Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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 zero-weight singular central character

Example

The weight −ρ is fixed by the dot action of every Weyl-group element: for every w∈W,

w⋅(−ρ)=−ρ.

Hence the central character of −ρ has full Weyl-group stabilizer and is singular.

Facts & Assumptions

Given: A complex semisimple Lie algebra with Weyl group W and the weight −ρ.

Verification

technique · direct
1.1

By definition of the dot action, w⋅(−ρ)=w((−ρ)+ρ)−ρ=w(0)−ρ=−ρ for every w∈W.

givenalgebra
2.1

Therefore the entire Weyl group stabilizes −ρ, and Central characters are dot-Weyl orbits shows that its central character is singular with full Weyl-group stabilizer.

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