LemmaStatement: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Jacobian of a characteristic strip at its initial surface

Statement

For the local strip $(X, Z)$ , the derivative of $Ψ (s, y) : = X (s, y)$ at $s = 0$ is

$D_{(s, y)} Ψ (0, y) = [a (γ (y), ϕ (y)), D γ (y)] .$

Consequently $D_{(s, y)} Ψ (0, y)$ is invertible exactly when this $n \times n$ matrix has rank $n$ .

Facts & Assumptions

Given: The $C^{1}$ local strip supplied by the preceding lemma.

Proof

technique · direct

1.1

Its initial condition gives $D_{y} X (0, y) = D γ (y)$ , and its ODE gives $\partial_{s} X (0, y) = a (γ (y), ϕ (y))$ .

givenalgebra

2.1

These are precisely the first and remaining columns of $D_{(s, y)} Ψ (0, y)$ , proving the formula; an $n \times n$ derivative is invertible exactly when it has rank $n$ .

step 1.1algebra∎

Depends on

Used by

Local quasilinear characteristic graph construction Theorem

Dependency tree · two levels

4 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)