ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

Analytic transport data

Example

For a fixed $a \in R^{d}$ and analytic g near zero, the problem $u_{t} + a \cdot D_{x} u = 0$ , $u (0, x) = g (x)$ has the analytic solution $u (t, x) = g (x - a t)$ near zero.

Facts & Assumptions

Given: The constant transport vector and analytic initial function in the Example. The proposed translated function must satisfy the equation and data.

[F1]

Analytic substitution is valid on a smaller polydisc. (Operations preserving coefficient majorisation).

[F2]

The derivative of a composite is the composite of the differentials. (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F3]

The analytic normal-form solution germ is unique. (Cauchy–Kovalevskaya for first-order analytic systems).

Verification

1.1givenF1F2

The map $(t, x) \mapsto x - a t$ is linear and sends zero to zero, so F1 makes $u = g (x - a t)$ analytic on a sufficiently small neighborhood. F2 gives $u_{t} = - a \cdot D g (x - a t)$ and $D_{x} u = D g (x - a t)$ ; thus $u_{t} + a \cdot D_{x} u = 0$ and $u (0, x) = g (x)$ .

2.1step 1.1F3algebra∎

The solved right side $F (t, x, u, p) = - a \cdot p$ is polynomial in its jet variables and therefore analytic at the required initial jet. F3 identifies the displayed solution with the unique analytic germ. For the explicit data $d = 1, a = 2, g (x) = x^{2}$ , this gives $u = (x - 2 t)^{2}$ , $u_{t} = - 4 (x - 2 t)$ and $2 u_{x} = 4 (x - 2 t)$ , displaying the cancellation directly.

Source notes

Gantumur, §5 transport discussion following Exercise 24, printed p. 14; the constant-vector solution is computed locally.

Depends on

Used by

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Sources

Gantumur, Math 580 Lecture Notes 2: The Cauchy-Kovalevskaya Theorem (standard reference, not scraped)
Ageno, Part III: Analysis of Partial Differential Equations (standard reference, not scraped)