ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 rank-two multiplicative Bernstein relation

Example

For $n = 2$ , the Bernstein cross-relations and quadratic relation give $T_{1} X_{1} - X_{2} T_{1} = - (Q - 1) X_{2}, T_{1} X_{2} - X_{1} T_{1} = (Q - 1) X_{2}, T_{1} X_{1} T_{1} = Q X_{2} .$

Facts & Assumptions

Given: The universal rank-two presentation with $Q$ a unit.

[F1]

The type-A weight lattice, Laurent ring, and divided differences: $D_{1} (f) = (f - s_{1} (f)) / (1 - X_{1} / X_{2})$ .

[F2]

The multiplicative Bernstein presentation of the affine Hecke algebra of type A: $T_{1} f - s_{1} (f) T_{1} = (Q - 1) D_{1} (f)$ and $T_{1}^{2} = (Q - 1) T_{1} + Q$ .

Verification

Given: $n = 2$ , $s_{1} (X_{1}) = X_{2}$ , $s_{1} (X_{2}) = X_{1}$ , and the displayed universal presentation.

1.1givenF1algebra

The divided-difference formula gives $D_{1} (X_{1}) = (X_{1} - X_{2}) / (1 - X_{1} / X_{2}) = - X_{2}$ and $D_{1} (X_{2}) = (X_{2} - X_{1}) / (1 - X_{1} / X_{2}) = X_{2}$ .

2.1F1F2step 1.1

Substituting these two values into [F2] gives $T_{1} X_{1} - X_{2} T_{1} = - (Q - 1) X_{2}$ and $T_{1} X_{2} - X_{1} T_{1} = (Q - 1) X_{2}$ .

3.1F2step 2.1algebra∎

Multiply the first equality on the right by $T_{1}$ and substitute the quadratic relation to get $T_{1} X_{1} T_{1} = X_{2} T_{1}^{2} - (Q - 1) X_{2} T_{1} = X_{2} ((Q - 1) T_{1} + Q) - (Q - 1) X_{2} T_{1} = Q X_{2}$ .

Depends on

Used by

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Dependency tree · two levels

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