CounterexampleConstruction: 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.

At order five, the rank-one form specialization map is not injective

Statement refuted

For every odd root order $ℓ > 2$ , the natural generator map $U_{ζ}^{D K} ({s l}_{2}) \to U_{ζ}^{L} ({s l}_{2})$ is injective.

Facts & Assumptions

Given: The asserted injectivity for all odd $ℓ > 2$ , and the rank-one root forms at a primitive fifth root.

[F1]

The identity-on-generators comparison map is well-defined and kills $E^{ℓ}$ although $E^{ℓ}$ is nonzero in its source, for each odd $ℓ > 2$ (The rank-one De Concini–Kac to Lusztig specialization map is not injective).

Counterexample

1.1givenF1algebra

The value $ℓ = 5$ is an odd integer greater than $2$ , and a primitive fifth root satisfies the root-order hypotheses in the refuted statement.

2.1givenF1step 1.1algebra∎

Apply [F1] at $ℓ = 5$ : the source element $E^{5}$ is nonzero and its image is zero. Therefore the map is not injective, contradicting the asserted universal injectivity.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Pavel Etingof and Mykola Semenyakin, A Brief Introduction to Quantum Groups (standard reference, not scraped)