ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 star-expansion of the four-vertex path contains an induced five-cycle

Example

In the star-expansion of $P_{4}$ , the vertices $v, a_{1}, b_{1}, b_{2}, a_{2}$ induce a five-cycle.

Facts & Assumptions

Given: The star-expansion of $P_{4}$ with root $v$ , teeth $a_{1}, a_{2}$ , and path vertices $b_{1}, b_{2}$ .

[L1]

The only edges using the new vertices are $v a_{i}$ and $a_{i} b_{i}$ (The star-expansion of a graph).

Verification

technique · direct finite check

1.1

Among the chosen five vertices, the edges are exactly $v a_{1}, a_{1} b_{1}, b_{1} b_{2}, b_{2} a_{2}, a_{2} v$ .

2.1

No chord is present: $a_{1}$ and $a_{2}$ are nonadjacent, neither tooth is adjacent to the wrong path vertex, and the root is not adjacent to $b_{1}$ or $b_{2}$ . Hence the chosen vertices induce a five-cycle.

step 1.1L1∎

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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 3 discussion (standard reference, not scraped)