CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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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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Mixedness of block pairs is not transitive

Statement refuted

If $(A, B)$ is mixed and $(B, C)$ is mixed, then $(A, C)$ must also be mixed.

Facts & Assumptions

Given: Three disjoint vertex sets $A = {a_{1}, a_{2}}, B = {b_{1}, b_{2}}, C = {c_{1}, c_{2}},$ with cross-edges $a_{1} b_{1}, a_{2} b_{2}, b_{1} c_{1}, b_{2} c_{2},$ and no other cross-edges between these sets.

[L1]

A pair is mixed when it is neither complete nor anticomplete (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

[L2]

The mixed-block reachability relation is built from chains of mixed pairs, precisely because mixedness itself need not be transitive (The mixed-block reachability relation on a blockade).

Counterexample

technique · direct

1.1

The pair $(A, B)$ is mixed: it has edges $a_{1} b_{1}$ and $a_{2} b_{2}$ , but also nonedges $a_{1} b_{2}$ and $a_{2} b_{1}$ . The same calculation shows that $(B, C)$ is mixed.

givenL1

1.2

By construction there are no edges between $A$ and $C$ , so $(A, C)$ is anticomplete and therefore not mixed.

givenL1

2.1

Thus mixedness can hold for $(A, B)$ and $(B, C)$ while failing for $(A, C)$ . This is why [L2] passes to the reachability closure $M$ rather than treating mixedness itself as an equivalence relation.

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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.