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

Example

Let $P_{4}$ have vertices $b_{1}, b_{2}, b_{3}, b_{4}$ in path order. Its star-expansion has additional vertices $a_{1}, a_{2}, a_{3}, a_{4}, v$ and edge set ${b_{1} b_{2}, b_{2} b_{3}, b_{3} b_{4}} \cup {a_{i} b_{i} : 1 \leq i \leq 4} \cup {v a_{i} : 1 \leq i \leq 4} .$

Facts & Assumptions

Given: The path $P_{4} = b_{1} - b_{2} - b_{3} - b_{4}$ .

[L1]

The star-expansion adds a tooth $a_{i}$ adjacent to its matched vertex $b_{i}$ and to the root $v$ , with no other edges incident with $a_{i}$ ; the root is adjacent only to the teeth (The star-expansion of a graph).

Verification

technique · direct finite check

1.1

Applying [L1] to the four path vertices gives the eight new edges $a_{i} b_{i}$ and $v a_{i}$ for $1 \leq i \leq 4$ , together with no others incident to the new vertices.

2.1

Keeping the original path edges $b_{1} b_{2}, b_{2} b_{3}, b_{3} b_{4}$ and adding exactly the edges from step 1.1 yields the displayed graph.

step 1.1∎

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

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