ExampleConstruction: AI-adaptedVerification: 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.

An upside-down comb with anticonnected blocks creates a $\overline{P_{5}}$

Example

In the mixed-pair obstruction from Claim 5.2.2 of the source, the five vertices $v, u, w, z, a_{i}$ induce $\overline{P_{5}}$ .

Facts & Assumptions

Given: Vertices $v, u, w, z, a_{i}$ such that $v u, v w, v z, u w, a_{i} w, a_{i} z \in E (G)$ and $v a_{i}, u a_{i}, u z, w z \notin E (G)$ .

Verification

technique · direct finite check

1.1

Among the five chosen vertices, the only nonedges are $v a_{i}$ , $a_{i} u$ , $u z$ , and $w z$ . All other pairs are adjacent by hypothesis.

given

2.1

The four nonedges from step 1.1 are exactly the edges of the path $v - a_{i} - u - z - w$ . Therefore the induced graph on ${v, u, w, z, a_{i}}$ is the complement of a five-vertex path.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. VII. The five-vertex path, Claim 5.2.2 pattern (standard reference, not scraped)

An upside-down comb with anticonnected blocks creates a P5‾P5​​