LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Blocks from distinct mixed-block classes are pure to each other

Statement

Let $A$ and $B$ be blocks of a blockade $L$ . If $A$ and $B$ lie in different blocks of the quotient blockade $L / M$ , then $(A, B)$ is a pure pair.

Facts & Assumptions

Given: A blockade $L$ with quotient blockade $L / M$ , and original blocks $A, B$ of $L$ lying in different quotient blocks.

[L1]

Two original blocks lie in the same quotient block exactly when they are related by the mixed-block reachability relation $M$ (The quotient blockade obtained from mixed-block reachability).

[L2]

By definition, if two blocks are mixed, then they are joined by a length-one mixed chain and hence are $M$ -related (The mixed-block reachability relation on a blockade, Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

Proof

technique · direct

1.1

Suppose for contradiction that $(A, B)$ is mixed. Then [L2] gives a mixed chain of length one from $A$ to $B$ , so $A M B$ .

L2assume-contra

2.1

By [L1], $M$ -related blocks lie in the same quotient block of $L / M$ . This contradicts the hypothesis that $A$ and $B$ lie in different quotient blocks.

step 1.1L1discharge-contradiction

3.1

Therefore $(A, B)$ is not mixed, hence it is pure.

step 2.1∎

Depends on

Used by

A quotient-level mixed-block witness descends to two mixed member blocks Lemma

Dependency tree · two levels

7 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, Lemma 6.1(2) (standard reference, not scraped)