TheoremStatement: Literature-sourcedProof: 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.

Grothendieck-ring multiplication is well-defined

Statement

The rule $[X] [Y] = [X \otimes Y]$ is a well-defined unital associative product on $K_{0} (C)$ .

Facts & Assumptions

Given: A tensor category $C$ .

[F1]

Tensoring is exact in each variable (Tensor product in a multitensor category is biexact).

[F2]

$K_{0} (C)$ is the quotient by short-exact-sequence relations (The Grothendieck ring of a tensor category).

Proof

technique · direct

1.1F1F2given

If $0 \to A \to B \to C \to 0$ is exact, then [F1] makes each of its tensors with $X$ exact. Thus $[B \otimes X] = [A \otimes X] + [C \otimes X]$ , and similarly in the other variable.

2.1step 1.1F2∎

Hence the bilinear rule on generators descends through the relations of [F2]. The associator and unitors identify $(X \otimes Y) \otimes Z$ with $X \otimes (Y \otimes Z)$ and $1 \otimes X$ with $X$ , so the descended product is associative with unit $[1]$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Lemma 4.5.1 (standard reference, not scraped)