CorollaryStatement: Literature-sourcedProof: 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.

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Monotone Burgers data have no forward characteristic crossing

Statement

If $u_{0}^{'} (ξ) \geq 0$ for all $ξ$ , then $1 + t u_{0}^{'} (ξ) \geq 1$ for $t \geq 0$ . Hence the Burgers projection has no forward crossing. This says nothing about any unrelated global lifespan condition.

Facts & Assumptions

Given: The characteristic formula and $u_{0}^{'} \geq 0$ .

Proof

technique · direct

1.1givenalgebra

For $t \geq 0$ , multiplication gives $t u_{0}^{'} (ξ) \geq 0$ .

2.1step 1.1given∎

Therefore $1 + t u_{0}^{'} (ξ) \geq 1$ , so the zero set defining $T_{*}$ is empty and the projection has no forward crossing.

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)