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.

Cauchy–Kovalevskaya normal form with analytic data

Example

For analytic $g_{0}, g_{1}$ , the wave equation $u_{t t} = u_{x x}$ with $u (0, x) = g_{0} (x)$ and $u_{t} (0, x) = g_{1} (x)$ is a second-order analytic Cauchy problem in normal form. This checks its hypotheses only; it does not invoke a recorded theorem.

Facts & Assumptions

Given: Analytic functions $g_{0}, g_{1}$ and the wave equation $u_{t t} = u_{x x}$ .

Verification

technique · direct

1.1

The equation is solved for the second normal derivative: $u_{t t} = u_{x x}$ , whose right side is analytic in the relevant jet variables.

givenalgebra

2.1

The normal line is $t$ , and the data supply exactly $\partial_{t}^{0} u ∣_{t = 0} = g_{0}$ and $\partial_{t} u ∣_{t = 0} = g_{1}$ , the orders $0$ and $1$ required for order $2$ .

step 1.1given

3.1

The coefficient of $u_{t t}$ is $1$ , so $t = 0$ is noncharacteristic for this solved normal form.

step 1.1algebra∎

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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