ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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 five-cycle is prime

Example

The cycle $C_{5}$ is prime.

Facts & Assumptions

Given: The five-cycle $C_{5}$ with vertices $0, 1, 2, 3, 4$ in cyclic order.

[L1]

A graph is prime when it has no nontrivial module (Prime graphs: those whose only modules are the trivial ones, Modules of a graph, and the trivial modules).

[L2]

In a connected graph, every nonempty proper module has some outside vertex complete to it (In a connected graph, some vertex outside a nonempty proper module is complete to it).

[L3]

In $C_{5}$ , every vertex has exactly two neighbours, and those two neighbours are nonadjacent (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Verification

technique · contradiction

1.1

Suppose, for contradiction, that $C_{5}$ has a nontrivial module $M$ . Then [L2] gives a vertex $v \notin M$ complete to $M$ .

assume-contraL2

2.1

By [L3], the set $M$ must be the two neighbours of $v$ .

step 1.1L3

3.1

The two remaining vertices of the cycle each see exactly one of those neighbours, so they split $M$ , contradicting that $M$ is a module.

step 2.1given

4.1

Therefore $C_{5}$ has no nontrivial module, and [L1] makes it prime.

step 3.1L1discharge-contradiction∎

Depends on

Used by

The modular decomposition of a five-cycle with each vertex blown up into an edgeless graph Example

Dependency tree · two levels

15 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

M. Chudnovsky, The Erdős–Hajnal Conjecture — A Survey, sec. 2 (standard reference, not scraped)