Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31
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: a shortest walk between two vertices may repeat a vertex and fail to be a path

Statement

FALSE. A shortest walk between two vertices may repeat a vertex and fail to be a path.

Facts & Assumptions

Given: Vertices u,vu,v in one connected component and a uu to vv walk WW of minimum length.

[L1]

Every walk contains a path between the same endpoints of no greater length, and if the walk repeats a vertex the path can be chosen strictly shorter (Every walk between two vertices contains a path between the same endpoints).

[F1]

The distance dG(u,v)d_G(u,v) is the minimum length of a uu to vv path (Graph distance within a component, eccentricity, diameter and girth, including the acyclic convention).

Refutation

technique · contradiction
1.1

Suppose, for contradiction, that the minimum-length walk WW repeats a vertex and is not a path.

assume-contragiven
2.1

By [L1], deleting a closed segment from WW yields a uu to vv path of length strictly less than the length of WW. This path is also a walk, contradicting the minimality of WW.

step 1.1L1F1
3.1

Therefore every shortest walk is a path, and the claimed possibility cannot occur.

step 2.1discharge-contradiction∎

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: 16 results over 12 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