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.

Inviscid Burgers gradient catastrophe

Example

For $u_{0} (ξ) = - \tanh ξ$ , the map is $X = ξ - t \tanh ξ$ . Its first crossing is $t = 1$ at $ξ = 0$ , and the slope there tends to $- \infty$ .

Facts & Assumptions

Given: The Burgers formula and $u_{0}^{'} (ξ) = - {sech}^{2} ξ$ .

Verification

technique · direct

1.1

$X_{ξ} = 1 - t {sech}^{2} ξ$ , which first vanishes at $t = 1$ , $ξ = 0$ , since $0 < {sech}^{2} ξ \leq 1$ .

givenalgebra

2.1

The Riccati formula gives $u_{x} (t, X (t, 0)) = - 1 / (1 - t)$ , which tends to $- \infty$ as $t ↑ 1$ .

step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 III: Analysis of Partial Differential Equations (standard reference, not scraped)