ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

A clique of size $s$ has self-density $1 - 1 / s$

Example

If $X$ is a clique of size $s \geq 1$ , then $d_{G} (X, X) = 1 - 1 / s$ .

Facts & Assumptions

Given: A clique $X$ of size $s$ in a finite simple graph $G$ .

[L1]

The self-density is $d_{G} (X, X) = e_{G} (X, X) / s^{2}$ (Edge counts and densities between nonempty vertex sets).

Verification

technique · direct

1.1given

Every ordered pair of distinct vertices of $X$ is an edge, and the diagonal contributes nothing, so $e_{G} (X, X) = s (s - 1)$ .

2.1step 1.1L1algebra

Dividing by $s^{2}$ as in [L1] gives $d_{G} (X, X) = s (s - 1) / s^{2} = 1 - 1 / s$ .

3.1step 2.1∎

In particular the self-density is always strictly less than $1$ , which is the reciprocal-size slack appearing in the dense half of A $c$ -sparse set has self-density at most $c$ , and a $c$ -dense set has self-density at least $1 - c - 1 / ∣ X ∣$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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A clique of size ss has self-density 1−1/s1−1/s