Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

First barycentric subdivision of a triangle

Example

For the oriented triangle [v0v1v2], write bi for the midpoint of the edge opposite vi and b for its barycenter. Its first subdivision is the sum of the six oriented triangles obtained by coning the two halves of each oriented boundary edge to b.

Facts & Assumptions

Given: An oriented affine 2-simplex.

Verification

technique · direct
1.1

Coning the three subdivided boundary edges to b produces exactly six small triangles with their cone orientations.

givenconstruct
2.1

Every interior radial or midpoint edge occurs twice with opposite orientations; the uncancelled edges are the once-subdivided boundary. Thus ∂S=S∂ in this case.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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