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.

Curvature ledger for a two cell diagram

Example

Take two vertices $u, v$ and three disjoint-in-the-interior arcs $e_{0}, e_{1}, e_{2}$ from $u$ to $v$ , in planar order. Fill the regions between $e_{0}, e_{1}$ and between $e_{1}, e_{2}$ by two faces. This is a disc with $V = 2, E = 3, F = 2$ . Give every face corner angle $1 / 2$ in units of $π$ . Both vertex curvatures are zero and both face curvatures are one.

One possible labelling is $a, b, c$ on the three arcs: the face words are $a b^{- 1}$ and $b c^{- 1}$ , up to orientation. This example is a curvature computation, not a $C^{'} (1 / 6)$ claim.

Facts & Assumptions

Given: The two-face disc and its four angles described in the Example.

[F1]

Curvature uses link Euler characteristic and face corner multiplicities (Arc reduction and combinatorial curvature of a disc diagram).

[F2]

Total curvature of a diagram is $2 (V - E + F) = 2$ (Euler curvature identity for an arc reduced disc diagram).

Verification

1.1

At each of $u, v$ the link is a path with three vertices (the three edge germs) and two edges (the two corners), so its Euler characteristic is $3 - 2 = 1$ . Its angle sum is $1 / 2 + 1 / 2 = 1$ , giving $k (u) = k (v) = 2 - 1 - 1 = 0$ by [F1].

givenF1algebra

2.1

Each face is a bigon with two corners of angle $1 / 2$ , giving $k (f) = 1 - (2 - 2) = 1$ . Thus total curvature is $0 + 0 + 1 + 1 = 2$ , agreeing with $2 (2 - 3 + 2) = 2$ in [F2]. These calculations include all four corners once in the face total and once in the vertex total with opposite sign.

F1F2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Touikan Theorem 3.4.6; explicit two-cell specialization (standard reference, not scraped)