False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

FALSE: reversing the order of the blocks never changes $x$ -sparsity

Statement

False claim: the order of the blocks is irrelevant for the property of being $x$ -sparse.

Facts & Assumptions

Given: Two blocks $X = {u}$ and $Y = {y_{1}, y_{2}}$ with the single edge $u y_{1}$ and with $c = \frac{1}{2}$ as the sparsity parameter.

[L1]

An ordered blockade $(B_{1}, B_{2})$ is $c$ -sparse exactly when the later block $B_{2}$ is $c$ -sparse to the earlier block $B_{1}$ (Complete, anticomplete, pure, weakly sparse, and $x$ -sparse blockades).

Proof

technique · direct

1.1

The one-vertex block $X$ is $c$ -sparse to $Y$ , because $u$ has one neighbor in $Y$ and $1 \leq c ∣ Y ∣ = \frac{1}{2} \cdot 2 = 1$ . By [L1], the ordered blockade $(Y, X)$ is therefore $c$ -sparse.

givenL1algebra

1.2

The block $Y$ is not $c$ -sparse to $X$ , because the vertex $y_{1}$ has one neighbor in $X$ but $1 > c ∣ X ∣ = \frac{1}{2}$ . By [L1], the reversed blockade $(X, Y)$ is therefore not $c$ -sparse.

givenL1algebra

2.1

Therefore reversing the order can change $x$ -sparsity, so the claim is false.

step 1.1step 1.2∎

Depends on

Complete, anticomplete, pure, weakly sparse, and $x$-sparse blockades

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, §2 (standard reference, not scraped)

FALSE: reversing the order of the blocks never changes xx-sparsity