CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

A nonorthogonal two-point flux gives a false transverse flux

Statement refuted

The Euclidean two-point flux formula $Q_{K, F} = ∣ F ∣ (u_{K} - u_{L}) / ∣ x_{L} - x_{K} ∣$ is consistent for isotropic diffusion on every cell-face mesh, including nonorthogonal meshes.

Witness

Let $K$ be the triangle with vertices $(- 1, 0), (0, 0), (0, 1)$ and $L$ the triangle with vertices $(0, 0), (1, 1), (0, 1)$ . They share the vertical face $F$ from $(0, 0)$ to $(0, 1)$ . The union is a parallelogram and the two cell centroids are $x_{K} = (- 1 / 3, 1 / 3)$ and $x_{L} = (1 / 3, 2 / 3)$ , so their difference is not normal to $F$ . Set $λ = 1$ and evaluate the naive algebraic extension of the TPFA formula on $u (x, y) = y$ .

Facts & Assumptions

Given: The two triangles, their common face, scalar diffusion coefficient $1$ , the affine function $u (x, y) = y$ , and the Euclidean-distance two-point expression in the refuted claim.

[F1]

Finite-volume cell-face meshes and geometric regularity: the common face has length $1$ , outward normal from $K$ equal to $(1, 0)$ , and the cell centers are the centroids of the triangles.

[F2]

Orthogonal TPFA diffusion with harmonic face transmissibility: when applied under its orthogonality hypothesis, the TPFA formula uses transmissibility $∣ F ∣ / ∣ x_{L} - x_{K} ∣$ for $λ = 1$ . Here we deliberately evaluate that same algebraic expression outside its admissible-mesh hypothesis.

Verification

1.1F1givenalgebra

The triangle-centroid formula gives $x_{K} = (- 1 / 3, 1 / 3)$ and $x_{L} = (1 / 3, 2 / 3)$ , hence $x_{L} - x_{K} = (2 / 3, 1 / 3)$ and $∣ x_{L} - x_{K} ∣ = \sqrt{5} / 3$ . The center line is not perpendicular to the vertical face, whose normal is $(1, 0)$ .

1.2F1givenalgebra

The exact normal derivative of $u (x, y) = y$ on $F$ is $\nabla u \cdot (1, 0) = 0$ , so the exact outward diffusive flux from $K$ is zero. The cell-center values are $u_{K} = 1 / 3$ and $u_{L} = 2 / 3$ .

2.1F2step 1.1step 1.2algebra∎

The nonorthogonal two-point expression gives $Q_{K, F} = \frac{1}{\sqrt{5} / 3} (\frac{1}{3} - \frac{2}{3}) = - \frac{1}{\sqrt{5}} \neq 0.$ It disagrees with the exact zero normal flux in step 1.2. Thus this formula is inconsistent on the stated nonorthogonal face, refuting the claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results within one dependency step 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

Jérôme Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods, 2014 (standard reference, not scraped)