Alphabeta Math
False statementConstruction: 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.

Centerless implies semisimple

Statement refuted

A centerless finite-dimensional Lie algebra is semisimple.

Facts & Assumptions

Given: A characteristic-zero field and the displayed affine algebra.

[L1]

Semisimplicity means vanishing solvable radical (Simple, semisimple, and reductive Lie algebras).

Counterexample

technique · the two-dimensional affine algebra
1.1

Over any field, let a have basis x,y and bracket [x,y]=y. For z=ax+by, the equations [z,x]=−by=0 and [z,y]=ay=0 give a=b=0. Thus Z(a)=0.

givenalgebra
2.1

Its derived algebra is ky and the next derived algebra is zero, so a is nonzero and solvable. Therefore its radical is all of a, not zero, and it is not semisimple by [L1]. This explicit witness has dimension two and refutes the implication even in characteristic zero.

L1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources