CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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.

A disconnected element outside every identity-component torus

Statement refuted

Every element of the compact Lie group $O (2)$ lies in a torus contained in its identity component.

Facts & Assumptions

Given: The group $O (2)$ of orthogonal $2 \times 2$ matrices with determinant $\pm 1$ , its identity component $S O (2)$ , and a reflection $r \in O (2)$ .

[L1]

$O (2)$ is a compact Lie group whose identity component is $S O (2)$ , and a torus is a compact connected abelian Lie group; a maximal torus of a group is a torus subgroup maximal under inclusion (Tori and maximal tori).

[L2]

The determinant is a continuous homomorphism $O (2) \to {\pm 1}$ ; every connected subset of ${\pm 1}$ is a singleton, so every connected subgroup of $O (2)$ lies in the kernel $S O (2)$ of the determinant (Tori and maximal tori).

Counterexample

technique · direct

1.1

A reflection is an orthogonal matrix with determinant $- 1$ , so $r \notin S O (2) = O (2)^{0}$ .

1.2

Every torus contained in the identity component is a connected subgroup of $O (2)$ lying in $S O (2)$ ; by [L2] every connected subgroup of $O (2)$ lies in the kernel of the determinant, so no torus of the identity component contains $r$ .

L1L2

2.1

Hence $r$ is an element of the compact Lie group $O (2)$ that lies in no torus of the identity component, refuting the asserted statement and showing that connectedness of the ambient group is a necessary hypothesis for the torus-containment theorems.

L1step 1.1step 1.2∎

Depends on

Tori and maximal tori

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Brian Conrad and Aaron Landesman, Compact Lie Groups (standard reference, not scraped)