Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

ex-ext-one-as-a-derived-category-morphism.md

Example

For positive integers m,n, Hom⁡D(Ab)(Z/m[0],Z/n[1])≅Z/gcd⁡(m,n).

Facts & Assumptions

Given: For positive integers m,n, Hom⁡D(Ab)(Z/m[0],Z/n[1])≅Z/gcd⁡(m,n).

[F1]

Ext computed by a supplied projective resolution is derived Hom into the positive shift (Ext is hom in the derived category).

Verification

1.1F1algebra

Resolve Z/m by P=(Z→mZ) in degrees −1,0, with its quotient augmentation. Since m>0 the first map is injective, so this is a projective resolution. Hom into Z/n has terms Z/n in degrees zero and one, and differential −m with the cochain Hom convention.

2.1F1step 1.1algebra∎

Degree-one cohomology is (Z/n)/m(Z/n)=Z/(mZ+nZ)=Z/gcd⁡(m,n). The Ext comparison identifies it with the claimed Hom group. If m=1 or n=1 the quotient is zero, as required for a zero input module.

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