False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Liouville–Arnold gives global action–angle coordinates on the entire manifold

Statement refuted

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through Liouville–Arnold action–angle theorem and Period-lattice monodromy obstructs global action–angle coordinates; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

Liouville–Arnold gives one global action–angle coordinate system on the whole phase space of every completely integrable system.

Facts & Assumptions

Given: ${A C}_{ω}$ , the proposed global conclusion.

[A1]

${A C}_{ω}$ is countable choice and is required here through Liouville–Arnold action–angle theorem and Period-lattice monodromy obstructs global action–angle coordinates; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

Liouville–Arnold is a local theorem near a compact connected regular fibre and assumes a regular locally proper fibration. Liouville–Arnold action–angle theorem.

[F2]

Nontrivial period-lattice monodromy forbids global action–angle coordinates. Period-lattice monodromy obstructs global action–angle coordinates.

Refutation

technique · direct

1.1

The spherical pendulum has a regular torus bundle around its focus--focus critical value whose period basis returns around a loop by the nonidentity matrix $(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array})$ , as computed in the cited Martynchuk--Broer--Efstathiou source.

given

2.1

By [F2], this system has no global action–angle coordinates on that regular-value region, while [F1] still supplies charts near each regular torus. Singular fibres also lie outside [F1]. Hence the claimed global conclusion is false.

A1F1F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)
N. Martynchuk, H. W. Broer, and K. Efstathiou, Hamiltonian Monodromy and Morse Theory (standard reference, not scraped)