Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

Cellular boundary matrix of a closed orientable surface

Example

For the genus-g closed orientable surface, the one-vertex, 2g-edge, one-face CW structure has zero cellular differentials. Thus H0≅Z, H1≅Z2g, and H2≅Z.

Facts & Assumptions

Given: The polygon word [a1,b1]⋯[ag,bg].

Verification

technique · direct
1.1given

Each oriented edge occurs once positively and once negatively in the abelianization of the word, so every incidence coefficient of the face is zero; the edge-to-vertex boundary is also zero.

2.1step 1.1∎

The cellular complex is 0→Z→Z2g→Z→0 with zero maps, and Cellular homology computes singular homology proves the calculation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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