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.

Free particle Hamiltonian flow

Example

Assume ${A C}_{ω}$ . On $T^{*} R^{d}$ let $H (q, p) = ∣ p ∣^{2} / (2 m)$ with $m > 0$ . The trajectory from $(q_{0}, p_{0})$ is

$q (t) = q_{0} + \frac{t}{m} p_{0}, p (t) = p_{0},$

and is defined for every $t \in R$ .

Facts & Assumptions

Given: The mass $m > 0$ , the initial point, and the library cotangent and Hamiltonian sign conventions.

[F1]

Under the stated choice assumption, Hamilton's equations are $\dot{q} = H_{p}$ and $\dot{p} = - H_{q}$ . Hamilton equations in canonical cotangent coordinates.

Verification

technique · direct

1.1

Here $H_{p} = p / m$ and $H_{q} = 0$ , so [F1] gives $\dot{q} = p / m$ and $\dot{p} = 0$ .

F1givenalgebra

2.1

The second equation gives $p (t) = p_{0}$ ; substituting into the first and integrating gives $q (t) = q_{0} + t p_{0} / m$ . These formulas exist for all real time and directly satisfy the initial condition.

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)