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.

Rarefying inviscid Burgers data

Example

For $u_{0} (ξ) = ξ$ , $X = (1 + t) ξ$ , so for $t \geq 0$ the solution is $u (t, x) = x / (1 + t)$ . The fan expands and has no forward caustic.

Facts & Assumptions

Given: The Burgers characteristic formula with $u_{0} (ξ) = ξ$ and $t \geq 0$ .

Verification

technique · direct

1.1

The formula gives $X = ξ + t ξ = (1 + t) ξ$ and $X_{ξ} = 1 + t > 0$ .

givenalgebra

2.1

Inverting gives $ξ = x / (1 + t)$ , while $u (t, X) = u_{0} (ξ) = ξ$ ; hence $u (t, x) = x / (1 + t)$ .

step 1.1algebra∎

Depends on

Inviscid Burgers characteristic formula and first crossing time

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)