Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09
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 subdivision chain of an oriented two simplex

Example

Abbreviate singleton face vertices by a,b,c, two-element face vertices by ab,ac,bc, and the triangle face vertex by T=abc. For the orientation [a,b,c], S[a,b,c]=[b,bc,T]−[c,bc,T]−[a,ac,T]+[c,ac,T]+[a,ab,T]−[b,ab,T]. Its boundary is [b,bc]−[c,bc]−[a,ac]+[c,ac]+[a,ab]−[b,ab]=S([b,c]−[a,c]+[a,b]).

Source locators

2.1 pp.121–122.

Facts & Assumptions

[F1]

Subdivision uses the cone recursion. Oriented simplicial subdivision operator.

[F2]

The general boundary identity is a chain-map identity. Oriented simplicial subdivision commutes with boundary.

Verification

Given: The full oriented triangle [a,b,c] and the displayed face-label abbreviations.

1.1F1

The augmented cone recursion gives S[u,v]=[uv,v]−[uv,u]=[u,uv]−[v,uv]. Hence S∂[a,b,c]=[b,bc]−[c,bc]−[a,ac]+[c,ac]+[a,ab]−[b,ab]. Prepending T to each term and moving it past the two other vertices changes sign by (−1)2=1, giving exactly the six displayed oriented triangles.

2.1F2step 1.1∎

For a term [u,e,T], its boundary is [e,T]−[u,T]+[u,e]. The first two triangle terms give radial contribution −[b,T]+[c,T] after the [bc,T] terms cancel. The next two give [a,T]−[c,T] after the [ac,T] terms cancel. The last two give −[a,T]+[b,T] after the [ab,T] terms cancel. The three remaining radial contributions sum to zero. The six base terms are exactly the displayed S∂[a,b,c], verifying the boundary identity directly, in agreement with the general chain-map lemma.

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