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 canonical-methods items: locality and topology of generating functions, caustics, and the E7 claims delivered by published root items

Remarks

Locality and topology. Generating functions exist locally under the mixed-invertibility hypotheses of Generating functions of the first and second kind for a local canonical map and A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function. Globally they may fail: The symplectic shift of the cylinder changes $λ$ by the closed, non-exact form $a d θ$ and has no global generating function exhibits a symplectic shift whose $Φ^{*} λ - λ = a d θ$ is closed but not exact, so no single-valued global generating function exists, although one exists on every chart. The distinction between the canonical map itself (defined at finite regularity in Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts, with the smooth intrinsic case in Symplectomorphisms, local symplectomorphisms, and symplectic embeddings and Canonical transformation) and the maps admitting global generating functions is a genuine topological one, not a technicality.
Caustics. The hypotheses $\det S_{q Q} \neq 0$ and $\det S_{q α} \neq 0$ are local no-caustic conditions: A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function states the converse direction, and A complete integral produces Hamilton characteristics, the Jacobi constants, a local converse and canonical $(β, α)$ labels shows that the Jacobi constants form local canonical coordinates exactly where the mixed derivative is invertible. A complete integral is a family of solutions with $n$ essential parameters, defined in The Hamilton-Jacobi equation and a complete integral of it; it is not a general solution, and no global Hamilton-Jacobi existence across a caustic is claimed.
Locality of the flow statements. The derivative of the Hamiltonian flow satisfies $Y^{T} J Y = J$ and $\det Y = 1$ and A Hamiltonian local flow on a phase domain preserves Lebesgue measure on its flow domain are stated on the actual flow domain of a $C^{2}$ autonomous or time-dependent Hamiltonian and assume no completeness; the nonautonomous evolution uses an initial time and a final time, with inverse given by reversing those endpoints. The published Hamiltonian flows preserve the symplectic form and Liouville volume preservation give the corresponding manifold-level statements.
Claims of this section delivered by published root items (used by reference, not re-authored). The tautological one-form and the canonical cotangent two-form, with the latter closed and nondegenerate: Tautological one-form on a cotangent bundle, The canonical cotangent two-form is symplectic. Hamiltonian vector fields, the Hamilton equations in canonical cotangent coordinates and the Poisson bracket with its coordinate formula, bilinearity, skew symmetry, derivation property and Jacobi identity: Hamiltonian vector field and Hamiltonian function, Hamiltonian vector fields exist uniquely for smooth functions, Hamilton equations in canonical cotangent coordinates, Poisson bracket on a symplectic manifold, Coordinate formula for the Poisson bracket, The Poisson bracket is bilinear, skew, and a derivation in each entry, The Poisson bracket satisfies the Jacobi identity. The smooth definitions of canonical maps and the manifold-level conservation statements are Canonical transformation, Symplectomorphisms, local symplectomorphisms, and symplectic embeddings, Hamiltonian flows preserve the symplectic form and Liouville volume preservation. This item records that division of labour for the reviewing steps; it asserts no new mathematics.

Scope of the canonical-methods items: locality and topology of generating functions, caustics, and the E7 claims delivered by published root items

Remarks

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