{\mathrm{\Psi }}}_{\nu }(x),{\widehat{\mathrm{\Psi }}}_{\nu }(y))$.

The limiting invariance identity. At $z=(0,\vartheta )$ with real $\vartheta$, [F1] supplies convergence of the phase maps, their angular derivatives and the parameter maps. Each extended image lies in $D_0$, so the action limit has maximum norm at most $r_0\mathrm{/}2<r_0$, and ${\mathrm{dist}}_{\mathrm{\infty }}({\chi }_{\mathrm{\infty }}(\nu ),\mathrm{\Lambda })\le h_1=h_0{4}^{-\mu }<h_0$; the latter follows from the triangle inequality for distance and convergence. All limiting coordinates are real by [F6] and the symmetry supplier, so the angle limit lies in $S_{s_0}$. The limit is therefore inside the open domain of $H_0$, where $X_H$ is continuous. Passing to the limit in step 2.3, with $W_0$ a fixed invertible matrix, gives $X_H({\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu );{\chi }_{\mathrm{\infty }}(\nu ))={\sum }_i{\nu }_i{\mathrm{\partial }}_{{\vartheta }_i}{\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu )$. The chain rule then makes $t\mapsto {\mathrm{\Psi }}_{\mathrm{\infty }}(0,{\vartheta }_0+t\nu ,\nu )$ an integral curve for every real $t$.

The embedding. Let $U\subseteq {\mathbb{T}}^n$ be open and $x_0\in U$; by [F3] there is $\delta >0$ with $\{x:d_{{\mathbb{T}}^n}(x,x_0)<\delta \}\subseteq U$, and by step 3.1 every $y$ with $d_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}({\widehat{\mathrm{\Psi }}}_{\nu }(y),{\widehat{\mathrm{\Psi }}}_{\nu }(x_0))<\delta \mathrm{/}(2n)$ satisfies $d_{{\mathbb{T}}^n}(y,x_0)<\delta$, hence lies in $U$; therefore ${\widehat{\mathrm{\Psi }}}_{\nu }(U)$ is relatively open in the image, so the continuous bijection ${\widehat{\mathrm{\Psi }}}_{\nu }:{\mathbb{T}}^n\to {\widehat{\mathrm{\Psi }}}_{\nu }({\mathbb{T}}^n)$ has continuous inverse, i.e. is a homeomorphism onto its image. This proves clause 4, together with the reality statement of [F6].

The limiting Lagrangian image. The inclusion ${\iota }_0(\vartheta ):=(0,\vartheta )$ has ${\iota }_0^{\ast }\omega =0$. At each fixed real target label the canonical phase map satisfies $({\mathrm{\Psi }}_j\circ {\iota }_0{)}^{\ast }\omega ={\iota }_0^{\ast }({\mathrm{\Psi }}_j^{\ast }\omega )=0$. Each coefficient of this pullback is a finite sum of products of first angular derivatives of ${\mathrm{\Psi }}_j\circ {\iota }_0$; their convergence in [F1] therefore gives $({\mathrm{\Psi }}_{\mathrm{\infty }}\circ {\iota }_0{)}^{\ast }\omega =0$. Together with the embedding of step 4.1 and the integral curves of step 3.2, this proves clause 5 under the stated $4nC{E}_0\le 1$, without assuming $C{E}_0\le 1\mathrm{/}16$.