Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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 sl2 Verma embedding chain

Example

For sl2 and m∈Z≥0, the dot reflection sends m to −m−2, and the singular vector fm+1vm gives M(−m−2)↪M(m). At m=0 this is M(−2)↪M(0); when m∉Z≥0 there is no nontrivial reflection embedding from M(s⋅m) into M(m).

Facts & Assumptions

Given: The simple-reflection embedding Simple-reflection embeddings of Verma modules and the BGG criterion The BGG criterion for homomorphisms between Verma modules.

Verification

technique · direct
1.1

Since ρ=1 and s⋅m=−m−2, the relevant pairing is m+1, which is positive integral exactly when m∈Z≥0.

givenalgebra
2.1

The embedding proposition gives the displayed inclusion, including m=0; if the pairing is not positive integral, the BGG criterion rules out that reflection map.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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