LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedaudited 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.

Compatibility of a characteristic strip with Cauchy data

Statement

If a $C^{1}$ solution has $u (γ (y)) = ϕ (y)$ and $p_{0} (y) : = D u (γ (y))$ , then

$p_{0} (y) D γ (y) = D ϕ (y), p_{0} (y) \cdot a (γ (y), ϕ (y)) = b (γ (y), ϕ (y)) .$

The first condition is tangential compatibility. The rank condition for the projected strip is a separate hypothesis; it is not derived from this identity.

Facts & Assumptions

Given: The $C^{1}$ Cauchy data and a $C^{1}$ solution attaining them.

Proof

technique · direct

1.1

Differentiate $u \circ γ = ϕ$ with respect to $y$ to obtain $D u (γ) D γ = D ϕ$ .

givenalgebra

2.1

Evaluate $a (x, u) \cdot D u = b (x, u)$ at $x = γ (y)$ and use $u (γ (y)) = ϕ (y)$ to obtain the second identity.

givenalgebra∎

Depends on

Used by

Local quasilinear characteristic graph construction Theorem

Dependency tree · two levels

8 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

First order PDE: The Methods of Characteristics (standard reference, not scraped)