ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

The quadratic Dirichlet solution is represented exactly

Example

Take $g (x) = 1$ and $u (x) = x (1 - x) / 2$ on $[0, 1]$ . Then $- u^{''} = g$ and $u (0) = u (1) = 0$ . We verify that the scheme in Centered finite differences for the zero-Dirichlet boundary-value problem returns its exact nodal values for every $N \geq 1$ .

Facts & Assumptions

Given: The function $u (x) = x (1 - x) / 2$ , data $g (x) = 1$ , and an integer $N \geq 1$ .

[F1]

Centered finite differences for the zero-Dirichlet boundary-value problem uses $x_{j} = j h$ , the interior equations $(- U_{j - 1} + 2 U_{j} - U_{j + 1}) / h^{2} = g (x_{j})$ , and the exact endpoint values $U_{0} = U_{N + 1} = 0$ .

[F2]

Discrete maximum principle and uniform inverse bound for the Dirichlet stencil proves that, for every admissible exact input, the Thomas recurrence returns the unique solution of this finite-difference system and neither linear-solve-failure branch is reached.

Verification

1.1givenalgebra

Differentiating $u (x) = x (1 - x) / 2$ gives $u^{''} (x) = - 1$ , so $- u^{''} = 1$ ; also $u (0) = u (1) = 0$ .

1.2givenF1algebra

Let $u_{j} = u (x_{j})$ . Since $u$ is quadratic, direct substitution gives $- u_{j - 1} + 2 u_{j} - u_{j + 1} = h^{2}$ for every interior $j$ ; its two endpoint values are zero. Thus the samples satisfy the system in [F1].

2.1step 1.2F2algebra∎

By [F2] the system has a unique solution, so the algorithm's output equals $(u_{1}, \dots, u_{N})$ and its nodal error is exactly zero for every $N \geq 1$ .

Depends on

Used by

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Dependency tree · one level

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, author-source revision 2025.01.27, Chapter 10, linear.md (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Lecture 5: Finite Difference Approximation, Spring 2009 (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Lecture 6: General Linear Second Order Equation, Spring 2009 (standard reference, not scraped)