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

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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.

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An irregularity witness raises the pair energy by more than $ϵ^{4} ∣ X ∣ ∣ Y ∣ / n^{2}$

Statement

Let $G$ have order $n > 0$ , let $X, Y \subseteq V (G)$ be nonempty, and suppose that $A \subseteq X$ , $B \subseteq Y$ witness that $(X, Y)$ is not $ϵ$ -regular. If $A = {A, X ∖ A}$ and $B = {B, Y ∖ B}$ after empty cells are omitted, then $q (A, B) - \frac{∣ X ∣ ∣ Y ∣}{n^{2}} d (X, Y)^{2} > ϵ^{4} \frac{∣ X ∣ ∣ Y ∣}{n^{2}} .$

Facts & Assumptions

Given: An irregular pair and witness sets as in the Statement.

[L1]

Such witnesses satisfy $∣ A ∣ \geq ϵ ∣ X ∣$ , $∣ B ∣ \geq ϵ ∣ Y ∣$ , and $∣ d (A, B) - d (X, Y) ∣ > ϵ$ ( $ϵ$ -regular pairs and self-regular vertex sets).

[L2]

Pair energy is the product-size-weighted mean square of the densities of the refined subpairs (The mean-square density, or energy, of a vertex partition).

Proof

technique · direct

1.1

Choose $(x, y)$ uniformly from $X \times Y$ , and let $Z$ be the density between the cells of $A$ and $B$ containing $x$ and $y$ . Double-counting gives $E Z = d (X, Y)$ , and [L2] identifies $∣ X ∣ ∣ Y ∣ E (Z^{2}) / n^{2}$ with $q (A, B)$ .

givenL2algebra

2.1

Therefore the energy gain in the Statement is $(∣ X ∣ ∣ Y ∣ / n^{2}) Var (Z) = (∣ X ∣ ∣ Y ∣ / n^{2}) E ((Z - d (X, Y))^{2})$ .

step 1.1algebra

2.2

On the event $(x, y) \in A \times B$ , which has probability $∣ A ∣ ∣ B ∣ / (∣ X ∣ ∣ Y ∣) \geq ϵ^{2}$ by [L1], the random variable equals $d (A, B)$ and differs from its mean by more than $ϵ$ .

L1step 1.1

3.1

Restricting the nonnegative expectation in step 2.1 to this event gives a strict lower bound $ϵ^{2} ϵ^{2} ∣ X ∣ ∣ Y ∣ / n^{2}$ , which is the asserted boost.

step 2.1step 2.2algebra∎

Depends on

Used by

Every nonregular k-part partition has a refinement with energy gain greater than ε⁵ and at most k2ᵏ⁺¹ parts Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 6 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, Lemma 2.1.13 (standard reference, not scraped)

An irregularity witness raises the pair energy by more than ϵ4∣X∣∣Y∣/n2ϵ4∣X∣∣Y∣/n2