CounterexampleConstruction: 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.

Characteristic initial data need not determine a fully nonlinear solution

Statement refuted

Characteristic initial data always determine a unique local fully nonlinear classical solution.

Counterexample

Given: The equation $F (x, y, z, p_{x}, p_{y}) = p_{x}^{2} = 0$ and data $u (x, 0) = 0$ .

Proof technique: direct.

1.1

Here $F_{p} = (2 p_{x}, 0) = 0$ on $F = 0$ , so $[F_{p}, D γ]$ has rank $1 < 2$ for $γ (y) = (y, 0)$ .

givenalgebra

1.2

Every $u (x, y) = f (y)$ with $f (0) = 0$ has $u_{x} = 0$ , hence solves $F = 0$ and attains the data.

givenalgebra

2.1

Choosing distinct such $f$ gives distinct local solutions, and step 1.1 identifies the failed rank hypothesis.

step 1.1step 1.2construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)