Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

FALSE: every tree action is without edge inversions

Statement

Every group action on a simplicial tree is automatically without edge inversions.

Facts & Assumptions

Given: The definitions of edge inversion and barycentric subdivision.

[L1]

An action is without inversions exactly when no element sends an oriented edge to its reverse. (Edge inversions and actions without inversions)

[L2]

Barycentric subdivision is used precisely to remove inversions when they are present. (Barycentric subdivision removes edge inversions while preserving the tree)

Refutation

technique · direct
1.1

Let T be a single geometric edge with endpoints u,v, and let the nontrivial element of C2 swap u and v. It sends one oriented edge e:u→v to its reverse eˉ:v→u, so the action has an inversion in the sense of [L1].

L1given
2.1

Since [L2] would be unnecessary if every action were already inversion-free, the example in step 1.1 disproves the statement.

L1L2step 1.1algebra∎

Depends on

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