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

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.

Harmonic oscillator and elliptic phase curves

Example

Assume ${A C}_{ω}$ . For $m, Ω > 0$ , the one-dimensional harmonic oscillator

$H (q, p) = \frac{p^{2}}{2 m} + \frac{m Ω^{2} q^{2}}{2}$

has period $2 π / Ω$ away from the equilibrium. Every positive-energy phase curve is the ellipse $p^{2} / (2 m E) + m Ω^{2} q^{2} / (2 E) = 1$ .

Facts & Assumptions

Given: Positive $m, Ω$ and canonical coordinates on $T^{*} R$ .

[F1]

Hamilton's equations hold under the stated choice assumption. Hamilton equations in canonical cotangent coordinates.

Verification

technique · direct

1.1

By [F1], $\dot{q} = p / m$ and $\dot{p} = - m Ω^{2} q$ , hence $\ddot{q} + Ω^{2} q = 0$ . Thus $q (t) = A \cos (Ω t) + B \sin (Ω t)$ and $p (t) = m \dot{q} (t)$ . Every nonconstant solution has period $2 π / Ω$ .

F1algebra

2.1

Conservation follows directly from $\frac{d}{d t} H = (p / m) \dot{p} + m Ω^{2} q \dot{q} = 0$ . Setting the constant value to $E > 0$ and dividing its level equation by $E$ gives the displayed ellipse. For $E = 0$ , positivity forces $(q, p) = (0, 0)$ , the stationary equilibrium.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)