ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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 trivial partition has energy $d (G)^{2}$ , while the singleton partition records every adjacency

Statement

Let $G$ be an $n$ -vertex graph with $n > 0$ and $m$ edges. The one-part partition has energy ${(\frac{2 m}{n^{2}})}^{2},$ whereas the partition into singletons has energy $2 m / n^{2}$ . The latter is at least the former. For the null graph both energies are $0$ by convention.

Facts & Assumptions

Given: A finite graph and its trivial and discrete partitions.

[L1]

Partition energy is the ordered part-pair weighted sum of squared densities, with null-graph value $0$ (The mean-square density, or energy, of a vertex partition).

[L2]

Energy cannot decrease under refinement (Energy lies in $[0, 1]$ and cannot decrease under refinement).

Verification

technique · direct

1.1

For the one-part partition, the ordered-pair density is $d (V, V) = 2 m / n^{2}$ , so [L1] gives energy $(2 m / n^{2})^{2}$ .

givenL1algebra

1.2

In the singleton partition, an ordered pair of distinct singleton parts has squared density $1$ exactly when its two vertices are adjacent; diagonal densities and nonedge densities are $0$ . Every edge contributes its two orientations, so [L1] gives energy $2 m / n^{2}$ .

givenL1algebra

2.1

Since $0 \leq 2 m / n^{2} \leq 1$ , its square is no larger than itself, agreeing with [L2] because the singleton partition refines the trivial one. The null case is the convention in [L1].

step 1.1step 1.2L1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Y. Zhao, Graph Theory and Additive Combinatorics, Definition 2.1.10 (standard reference, not scraped)

The trivial partition has energy d(G)2d(G)2, while the singleton partition records every adjacency