ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge 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.

The Heisenberg Lie algebra is two-step nilpotent

Example

Let $h_{3} (k)$ have basis $x, y, z$ , with $[x, y] = z$ and $z$ central. Then $h_{3} (k)$ is nilpotent of class two.

Facts & Assumptions

Given: The displayed three-dimensional Lie algebra over a field $k$ .

[L1]

The nilpotency class is the least $c$ with $γ_{c + 1} = 0$ (Nilpotency class of a Lie algebra).

Verification

technique · direct

1.1

Bilinearity, alternation, and centrality of $z$ show that every bracket is a scalar multiple of $z$ , and the bracket $[x, y] = z$ realizes every such multiple. Hence $γ_{2} (h_{3}) = [h_{3}, h_{3}] = k z \neq 0$ .

givenalgebra

2.1

Since $z$ is central, $γ_{3} (h_{3}) = [h_{3}, k z] = 0$ . Thus the lower central series is $h_{3}, k z, 0$ ; its second term is nonzero and its third is zero, so [L1] gives nilpotency class exactly two. The computation works in every characteristic and uses no choice.

L1step 1.1algebra∎

Depends on

Nilpotency class of a Lie algebra

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Milne, Lie Algebras, Heisenberg example (standard reference, not scraped)