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.

Smooth nonanalytic transport data give a smooth nonanalytic solution

Example

Let $g (x) = e^{- 1 / x^{2}}$ for $x \neq 0$ and $g (0) = 0$ . The transport solution $u (t, x) = g (x - t)$ is smooth but not analytic at each point $x = t$ .

Facts & Assumptions

Given: The displayed flat function $g$ and $u (t, x) = g (x - t)$ .

Verification

technique · direct

1.1

Every derivative of $g$ at $0$ is $0$ , while $g (x) > 0$ for $x \neq 0$ ; thus $g$ is $C^{\infty}$ but cannot equal its Taylor series near $0$ .

givenalgebra

1.2

The chain rule gives $u_{t} = - g^{'} (x - t)$ and $u_{x} = g^{'} (x - t)$ , hence $u_{t} + u_{x} = 0$ and $u (0, x) = g (x)$ .

givenalgebra

2.1

Translation carries the flat nonanalytic point from $0$ to $x = t$ , so $u$ is nonanalytic there despite being smooth.

step 1.1step 1.2given∎

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 III: Analysis of Partial Differential Equations (standard reference, not scraped)