Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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 small-cancellation presentation gives a hyperbolic group

Example

Consider the one-relator presentation

G=⟨x1,x2,x3,x4,x5,x6,x7∣x1x2x3x4x5x6x7⟩.

This is a finite C′(1/6) presentation and hence defines a hyperbolic group.

Facts & Assumptions

Given: The displayed presentation of G.

[A1]

In the single relator x1x2x3x4x5x6x7, no nonempty subword occurs as an initial segment of two distinct cyclic conjugates or inverse cyclic conjugates, so the symmetrized presentation has no nontrivial pieces and therefore satisfies C′(1/6) vacuously.

[L1]

Finite C′(1/6) presentations define hyperbolic groups (Finite C'(1/6) presentations define hyperbolic groups).

Verification

technique · direct
1.1

By [A1], the displayed finite presentation satisfies the C′(1/6) condition.

givenA1
2.1

Therefore [L1] shows that the presented group G is hyperbolic.

L1step 1.1∎

Depends on

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