ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-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.

An edge inversion and its barycentric subdivision

Example

Let $T$ be a single geometric edge with endpoints $u$ and $v$ , and let the nontrivial element of $C_{2}$ swap $u$ and $v$ . This action inverts the unique edge of $T$ , but after barycentric subdivision it fixes the midpoint vertex and acts without inversions.

Facts & Assumptions

Given: The reflected one-edge tree.

[L1]

Barycentric subdivision preserves the tree and removes edge inversions. (Barycentric subdivision removes edge inversions while preserving the tree)

Verification

technique · direct

1.1

Before subdivision, the nontrivial element sends the oriented edge $u \to v$ to the reverse edge $v \to u$ , so there is an inversion.

L1given

2.1

After subdivision, the midpoint is a vertex fixed by the action, and each half-edge is sent to the other half-edge with the same orientation type rather than to its reverse. This is exactly the mechanism asserted in [L1].

L1step 1.1algebra∎

Depends on

Barycentric subdivision removes edge inversions while preserving the tree

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Yuriy Tumarkin, Groups Acting on Trees (standard reference, not scraped)