ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Reducible and generic sl2 Verma modules

Example

$M (λ)$ for ${s l}_{2}$ is reducible exactly when $λ = m \in Z_{\geq 0}$ ; then $f^{m + 1} v_{m}$ is a singular vector. When $λ \notin Z_{\geq 0}$ it is simple.

Facts & Assumptions

Given: The criterion The Verma irreducibility criterion from Shapovalov determinants and the action formula The sl2 Verma action in the PBW basis.

Verification

technique · direct

1.1

The only positive root has coroot pairing $⟨ λ + ρ, α^{\lor} ⟩ = λ + 1$ , so the criterion says reducible precisely for $λ \in Z_{\geq 0}$ .

givenalgebra

2.1

For $λ = m$ , the action formula gives $e f^{m + 1} v_{m} = (m + 1) (m - (m + 1) + 1) f^{m} v_{m} = 0$ ; its weight is below $m$ , so it is the indicated nonzero singular vector.

givenalgebra∎

Depends on

Used by

The finite-dimensional sl2 quotient of a Verma module Example

Dependency tree · two levels

6 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

Mrudul Thatte, Category O: Verma's Thesis, Example 4.3 (standard reference, not scraped)