Compact connected regular fibres of a completely integrable datum of the presentation are Lagrangian tori with a full period lattice

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with the momentum charts and two-form datum of The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, the Hamiltonian fields of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian and the branch Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions; by clauses 1–2 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ (clause 1 for Cartesian domains and clause 2 for generalized-coordinate domains), each momentum phase domain is a symplectic manifold for the presentation's two-form $\omega$ (Symplectic form and symplectic manifold). Let $W_0\subseteq W$ be a nonempty open phase domain of dimension $2n$ and let $F=(F_1=H{\mathrm{\mid }}_{W_0},\dots ,F_n)$ be a completely integrable datum of the presentation on $W_0$ (Completely integrable data of a declared presentation: $n$ smooth phase functions with $F_1=H$, pairwise in involution and independent on the regular locus). Let $b_0$ be a regular value of the integral map whose regular fibre $N:=F^{-1}(b_0)\subseteq W_0$ is nonempty, compact and connected, and assume there are a saturated open neighbourhood $U\subseteq W_0$ of $N$ and a ball $B\subseteq {\mathbb{R}}^n$ of regular values with $F(U)=B$ such that $\pi :=F{\mathrm{\mid }}_U:U\to B$ is a proper submersion with connected fibres. Then:

1. (Lagrangian fibre) $N$ is a compact connected embedded $n$-dimensional Lagrangian submanifold of $(W_0,\omega )$, and for every $z\in N$ $T_{z}N=\mathrm{span}\{X_{F_1}(z),\dots ,X_{F_n}(z)\},$ the fields being those of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian.
2. (commuting flows and period lattice) The fields $X_{F_1},\dots ,X_{F_n}$ have pairwise commuting flows; these flows combine into a local ${\mathbb{R}}^n$ action on $W_{\mathrm{r}\mathrm{e}\mathrm{g}}$, and on the compact invariant fibre $N$ their restrictions are complete and generate a transitive action of ${\mathbb{R}}^n$ whose stabilizer at every point is one and the same full lattice $\mathrm{\Gamma }\subseteq {\mathbb{R}}^n$, with $N\cong {\mathbb{R}}^n\mathrm{/}\mathrm{\Gamma }$.
3. (torus) $N$ is diffeomorphic to the torus $T^n={\mathbb{R}}^n\mathrm{/}{\mathbb{Z}}^n$.

The statement is conditional on the adopted force branch, the declared presentation, the smooth completely integrable datum and the listed hypotheses on the fibre; it asserts no global product structure of $U$, no action-angle coordinates, no completeness of any flow outside the compact fibre and no empirical content.

• Where each hypothesis is used. Compactness is used twice: for the completeness of the restricted fields and for the full-lattice and torus conclusions; connectedness is used for transitivity; regularity at the base value is used to make $N$ a submanifold and to make the fields independent along it; the proper-submersion neighbourhood with connected fibres is not used by this theorem but is exactly the hypothesis needed by Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H=h(I)$ and linear motion on the tori to globalize the action coordinates over a base ball.

• What is not claimed. No statement is made about the topology of a neighbourhood of $N$ beyond the local action of clause 2; in particular no global trivialization $U\cong B\times T^n$, no global action-angle coordinates on a multiply connected phase domain and no statement about critical or noncompact fibres. The companion page exhibits a compact critical fibre and a noncompact regular fibre of explicit models, showing that compactness and regularity cannot be dropped, and cites the published monodromy obstruction to global action-angle coordinates.