Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-04
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.

With trivial action, H^1(C_2,C_3) is zero

Example

For the trivial action of C2 on C3,

H1(C2,C3)=0.

Facts & Assumptions

Given: The trivial action of C2 on C3.

[L1]

For a trivial action, H1(G,A)≅Hom⁡(G,A) (For a trivial action, first cohomology is Hom).

[L2]

A group of order 3 is cyclic, and every nonidentity element has order 3 (Every cyclic group is isomorphic to (Z,+) or to (Z/n,+) for its finite order n≥1).

Verification

technique · direct
1.1

By [L1], it is enough to compute homomorphisms C2→C3.

givenL1
2.1

The generator of C2 must map to an element whose order divides 2, but [L2] says the only such element of C3 is the identity. So every homomorphism is trivial, and therefore H1(C2,C3)=0.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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