ExampleConstruction: 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.

Interface width and a grid that misses the transition

Example

For the front in The one-dimensional Allen–Cahn double-well front, the interval where $∣ u_{ε} (x) ∣ \leq \frac{1}{2}$ has width $W_{ε} = 2 \sqrt{2 κ} ε artanh (1 / 2) .$ If a uniform sample grid has spacing $h > W_{ε}$ and the front is centered halfway between two sample points, no grid point lies in that transition interval. This gives a precise unresolved-interface configuration; it is not a sufficient resolution criterion.

Facts & Assumptions

Given: The exact front profile and positive parameters $κ, ε$ .

[F1]

The one-dimensional Allen–Cahn double-well front: $u_{ε} (x) = \tanh (x / (\sqrt{2 κ} ε))$ and its transition is centered at $x = 0$ .

Verification

1.1F1givenalgebra

Since $\tanh$ is strictly increasing and $\tanh (artanh (1 / 2)) = 1 / 2$ , the endpoints of $∣ u_{ε} ∣ \leq 1 / 2$ are $x = \pm \sqrt{2 κ} ε artanh (1 / 2)$ . Subtracting them gives $W_{ε}$ as stated.

2.1step 1.1givenalgebra∎

Place sample points at $(j + \frac{1}{2}) h$ for $j \in Z$ . Their distance from the front center is at least $h / 2 > W_{ε} / 2$ , so every sample point lies outside the transition interval from step 1.1. The grid therefore records no sample with $∣ u_{ε} ∣ \leq 1 / 2$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Jie Shen and Xiaofeng Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete and Continuous Dynamical Systems A 28 (2010), 1669–1691 (standard reference, not scraped)