ExampleConstruction: 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.

Quadratic Hamilton–Jacobi data produce an explicit caustic time

Example

For $u_{t} + (u_{x})^{2} / 2 = 0$ with $u_{0} (ξ) = - ξ^{2} / 2$ , the characteristic map is $X (t, ξ) = (1 - t) ξ$ . Its first caustic time is $1$ .

Facts & Assumptions

Given: The Hamiltonian $H (p) = p^{2} / 2$ and initial momentum $p_{0} (ξ) = - ξ$ .

Verification

technique · direct

1.1

Since $H_{x} = 0$ , the momentum equation gives $\dot{p} = 0$ , hence $p (t, ξ) = - ξ$ .

givenalgebra

2.1

As $\dot{X} = H_{p} (p) = p$ , integration from $X (0, ξ) = ξ$ gives $X = (1 - t) ξ$ .

step 1.1givenalgebra

3.1

$X_{ξ} = 1 - t$ is nonzero for $t < 1$ and zero at $t = 1$ , which is the first caustic by the stated convention.

step 2.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)