One KAM step: the affine generator, its canonical time-one flow, and the quadratic remainder bound

Generator and domains. The datum $R-\widehat{R}$ is affine, zero-mean, and has norm at most $C{\sigma }^{-n}\epsilon$ on $D_{r,s-2\sigma ,h}$. Its Fourier sum is finite and its supported divisors never vanish, so $F$ is holomorphic even on $D_{r,s-2\sigma ,h}$, is affine in $I$, and satisfies the equation there. Apply [F3] with $u=s-2\sigma$ and action margin $r\mathrm{/}4$. On the appropriate nested domains this gives $\mathrm{\mid }{F}_{\vartheta }\mathrm{\mid }\le C\epsilon {\alpha }^{-1}{\sigma }^{-(\tau +2n+1)}$, $\mathrm{\mid }{F}_I\mathrm{\mid }\le C\epsilon (\alpha r{)}^{-1}{\sigma }^{-(\tau +2n)}$, $\mathrm{\mid }{F}_{\vartheta \vartheta }\mathrm{\mid }\le C\epsilon {\alpha }^{-1}{\sigma }^{-(\tau +2n+2)}$, and $\mathrm{\mid }{F}_{I\vartheta }\mathrm{\mid }\le C\epsilon (\alpha r{)}^{-1}{\sigma }^{-(\tau +2n+1)}$. In particular, all four bounds hold on $D_{3r\mathrm{/}4,s-6\sigma ,h}$. Since $F_{II}=0$, the blocks of $W(D{X}_F)W^{-1}$ are $-F_{\vartheta I}$, $-(\sigma \mathrm{/}r)F_{\vartheta \vartheta }$, zero, and $F_{I\vartheta }$. Substitution of $E$ and $\mu$ bounds each nonzero block by $CE{\sigma }^{n+1}$ and also gives $\mathrm{\mid }W{X}_F\mathrm{\mid }\le CE{\sigma }^{n+1}$. The defining equation is precisely $\{F,N\}=R-\widehat{R}$.

Existence through time one. Set $\mathcal{U}=B_{3r\mathrm{/}4}\times S_{s-6\sigma }$. Fix $\nu \in O_h$ and an initial point in $B_{r\mathrm{/}2}\times S_{s-8\sigma }$. In the norm $\mathrm{\mid }Wz{\mathrm{\mid }}_{\mathrm{\infty }}$, its distance to the action boundary of $\mathcal{U}$ is at least $1\mathrm{/}4$ and to the angle boundary at least $2$. Step 1.1 gives field and Lipschitz bounds $M,L\le CE{\sigma }^{n+1}$ there. Choose $c$ so that $CE\le 1\mathrm{/}8$. While a solution remains in $\mathcal{U}$, integration gives $\mathrm{\mid }W({\mathrm{\Phi }}_t-z)\mathrm{\mid }\le Mt$ for $t\le 1$. It therefore cannot approach the boundary before time one. To justify continuation without a compact domain assumption, a putative finite endpoint has a limit because the velocity is bounded; the limit has the same positive boundary margins, and local existence there extends the trajectory, contradicting maximality. Thus ${\mathrm{\Phi }}_t$ exists for $0\le t\le 1$, has $\mathrm{\mid }{I}_t\mathrm{\mid }<3r\mathrm{/}4$ and $\mathrm{\mid }\mathrm{Im}{\vartheta }_t\mathrm{\mid }<s-7\sigma$, and its weighted displacement is at most $CE$. For initial $\mathrm{\mid }I\mathrm{\mid }<\eta r$, the sharper component estimate gives $\mathrm{\mid }{I}_t-I\mathrm{\mid }\le CEr{\sigma }^{n+1}\le \eta r\mathrm{/}2$, after decreasing $c$ using $E\le c\eta$. Consequently all these trajectories lie in $B_{2\eta r}\times S_{s-7\sigma }$.

Phase derivatives. Apply the local smooth-dependence result in [F4] on short time intervals in Euclidean balls with closure inside the phase domain. Each trajectory on $[0,1]$ is compact and stays inside that open domain by step 2.1, so finitely many such intervals suffice; the flow composition law transfers their $C^1$ dependence to the entire trajectory. Smooth dependence then gives $A_t=D{\mathrm{\Phi }}_t$ with $A_0=\mathrm{i}\mathrm{d}$ and $A_t^{\mathrm{\prime }}=D{X}_F({\mathrm{\Phi }}_t)A_t$. Thus ${\tilde{A}}_t=W{A}_t{W}^{-1}$ solves ${\tilde{A}}_t^{\mathrm{\prime }}=(WD{X}_F{W}^{-1})({\mathrm{\Phi }}_t){\tilde{A}}_t$. Gronwall gives $\mathrm{\mid }{\tilde{A}}_t\mathrm{\mid }\le e^{Lt}$ and, by integrating the equation, $\mathrm{\mid }{\tilde{A}}_t-\mathrm{i}\mathrm{d}\mathrm{\mid }\le e^{Lt}-1\le CE$ for $t\le 1$. These are the asserted phase derivative bounds uniformly over all initial points and parameters.

Holomorphy and affine structure. Treat the parameter as additional state with equation ${\nu }^{\mathrm{\prime }}=0$. This extended field is holomorphic, so differentiating its real $C^1$ flow with respect to any conjugate initial coordinate gives a homogeneous complex linear variational equation with zero initial value; all conjugate-coordinate derivatives vanish by uniqueness. Hence [F6] gives holomorphy in both phase and parameter variables. Since $F$ is affine in $I$, the equation ${\vartheta }^{\mathrm{\prime }}=F_I(\vartheta ,\nu )$ is independent of $I$. With this angle trajectory fixed, $I^{\mathrm{\prime }}=-F_{\vartheta }(I,{\vartheta }_t,\nu )$ is a linear inhomogeneous equation in $I$, so its solution depends affinely on the initial action. This proves the stated structure.

Exact transformed identity. For a fixed initial point set $G(t)=(N+{\chi }_t)\circ {\mathrm{\Phi }}_t$, where ${\chi }_t=\widehat{R}+t(R-\widehat{R})$. Using [F5] and $\{N,F\}=-(R-\widehat{R})$, one obtains $G^{\mathrm{\prime }}(t)=(R-\widehat{R}+\{N,F\}+\{{\chi }_t,F\})\circ {\mathrm{\Phi }}_t=\{{\chi }_t,F\}\circ {\mathrm{\Phi }}_t$. The endpoint values are $G(0)=N+\widehat{R}$ and $G(1)=(N+R)\circ \mathrm{\Phi }$. Integrating and then adding $(P-R)\circ \mathrm{\Phi }$ proves the exact identity in clause 3 on the full time-one domain.

Parameter derivatives. If $\nu \in O_{h\mathrm{/}2}$, every coordinate disc of radius $h\mathrm{/}2$ about $\nu$ lies in $O_h$: this follows directly from the triangle inequality for distance to the label set. Cauchy's estimate [F1], applied to each component of $W{X}_F$ on the same phase domain, therefore gives $\mathrm{\mid }W{\mathrm{\partial }}_{{\nu }_j}{X}_F\mathrm{\mid }\le CE\mathrm{/}h$. The derivative $b_t={\mathrm{\partial }}_{{\nu }_j}{\mathrm{\Phi }}_t$ solves $b_t^{\mathrm{\prime }}=D{X}_F({\mathrm{\Phi }}_t)b_t+{\mathrm{\partial }}_{{\nu }_j}{X}_F({\mathrm{\Phi }}_t)$, $b_0=0$, by [F4]. Multiplying by $W$ and using the bounds already proved, Gronwall gives $\mathrm{\mid }W{b}_t\mathrm{\mid }\le (CE\mathrm{/}h)e^L$ for $t\le 1$, which is the required estimate after increasing $C$.

Periodicity, reality, and canonicity. The Fourier polynomial $F$ is periodic, and consequently $X_F(I,\vartheta +2\pi k,\nu )=X_F(I,\vartheta ,\nu )$. Translating a trajectory by $(0,2\pi k)$ and invoking uniqueness proves equivariance. On real points the data satisfy $R_{-k}=\overline{R_k}$ and the real frequencies give $F_{-k}=\overline{F_k}$; hence $F$ and its field are real there, and uniqueness preserves reality. By [F5] the real flow is canonical. The same symplectic identity holds on the complex domain directly: in the constant matrix representing $\omega$, the Hamiltonian Jacobian satisfies $(D{X}_F{)}^{\mathsf{T}}J+JD{X}_F=0$, by symmetry of the Hessian of $F$; differentiating $(D{\mathrm{\Phi }}_t{)}^{\mathsf{T}}J(D{\mathrm{\Phi }}_t)$ using step 3.1 therefore gives zero derivative and initial value $J$. This proves canonicity throughout the complex domain as well.

Remainder on the shrinking action domain. On $D_{\eta r,s-8\sigma ,h}$ the trajectories lie in $D_{2\eta r,s-7\sigma ,h}$ by step 2.1. Thus [F2] gives $\mathrm{\mid }(P-R)\circ \mathrm{\Phi }\mathrm{\mid }\le C[{\eta }^2+{\sigma }^{-n}{e}^{-\sigma K\mathrm{/}2}]\epsilon$. Moreover, on every trajectory, $\mathrm{\mid }{R}_{\vartheta }\mathrm{\mid }\le C{\sigma }^{-n-1}\epsilon$, $\mathrm{\mid }{R}_I\mathrm{\mid }\le C{\sigma }^{-n}\epsilon \mathrm{/}r$, and $\mathrm{\mid }{\widehat{R}}_I\mathrm{\mid }\le C\epsilon \mathrm{/}r$ because $[R]=[Q]$ and each action-Taylor coefficient at zero is bounded by $\epsilon \mathrm{/}r$ by Cauchy. Since ${\widehat{R}}_{\vartheta }=0$, these imply $\mathrm{\mid }{\chi }_{t,\vartheta }\mathrm{\mid }\le C{\sigma }^{-n-1}\epsilon$ and $\mathrm{\mid }{\chi }_{t,I}\mathrm{\mid }\le C{\sigma }^{-n}\epsilon \mathrm{/}r$. Step 1.1 gives $\mathrm{\mid }{F}_I\mathrm{\mid }\le CE{\sigma }^{n+2}$ and $\mathrm{\mid }{F}_{\vartheta }\mathrm{\mid }\le CEr{\sigma }^{n+1}$. Therefore $\mathrm{\mid }\{{\chi }_t,F\}\mathrm{\mid }\le \mathrm{\mid }{\chi }_{t,\vartheta }{\mathrm{\mid }}_1\mathrm{\mid }{F}_I{\mathrm{\mid }}_{\mathrm{\infty }}+\mathrm{\mid }{\chi }_{t,I}{\mathrm{\mid }}_1\mathrm{\mid }{F}_{\vartheta }{\mathrm{\mid }}_{\mathrm{\infty }}\le CE\epsilon \sigma \le CE\epsilon$. Integrating this bound in the identity of step 3.3 proves clause 3.

Mean frequency shift. Since the cutoff preserves the zero Fourier mode, $\widehat{R}=[Q]$; hence $v_j=[{\mathrm{\partial }}_{I_j}P(0,\cdot ,\nu )]$. Cauchy's estimate at the centre of $B_r$ gives $\mathrm{\mid }{\mathrm{\partial }}_{I_j}P(0,\vartheta ,\nu )\mathrm{\mid }\le \epsilon \mathrm{/}r$ (take radii tending upward to $r$), and taking a real angular mean does not increase this bound. Thus $\mathrm{\mid }v\mathrm{\mid }\le C\epsilon \mathrm{/}r=B$ on $O_h$, without a Fourier-loss factor. The mean is holomorphic in $\nu$ and real on real points. On $O_{h\mathrm{/}2}$, coordinate Cauchy estimates with radius $h\mathrm{/}2$ bound each entry of $Dv$ by $2B\mathrm{/}h$; the induced maximum-norm operator bound is consequently $2nB\mathrm{/}h$. Finally $\epsilon \mathrm{/}r=\alpha E{\sigma }^{\mu }$ by definition. This proves clause 4 and completes the step.