ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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 two-strand braiding in the braid category has infinite order

Example

In the braid category, the braiding on the object $1 \otimes 1 = 2$ has infinite order.

Facts & Assumptions

Given: The braid category and the computation of $B_{2}$ .

[L1]

In the braid category, the braiding on $1 \otimes 1$ is the generator $σ_{1} \in B_{2}$ (The braid category).

[L2]

The group $B_{2}$ is infinite cyclic, generated by $σ_{1}$ (The two-strand braid group is infinite cyclic).

Verification

technique · direct

1.1

By [L1], the braiding on $2$ is the element $σ_{1}$ .

L1given

2.1

By [L2], all nonzero powers of $σ_{1}$ are distinct from the identity. Hence the braiding on $2$ has infinite order.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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 Muger, Tensor Categories: A Selective Guided Tour, Section 4 (standard reference, not scraped)