False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

FALSE: every diagram built from the associator and the braiding commutes

Statement

False claim: in every braided monoidal category, every diagram whose edges are built only from associators, unitors, braidings, and their inverses commutes.

Facts & Assumptions

Given: The failure theorem for braided coherence in the symmetric form.

[L1]

There is a braided monoidal category with pairwise distinct canonical endomorphisms $1, c_{X, X}^{2}, c_{X, X}^{4}, \dots$ of one tensor square (Braided coherence fails in the symmetric form).

Refutation

technique · direct

1.1

If the displayed claim were true, then any two canonical endomorphisms of the same tensor word built from associators and braidings would agree.

givenassume-contra

2.1

The morphisms $1, c_{X, X}^{2}, c_{X, X}^{4}, \dots$ from [L1] are all such canonical endomorphisms of $X \otimes X$ , yet [L1] says they are pairwise distinct in a specific braided monoidal category. This contradicts step 1.1.

L1step 1.1contradiction

3.1

Therefore the claim is false.

step 2.1discharge-contradiction∎

Depends on

Braided coherence fails in the symmetric form

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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Remark 8.2.5 (standard reference, not scraped)