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 hexagon checked for cartesian products

Example

For cartesian products, both routes around the first braiding hexagon send $(x, y, z)$ to $(y, z, x)$ .

Facts & Assumptions

Given: A category with finite products and its cartesian swap braiding.

[L1]

The cartesian swap maps form a braiding (The cartesian swap braiding is a symmetry).

Verification

technique · direct

1.1

Write the source of the first hexagon as $X \times (Y \times Z)$ and evaluate the left-hand route on $(x, y, z)$ . The inner swap sends $(x, (y, z))$ to $((y, z), x)$ after the evident rebracketing.

givenL1algebra

2.1

Evaluate the right-hand route: first swap $x$ past $y$ , then swap $x$ past $z$ . The result is again $(y, z, x)$ after the same rebracketing identifications.

step 1.1algebra

3.1

Since the two routes agree on every triple $(x, y, z)$ , the first hexagon commutes in this cartesian case. The second hexagon is checked by the same coordinate calculation with the factors regrouped as $((x, y), z)$ .

L1step 1.1step 2.1∎

Depends on

The cartesian swap braiding is a symmetry

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, Example 8.2.1 (standard reference, not scraped)