CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

A boundary spur when free reduction is omitted

Statement refuted

Every nonempty null boundary word of a diagram over a $C^{'} (1 / 6)$ presentation has a shell, even without requiring free reduction.

Facts & Assumptions

Given: The presentation $⟨ a ∣ \emptyset ⟩$ , satisfying $C^{'} (1 / 6)$ vacuously, and a single edge oriented from $u$ to $v$ labelled $a$ , with no faces.

[F1]

A finite tree is a zero-face diagram, and its outer walk traverses a bridge in both directions (Sc toolkit labelled planar disc diagram).

[F2]

A spur tip has singleton link, no corners and curvature one (Arc reduction and combinatorial curvature of a disc diagram).

Counterexample

1.1

The single closed edge is finite, connected, planar and contractible, hence is a diagram by [F1]. Its face-label condition is vacuous. Its outer word from $u$ is $a a^{- 1}$ , a nonempty word freely reducing to the empty word and therefore null in the given free group. It is not freely reduced.

givenF1

2.1

There are no faces, so there cannot be an exposed face or shell. Both endpoints have one edge germ and no corners, giving curvature one each by [F2], and the total is two. Thus positive curvature is entirely carried by spur tips, and the nonempty null boundary in step 1.1 refutes the assertion.

step 1.1F2∎

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

Touikan Lemma 3.4.3, zero-face case (standard reference, not scraped)