CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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 overlap of exactly one sixth shows that the strict C prime(1/6) inequality is not cosmetic

Statement refuted

An overlap of exactly one sixth still counts as satisfying the strict $C^{'} (1 / 6)$ convention.

Facts & Assumptions

Given: The relator set $R = {a b c d e f, a g h i j k}$ .

[L1]

The page's convention is strict: $C^{'} (1 / 6)$ demands $∣ p ∣ < ∣ r ∣ / 6$ for every piece $p$ in every relator $r$ (The small-cancellation conditions C(lambda) and C prime(lambda)).

Counterexample

technique · direct

1.1

The common initial segment of the two relators is the one-letter word $a$ , so $a$ is a piece. Both relators have length $6$ .

given

2.1

Hence the piece length is exactly $1 = 6 / 6$ . This meets the weak inequality $∣ p ∣ \leq ∣ r ∣ / 6$ , but it does not satisfy the strict inequality required by [L1].

L1step 1.1algebra

3.1

So the strict convention is genuinely stronger: exact one-sixth overlap fails it. This refutes the statement.

step 1.1step 2.1∎

Depends on

The small-cancellation conditions C(lambda) and C prime(lambda)

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

GAP SmallCancellation manual, Chapter 1: Small Cancellation Theory — the classical conditions (standard reference, not scraped)
Jay Williams, Universal Countable Borel Quasi-Orders (standard reference, not scraped)
Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Section 3.5 (standard reference, not scraped)
Clara Löh, Geometric Group Theory: An Introduction, Section 7.4.1 (standard reference, not scraped)