Alphabeta Math
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.

PBW reordering in sl_2

Example

For the basis e,f,h of sl2 with

[h,e]=2e,[h,f]=−2f,[e,f]=h,

choose the order f<h<e. Then fahbec, with a,b,c≥0, is a PBW basis of U(sl2).

Facts & Assumptions

Given: The displayed Lie algebra and supplied order f<h<e.

[L1]

PBW supplies the ordered-monomial basis (Poincaré–Birkhoff–Witt theorem).

Verification

technique · direct reordering
1.1

The defining enveloping relations give eh=(h−2)e, hf=f(h−2), and ef=fe+h. Each formula replaces an adjacent inversion for f<h<e by an ordered pair plus a shorter term.

givenalgebra
1.2

Every weakly increasing word has all f's first, then all h's, then all e's, hence is uniquely fahbec.

givenalgebra
2.1

Step 1.1 rewrites every word into a linear combination of the forms in step 1.2, and [L1] makes those forms linearly independent, so the reordering result is unique.

step 1.1step 1.2L1∎

Depends on

Used by

Dependency tree · two levels

5 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