Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01
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.

Every edge of a connected graph lies in every spanning tree

False Statement

Every edge of a connected graph belongs to every spanning tree of that graph.

Facts & Assumptions

Given: The claimed universal assertion about connected graphs and their spanning trees.

[L1]

The triangle C3C_3 has distinct spanning trees obtained by deleting different edges (A connected graph with two distinct spanning trees).

[L2]

An edge outside a spanning tree closes a fundamental cycle (Every edge outside a spanning tree determines a unique fundamental cycle).

[F1]

A spanning tree need only use a subset of the graph's edges (Spanning trees of a graph).

Refutation

technique · direct
1.1

In C3C_3, fix an edge ee and delete it. The remaining two edges form a spanning tree.

L1F1
2.1

This spanning tree omits ee; adding ee back closes the triangle as its fundamental cycle.

step 1.1L2
3.1

Hence an edge of a connected graph need not lie in every spanning tree.

step 1.1step 2.1∎

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: 19 results over 9 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