Alphabeta Math
ExampleConstruction: 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.

Symmetrising a one-relator presentation adds the cyclic conjugates and inverse cyclic conjugates

Example

For the one-relator presentation ⟨a,b,c∣abc⟩, the symmetrisation of the relator set is

{abc,bca,cab,c−1b−1a−1,b−1a−1c−1,a−1c−1b−1}.

Facts & Assumptions

Given: The cyclically reduced relator r=abc.

[L1]

Symmetrisation adds all cyclic conjugates of r and of r−1 (The symmetrisation of a relator set closes under inverses and cyclic conjugates).

Verification

technique · direct
1.1given

The cyclic conjugates of r=abc are abc, bca, and cab.

2.1L1step 1.1∎

The inverse word is r−1=c−1b−1a−1, whose cyclic conjugates are c−1b−1a−1, b−1a−1c−1, and a−1c−1b−1. Applying [L1] gives exactly the displayed six-element set.

Depends on

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