Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 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.

The boundary of a triangle has first homology Z

Example

Let K have vertices v0,v1,v2 and edges e01,e12,e20, with no 2-simplex. Then C2(K)=0 and C1(K)≅Z3. If the edges are oriented cyclically, the boundary map is ∂1(e01,e12,e20)=(v1−v0, v2−v1, v0−v2). The sum e01+e12+e20 is a 1-cycle, and every 1-cycle is an integer multiple of this one because the three vertex equations force the three edge coefficients to agree. Since C2(K)=0, one has B1(K)=0, so H1simp(K)≅Z. The complex is connected, so H0simp(K)≅Z, and all higher homology groups vanish.

Depends on

Used by

Dependency tree · two levels

7 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