ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Eikonal cones are not classical at the vertex

Example

For $n \geq 1$ , on $R^{n}$ the function $u (x) = ∣ x ∣$ solves $∣ D u ∣ = 1$ away from $x = 0$ , but is not a classical solution at its vertex.

Facts & Assumptions

Given: An integer $n \geq 1$ and the Euclidean norm function $u (x) = ∣ x ∣$ .

Verification

technique · direct

1.1

For $x \neq 0$ , $D u (x) = x / ∣ x ∣$ , so $∣ D u (x) ∣ = 1$ .

givenalgebra

1.2

Along a unit vector $e$ , the directional quotients at zero are $u (t e) / t = 1$ for $t > 0$ and $u (t e) / t = - 1$ for $t < 0$ .

givenalgebra

2.1

They have no common limit, so $u$ is not differentiable at zero and cannot be classical there.

step 1.2given∎

Depends on

Fully nonlinear first-order PDEs and complete integrals

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Part I: Explicit methods — Lecture notes for MA342H (standard reference, not scraped)