Theorems.KAMMainCorrected / KAMMainCorrected.poincareTreshchevPersistence

For every $n,m\in\mathbb N$ and every model $M$ of those dimensions, the theorem asserts the following. Here a model consists of an integer unimodular resonance frame $K_0$ whose last $m$ columns generate a specified resonance subgroup, a parameter set $G\subseteq\mathbb R^{n+m}$, a real function $N:G\text{-ambient}\to\mathbb R$, an arbitrary external frequency family $\omega:\mathbb Z\to\mathbb R$, shell-indexed complex Fourier coefficients $P_{A,k,\ell,r}(y,\epsilon)$, an approximation function $\Delta$, a real divisor constant $\gamma$, and a spatial structure $S$. The assertion is universal over every possible proof package $h$ satisfying all of these conditions:

• $n>0$ and $m>0$; the resonance subgroup is primitive; $G$ is closed and bounded; and $N$ is real analytic on a neighborhood of $G$.
• Writing $\nabla N$ for the coordinate Fréchet derivative, $K_2$ for the last $m$ columns of $K_0$, and $K_1$ for the first $n$, the set of $y\in G$ such that $K_2^T\nabla N(y)=0$ and for which there exists a circle angle $\phi\in\mathbb T^m$ satisfying

$$\nabla_\phi h_0(\phi,y)=0,\qquad \det D_\phi(\nabla_\phi h_0)(\phi,y)\ne0$$

is nonempty. The averaged function here is the explicit real Fourier sum

$$h_0(\phi,y)=\operatorname{Re}\sum_{A,r} P_{A,0,0,r}(y,0)e^{i\langle r,\widetilde\phi\rangle},$$

evaluated at the standard real representative $\widetilde\phi$ of the circle angle.

• The package supplies $r_*>0$. For every $0<\xi\le r_*$, form the reduced-frequency image

$$\mathcal O=K_1^T\nabla N\bigl(\{y\in G:K_2^T\nabla N(y)=0 \text{ and some associated nondegenerate }\phi\text{ exists}\}\bigr),$$

retain those $\Omega\in\mathcal O$ with

$$\xi\le\operatorname{infDist}(\Omega,\operatorname{frontier}\mathcal O),$$

and pull this trim back to the preceding resonant set. That pulled-back set is measurable and compact, while the reduced trim has strictly positive $n$-dimensional Lebesgue volume. At every retained $y$, the full derivative $D(\nabla N)(y):\mathbb R^{n+m}\to\mathbb R^{n+m}$ is injective. The map $y\mapsto K_1^T\nabla N(y)$ is analytic near the pulled-back trim, maps it bijectively onto the reduced trim, has an analytic two-sided inverse on those sets, and admits one lower Lipschitz constant $\tau>0$ on the source. Every $\phi$ that is averaged-critical at every retained $y$, not only one chosen critical point, has nonzero averaged Hessian determinant.

• The divisor constant satisfies $\gamma>0$, and every nonzero finitely supported integer mode $k$ whose support lies in an allowed spatial shell obeys

$$|\langle k,\omega\rangle| \ge \frac{\gamma}{\Delta([k]_S)\Delta(|k|_1)}.$$

The spatial structure covers every integer site, so all unit modes occur in this quantifier.

• There exist positive angular, spatial, and complex-parameter widths and nonnegative shell bounds $B_A$, zero off the allowed spatial shells, with $\sum_A B_Ae^{s[A]_S}$ summable. Coefficients vanish unless their external support lies in their indexing allowed shell, are complex differentiable on the corresponding complex neighborhoods of $G$ and $[-1,1]$, and satisfy the uniform exponential Fourier bound $|P_{A,\mu}|e^{a|\mu|_1}\le B_A$. Each shell’s coefficient norms are summable; coefficients satisfy conjugate symmetry on real $G\times[-1,1]$; the actually summed perturbation is real analytic in all lifted finite angle, parameter, and perturbation variables for every real external lift; and the actually summed averaged potential above is analytic near $\mathbb R^m\times G$.

For every such package $h$ and every real $0<\xi\le r_*$, there must exist $\epsilon_0\in\mathbb R$ with $0<\epsilon_0\le1$, a function $r:\mathbb R\to\mathbb R$, and a family of parameter sets $\Lambda_\epsilon\subseteq\mathbb R^{n+m}$ satisfying the following. The rate obeys $r(\epsilon)\ge0$ for every $0<\epsilon\le\epsilon_0$ and tends to $0$ as $\epsilon\to0$ through positive values. For every such $\epsilon$, $\Lambda_\epsilon$ is closed in the ambient space, measurable, nonempty, and contained in the $\xi$-trimmed nondegenerate resonant set. Moreover,

$$\operatorname{vol}_n\!\left( (K_1^T\nabla N)\bigl( \operatorname{trim}_\xi(M)\setminus\Lambda_\epsilon \bigr)\right)\longrightarrow0 \quad\text{as }\epsilon\to0^+.$$

This is the volume of the reduced-frequency image of the removed parameter set, not the ambient $(n+m)$-dimensional volume of that set; the statement gives only the limit, not a pointwise quantitative estimate for each $\epsilon$.

For every $0<\epsilon\le\epsilon_0$, every $y\in\Lambda_\epsilon$, and every $\phi\in\mathbb T^m$, if the explicit averaged gradient at $(\phi,y)$ is zero and its explicit Hessian determinant is nonzero, then there exists a map

$$\iota:\bigl(\mathbb T^{\mathbb Z}\times\mathbb T^n\bigr) \longrightarrow \bigl((\mathbb T^{\mathbb Z}\times\ell^1_{\mathbb R}) \times(\mathbb T^{n+m}\times\mathbb R^{n+m})\bigr)$$

with all of these properties:

• $\iota$ is a topological embedding.
• There exist widths $a,s,w$ with $a>0$, $s>0$, and $0\le w<s$ such that every external-angle circle coordinate, every finite-angle circle coordinate, and every finite action coordinate has an actual shell-indexed Fourier expansion with external and internal characters, allowed-shell support, absolute mode summability, bounds $|c_{A,\mu}|e^{a|\mu|_1}\le B_A$, and spatially weighted shell-bound summability. The full external $\ell^1$-action component has one complex $\ell^1$-valued expansion whose coefficients and output are summable with weight $e^{w[j]_S}$, whose weighted norms are globally summable over shell modes, and whose Banach-valued Fourier series converges pointwise to the complexification of the real output.
• Let the standard torus be

$$\iota_0(\theta,\psi)= ((\theta,0),(K_0^{-T}(\psi,\phi),y)),$$

where the code uses the integer adjugate formula. There exists a local coordinate transformation $F$ between open subsets of suspended phase space, with continuous two-sided inverse on those subsets, whose source contains the whole range of $\iota_0$, which fixes every external angle, and for which $\iota=F\circ\iota_0$. For every source point and every two cylinder directions—directions with finitely supported external-angle part, arbitrary $\ell^1$ external-action part, and arbitrary finite components—the relevant coordinate curves of $F$ are differentiable, the external series of the two output tangents is summable, and

$$\sum_j(d\theta_j\wedge dJ_j)+\sum_i(dx_i\wedge dy_i)$$

has the same value on the output tangents as on the input directions.

• Put $\Omega=K_1^T\nabla N(y)$ and translate the source torus rigidly by $(\omega,\Omega)$. At every source point $q$ and every time $t$, with $z=\iota(T_tq)$, the external pairing $\sum_j\omega_jJ_j$ is summable, the raw external action velocity belongs to $\ell^1$, every one-coordinate external-angle derivative of the Hamiltonian exists, and the lifted Hamiltonian is differentiable in all finite action and angle variables at $z$. Every external-angle circle coordinate of $s\mapsto\iota(T_sq)$, its full external-action component, every finite-angle circle coordinate, and its finite-action component has derivative at $s=t$ equal to the corresponding component of the explicit Hamiltonian vector field of

$$\mathcal H_\epsilon(\theta,J,x,y) =\sum_j\omega_jJ_j+N(y)+\epsilon P(\theta,x,y,\epsilon).$$

Thus invariance is pointwise in every coordinate and for all $q,t$, rather than almost everywhere.

• For every torus point $q$, every external output angle lies within $r(\epsilon)$ of $q$’s external angle; the norm of the full external-action output is at most $r(\epsilon)$; every finite output angle lies within $r(\epsilon)$ of the corresponding angle of $K_0^{-T}(q_{\rm int},\phi)$; and every finite output action coordinate differs from $y$ by at most $r(\epsilon)$.

The threshold, rate, and family $\Lambda$ may depend on $M$, the entire hypothesis witness, and $\xi$; the embedding may further depend on $\epsilon,y,\phi$. Each retained $y$ belongs to the nondegenerate resonant set and therefore has at least one associated nondegenerate $\phi$, so the embedding implication cannot be false-premised for every $\phi$ at that $y$. Nothing is asserted at $\epsilon=0$, for negative $\epsilon$, or above $\epsilon_0$. The theorem takes an arbitrary model and concludes a proposition universally quantified over CorrectedHypotheses witnesses; consequently, if no such witness exists for $M$, the theorem is vacuously true for that model. If a witness does exist, its positive trim radius ensures that the subsequent $\xi$-quantifier has admissible values.