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 primitive product-action wreath product

Example

Let H=S5 acting naturally on Δ={1,2,3,4,5} and let K=S2 act on two coordinates. Then H≀K is primitive on Δ2 and is of O'Nan--Scott product-action type.

Facts & Assumptions

Given: The standard product action of S5≀S2 on {1,2,3,4,5}2.

[L1]

Under the standard hypotheses, product-action wreath products are primitive (Product-action wreath products are primitive under the standard hypotheses).

[L2]

Product action is one of the five coarse O'Nan-Scott types (Affine, almost simple, diagonal, product action, and twisted wreath types).

Verification

technique · direct
1.1

The action of S5 on five points is primitive and not regular, and the action of S2 on the two coordinates is transitive. Therefore [L1] applies to S5≀S2 on {1,2,3,4,5}2.

givenL1
2.1

The socle of the wreath product is A52, which is nonabelian and acts coordinatewise. Thus the action is primitive and has the product-action socle data in [L2], rather than affine socle data.

L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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