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.

A layout with a single wrong decided pair

Example

For this example, a layout consists of a pattern graph $J$ and pairwise disjoint vertex blocks $(A_{x} : x \in V (J))$ . Pairs inside one block are undecided. A pair with endpoints in distinct blocks $A_{x}, A_{y}$ is decided, with the pattern predicting an edge exactly when $x y \in E (J)$ ; the decided pair is wrong when its actual adjacency disagrees with that prediction.

Take such a layout whose pattern graph is a single edge $u v$ with blocks $A_{u} = {1, 2}$ and $A_{v} = {3, 4}$ . If the graph on ${1, 2, 3, 4}$ has all cross edges except $24$ , then the decided pair ${2, 4}$ is the unique wrong pair.

Facts & Assumptions

Given: The described two-block layout.

Verification

technique · direct finite check

1.1

The pairs inside $A_{u}$ and $A_{v}$ are undecided by definition of layout, so only cross-pairs are decided.

given

2.1

Because the pattern graph makes every cross-pair expected to be an edge, the missing edge $24$ is wrong, while all other cross-pairs agree with the pattern. Hence there is exactly one wrong decided pair.

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, Theorem 6.1 layout setup (standard reference, not scraped)