LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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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.
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C prime one sixth interior faces have at least seven arcs

Statement

Every interior face of an arc-reduced reduced $C^{'} (1 / 6)$ diagram has at least seven incident arcs, counted with multiplicity.

Facts & Assumptions

Given: An interior face with boundary relator $r$ and $d$ incident arcs $a_{1}, \dots, a_{d}$ .

[F1]

Each internal arc, including one with the same face on both sides, is a piece and has length strictly below one sixth of each incident face perimeter (Internal arcs of a reduced small cancellation diagram are pieces).

Proof

1.1givenF1

Every boundary edge of an interior face is internal. The arcs partition its boundary occurrences, so $∣ r ∣ = \sum_{j = 1}^{d} ∣ a_{j} ∣$ . If an arc occurs twice at this face it contributes twice to this sum, and [F1] applies to each occurrence; no distinct-face or simple-boundary hypothesis is needed. The relator is nonempty, hence $∣ r ∣ > 0$ and $d \geq 1$ . By [F1], each summand is strictly below $∣ r ∣ / 6$ .

2.1step 1.1algebra∎

Summing gives $∣ r ∣ < d ∣ r ∣ / 6$ . Dividing by the positive number $∣ r ∣$ gives $d > 6$ ; because $d$ is an integer, $d \geq 7$ . In particular equality at six arcs is excluded by strictness.

Depends on

Internal arcs of a reduced small cancellation diagram are pieces

Used by

Boundary spur or at most three shell from curvature Lemma

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 §3.5.1 paragraph preceding Definition 3.5.3 (standard reference, not scraped)