TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

Ribbon evaluation and framed anyon links

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $C$ be a unitary ribbon fusion category, let $X$ be a simple color, and let $β \in B_{n}$ and $β^{'} \in B_{m}$ with $n, m \geq 1$ . If their blackboard-framed, $X$ -colored closures are isotopic as framed oriented links, then their root ribbon evaluations satisfy $t_{n} (β) = t_{m} (β^{'})$ . No modularity assumption is needed. Ordinary Markov stabilization is not a framed isotopy because it changes the framing.

Facts & Assumptions

Given: ${A C}_{ω}$ , a unitary ribbon fusion category $C$ , a simple object $X$ , and framed-isotopic blackboard closures of $β$ and $β^{'}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ asserts that every countable family of nonempty sets has a choice function.

[F1]

Unitary ribbon data: the underlying category has a ribbon structure, so it is a ribbon category in the root sense.

[F2]

The ribbon evaluation is an invariant of framed colored links: under ${A C}_{ω}$ , in any ribbon category, framed-isotopic blackboard closures of braids with the same color have equal root ribbon evaluations.

[F3]

The ribbon evaluation of an $X$ -colored closed braid: for $n \geq 1$ , $t_{n} (β)$ is the left categorical trace of the pivotal comparison composed with the braid action; this is the root ribbon-evaluation notation.

Proof

1.1F1F2given

By [F1], $C$ is a ribbon category with the same root duality, braiding, and twist structure required by [F2]. The color $X$ and the braids $β, β^{'}$ meet [F2]'s common-color and positive-strand hypotheses.

2.1A1F2F3givenstep 1.1∎

The assumed ${A C}_{ω}$ supplies [F2]'s choice hypothesis by [A1]. Since the blackboard-framed closures are framed-isotopic, [F2] gives $t_{n} (β) = t_{m} (β^{'})$ ; by [F3] these are exactly the root ribbon evaluations named in the Statement. The same root theorem states that ordinary Markov stabilization changes framing, and it uses no modularity hypothesis.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, author final text (standard reference, not scraped)