ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Finite-dimensional vector spaces form a fusion category

Example

For a field $k$ , the category ${V e c t}_{k}^{f d}$ is a fusion category.

Facts & Assumptions

Given: A field $k$ .

[F1]

$k$ -modules are monoidal under $\otimes_{k}$ with unit $k$ (Modules over a commutative ring form a monoidal category).

[F2]

Every finite-dimensional $k$ -vector space is rigid (Finite-dimensional vector spaces are rigid).

[F3]

Fusion means finite semisimple tensor category (Fusion and multifusion categories).

Verification

technique · direct

1.1F1F2given

By [F1] and [F2], the finite-dimensional subcategory is a rigid $k$ -linear monoidal category.

2.1step 1.1F3

Every finite-dimensional vector space is a finite direct sum of copies of $k$ , so $k$ is the only simple isomorphism class and the category is finite semisimple. Its unit has endomorphism algebra $k$ .

3.1step 2.1F3∎

These clauses are exactly those of [F3], so ${V e c t}_{k}^{f d}$ is fusion.

Depends on

Used by

Dependency tree · two levels

16 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

Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Example 4.1.2 (standard reference, not scraped)