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.

Poisson brackets in canonical coordinates

Example

Assume ${A C}_{ω}$ . In canonical cotangent coordinates,

${q^{i}, q^{j}} = 0, {p_{i}, p_{j}} = 0, {q^{i}, p_{j}} = δ_{j}^{i} .$

Consequently ${p_{j}, q^{i}} = - δ_{j}^{i}$ .

Facts & Assumptions

Given: The library convention ${F, G} = \sum_{k} (F_{q^{k}} G_{p_{k}} - F_{p_{k}} G_{q^{k}})$ .

[F1]

That coordinate formula follows from $ω_{c a n} = \sum_{k} d q^{k} \land d p_{k}$ and $ι_{X_{H}} ω = d H$ . Coordinate formula for the Poisson bracket.

Verification

technique · direct

1.1

The only nonzero derivatives are $\partial q^{i} / \partial q^{k} = δ_{k}^{i}$ and $\partial p_{j} / \partial p_{k} = δ_{j k}$ . Substitution in [F1] gives zero for the $q$ -- $q$ and $p$ -- $p$ brackets and $\sum_{k} δ_{k}^{i} δ_{j k} = δ_{j}^{i}$ for ${q^{i}, p_{j}}$ .

F1algebra

2.1

Skew-symmetry, also visible by reversing the two terms in the coordinate formula, gives ${p_{j}, q^{i}} = - δ_{j}^{i}$ .

F1step 1.1∎

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)