Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-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.

Multiplier of a finite abelian group

Example

Assume the Axiom of Choice and supplied projective-resolution data for group homology. For A≅Cn1×⋯×Cnr, M(A)≅⨁1≤i<j≤rCgcd⁡(ni,nj).

Facts & Assumptions

Given: The stated choice and resolution hypotheses and the invariant-factor decomposition of A.

Verification

technique · direct
1.1givenalgebra

For two abelian groups B,C, the alternating universal property gives ⋀2(B⊕C)≅⋀2B⊕(B⊗C)⊕⋀2C: an alternating pairing on the direct sum is exactly a pairing on each summand plus an arbitrary bilinear cross-pairing. Iterate over the finite summands. Each cyclic summand has zero exterior square, while Cm⊗Cn≅Cgcd⁡(m,n) by the relations m(1⊗1)=n(1⊗1)=0.

2.1step 1.1algebra∎

Applying Multiplier of an abelian group gives the displayed formula, including the empty sum 0 when r≤1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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