False statementConstruction: AI-adaptedVerification: 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.

The exponential map of a Lie group is a group homomorphism

Statement refuted

The exponential map of a Lie group is a group homomorphism from its additive Lie algebra.

Facts & Assumptions

Given: The upper-unitriangular real $3$ -by- $3$ matrix group and $X = E_{01}$ , $Y = E_{12}$ .

[F1]

Matrix units have their standard entries, and the row-by-column product therefore gives $E_{i j} E_{k l} = δ_{j k} E_{i l}$ . Matrix units $E_{i j}$ and the Kronecker delta. Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes.

[F2]

Matrix multiplication and the identity matrix have their usual entrywise meaning. Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes.

[F3]

The scalar exponential series converges absolutely. The exponential series converges absolutely for every real argument.

Refutation

technique · counterexample

1.1

By [F1], $X^{2} = Y^{2} = 0$ , $X Y = E_{02}$ , $Y X = 0$ , and $(X + Y)^{2} = E_{02}$ while $(X + Y)^{3} = 0$ . The finite matrix exponential series therefore gives $e^{X} = I + X$ , $e^{Y} = I + Y$ , and $e^{X + Y} = I + X + Y + \frac{1}{2} E_{02}$ .

F1F2F3algebra

2.1

Direct multiplication gives $e^{X} e^{Y} = (I + X) (I + Y) = I + X + Y + E_{02}$ , which differs from $e^{X + Y}$ in its $(0, 2)$ entry. Thus exponential does not preserve addition.

F1F2step 1.1algebra

3.1

The witness is a nonempty connected three-dimensional matrix Lie group. No zero- or one-dimensional group can exhibit this particular noncommutative failure. No metric, nondegeneracy, interval, endpoint, choice principle, or biconditional occurs.

F1F2F3step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Michael Müger, Notes on the Baker-Campbell-Hausdorff-Dynkin theorem (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)