Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)
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.

Low-degree admissible Steenrod monomials

Example

Assume AC. The admissible bases in degrees 0–4 are respectively {1}, {Sq¹}, {Sq²}, {Sq³,Sq²Sq¹}, and {Sq⁴,Sq³Sq¹}. The Adem relations give Sq¹Sq¹=0, Sq¹Sq²=Sq³, and Sq²Sq²=Sq³Sq¹.

Facts & Assumptions

Given: AC; the admissible words of the mod-two square algebra in total degrees 0 through 4; and the three displayed composite pairs Sq1Sq1, Sq1Sq2, Sq2Sq2.

[F1]

The Adem-reduction lemma spans each homogeneous degree by admissible words, and the admissible composites form a basis of the square algebra in each degree (Adem reduction spans by admissible square composites, Admissible composites present the mod-two square algebra).

[F2]

The Adem relations hold in the square algebra for 0<a<2b, and the square algebra and its excess calculus are the local definition.

Verification

1.1givenF1

List the positive sequences of each total degree and retain those satisfying i_j≥2i_{j+1}; this gives exactly the displayed rows. The Adem-reduction item shows every other word reduces to an admissible combination.

2.1step 1.1F2algebra∎

The basis theorem proves the listed words are independent, so the table is a basis calculation rather than a dimension guess. Substitution in the displayed Adem relation gives the three sample reductions.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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