ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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.

Finite triangulation angle bookkeeping

Example

For the tetrahedral combinatorial pattern $V = 4$ , $E = 6$ , $F = 4$ , the incidence identity $3 F = 2 E$ holds and the Euler expression is $2 π (V - E + F) = 4 π$ . If this pattern is equipped with face-corner angles from a smooth closed surface triangulation, the angles around each vertex sum to $2 π$ ; then the total exterior-angle correction is $3 π F - 2 π V = 4 π$ , leaving $2 π F - 4 π = 4 π$ in the local-to-global count. The incidence pattern alone supplies no angle measurements or curvature integral.

Facts & Assumptions

Given: The finite combinatorial data of a closed triangulation with four vertices, six edges and four triangular faces, each vertex incident with three face corners, each edge incident with two faces, and each face bounded by three distinct edges.

[F1]

For a curvilinear triangulation of a compact surface, the number $m_{v}$ of face corners at a vertex satisfies $\sum_{v} m_{v} = 3 F$ , the face-edge incidences satisfy $3 F = 2 E_{i n t} + E_{b d}$ , and boundary incidences satisfy $E_{b d} = V_{b d}$ ; in the closed case these reduce to $\sum_{v} m_{v} = 3 F = 2 E$ (Curvilinear face-to-face triangulation).

[F2]

Under full AC, the supplied local-to-global lemma applies to a closed oriented Riemannian surface with a finite face-to-face geodesic triangulation whose faces are compact regular oriented disks in frameable charts, and gives $\int_{M} K d A = 2 π (V - E + F)$ ; the incident face angles sum to $2 π$ at each interior vertex (Summing local Gauss-Bonnet over a supplied triangulation).

[F3]

For a curvilinear triangulation of a compact surface the Euler characteristic of the triangulated surface is $χ (M; T) = V - E + F$ (Euler characteristic of a finitely triangulated compact surface).

Verification

technique · enumerate the incidences of the tetrahedral pattern, count the vertex angles and substitute into the finite local-to-global identity

1.1F1F3given

In the given pattern $V = 4$ , $E = 6$ and $F = 4$ ; every edge is interior and incident with two faces, so counting face-edge incidences gives $3 F = 12 = 2 E$ , and counting face corners gives $\sum_{v} m_{v} = 3 F = 12$ , compatible with three corners at each of the four vertices. Hence $V - E + F = 4 - 6 + 4 = 2$ .

2.1F1step 1.1algebra

For the conditional angle count, suppose face-corner angles $β_{c}$ are supplied and satisfy the smooth closed-surface normalization $\sum_{c ∋ v} β_{c} = 2 π$ at each of the four vertices. This is additional geometric data, not a consequence of the incidence pattern. With exterior angles $α_{c} = π - β_{c}$ , there are $3 F$ face corners by [F1], so $\sum_{f} \sum_{c} α_{c} = 3 π F - \sum_{v} 2 π = 12 π - 8 π = 4 π$ .

3.1F2F3step 1.1step 2.1algebra∎

Under the angle normalization in step 2.1, the formal remainder is $2 π F - \sum_{f} \sum_{c} α_{c} = 8 π - 4 π = 4 π$ . Since $3 F = 2 E$ and $V = 4$ , this is $2 π (V - E + F)$ . If AC and a smooth oriented Riemannian realization with a supplied finite geodesic triangulation satisfying [F2]'s hypotheses were separately supplied, [F2] would identify this count with its curvature integral. The combinatorial pattern alone supplies neither the angle normalization nor a Riemannian surface, so it asserts no curvature integral.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, printed pp. 167-172, reduces the global theorem to the vertex-edge-face count for a finite triangulation; Datar, Lectures on Riemannian Geometry, Lecture 2, Section 2.2, printed pp. 13-15, uses the same incidence identities. The tetrahedral enumeration is carried out here purely as finite arithmetic against the conventions of Curvilinear face-to-face triangulation and the count of Summing local Gauss-Bonnet over a supplied triangulation, with no geometric realization claimed.

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Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)