RemarkRemark: AI-adaptedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Scope of the action-angle and complete-integrability items: hypotheses, choice, singular and noncompact fibres, and the published proof gap

Remarks

Hypotheses actually used. Action-angle coordinates on a saturated proper submersion neighbourhood of a compact connected regular fibre, with the two-form primitive supplied locally assumes ${A C}_{ω}$ , an involutive system of $n$ independent smooth integrals, a compact connected regular fibre, and a saturated neighbourhood over a ball of regular values on which the integral map is a proper submersion with connected fibres. Compactness supplies complete flows on the fibre (through Every smooth vector field on a compact manifold is complete); connectedness makes the orbit of the abelian action all of the fibre; properness keeps the nearby fibres compact and the period structure locally uniform. The assumption ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) is inherited through the compact-flow corollary and the lattice and torus suppliers, and is also used directly to select the counterexample sequence of periods in step 4.1. None of these conclusions follows from independence alone.
What is not claimed. No statement is made about singular fibres, about noncompact or disconnected fibres, or about a global action-angle chart on all of $M$ ; the theorem is local over a shrunken ball of regular values, in accordance with Completely integrable Hamiltonian system and Action and angle coordinates. A uniform parameter-dependent action-angle chart, as used by adiabatic estimates, is an additional hypothesis and is not a consequence of one fibre's chart.
The published proof gap and the local repair. Liouville–Arnold action–angle theorem states the same conclusion. Its proof invokes the one-form Poincare lemma Poincare's lemma on a star-shaped domain: every closed C1 field is exact for the closed two-form $σ^{*} ω$ in its step 5.2; a closed two-form needs a two-form primitive, which is what A closed smooth two-form on a star-shaped domain has the radial primitive $α_{x} (v) = \int_{0}^{1} t τ_{t x} (x, v) d t$ supplies. The local theorem's step 3.2 uses that primitive; the published item is not consumed as a premise in the local proof, and no published or imported item is edited. The sound published suppliers consumed by the local proof are Commuting Hamiltonian vector fields integrate to a local $R^{n}$ -action, Stabilizer of the $R^{n}$ -action on a compact connected regular fibre is a full lattice, Compact connected regular fibres are tori, Cartan's magic formula, Local normal form for submersions, The smooth inverse function theorem on manifolds and Poincare's lemma on a star-shaped domain: every closed C1 field is exact (the last only for one-forms).
Downstream motion. Once action-angle coordinates exist, the linear motion $I^{'} = 0$ , $θ^{'} = \nabla h (I)$ for a fibre-constant Hamiltonian is the published Motion of a completely integrable Hamiltonian is linear on invariant tori, and the period-one to radian conversion is Period-one to radian action-angle rescaling: $θ^{r a d} = 2 π θ$ and $I^{r a d} = I / (2 π)$ preserve $ω$ . The scope remark records the division of labour for the reviewing steps; it asserts no new mathematics.

Scope of the action-angle and complete-integrability items: hypotheses, choice, singular and noncompact fibres, and the published proof gap

Remarks

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