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.

The Charpit flow preserves the PDE constraint

Statement

Along a Charpit curve,

$\frac{d}{d s} F (X, Z, P) = 0.$

Consequently an initial point in $F = 0$ remains in $F = 0$ while the curve exists.

Facts & Assumptions

Given: A $C^{1}$ Charpit curve for a $C^{2}$ function $F$ .

Proof

technique · direct

1.1

The chain rule gives $\dot{F} = F_{x} \cdot \dot{X} + F_{z} \dot{Z} + F_{p} \cdot \dot{P}$ .

givenalgebra

2.1

Substitute $\dot{X} = F_{p}$ , $\dot{Z} = P \cdot F_{p}$ , and $\dot{P} = - F_{x} - P F_{z}$ ; the three terms cancel pairwise.

step 1.1givenalgebra

3.1

Hence $F (X, Z, P)$ is constant, so an initial zero stays zero.

step 2.1given∎

Depends on

Used by

Dependency tree · two levels

7 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)