ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

A negative finite-difference PDE state is rejected before solving

Example

Discretize $- u^{''} = f$ on $(0, 1)$ with $u (0) = u (1) = 0$ using grid points $x_{i} = i / 3$ for $i = 0, 1, 2, 3$ and the two interior values $u_{i} = u (x_{i})$ , $z = (u_{1}, u_{2})$ . Put $h = 1 / 3$ and define the centered stencil at $i = 1, 2$ by $D_{h} u_{i} = (2 u_{i} - u_{i - 1} - u_{i + 1}) / h^{2}$ , with boundary values $u_{0} = u_{3} = 0$ . The resulting finite-difference system is $A_{h} z = f_{h}$ , where $A_{h} = h^{- 2} (\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array})$ and $f_{h} = (f (x_{1}), f (x_{2}))$ . Use parameter and data spaces $X = Y = R^{2}$ with their Euclidean metrics, take $U = X$ , and use the linear forward map $A_{h} ∣_{U} : U \to Y$ . Impose the physical constraint $C = {(z_{1}, z_{2}) \in R^{2} : z_{1} \geq 0, z_{2} \geq 0}$ for nonnegative nodal concentration. An input validator that checks both coordinates before invoking the linear solver rejects the candidate $z = (- 1, 1)$ before solving because its first coordinate is negative.

Facts & Assumptions

Given: The two-node centered finite-difference model, $X = Y = R^{2}$ with Euclidean metrics, $U = X$ , forward map $A_{h} ∣_{U}$ , and the nonnegative nodal constraint in the Example, with candidate state $z = (- 1, 1)$ .

[F1]

Admissible constraint classes and restricted identifiability: a constrained truth class is a nonempty subset of the parameter space, and its forward map is the restriction to that class.

[F2]

Linear map between vector spaces over the same field: a linear map satisfies $A (a u + b v) = a A (u) + b A (v)$ for scalar coefficients and vectors.

Verification

1.1givenF2algebra

By the definition of $D_{h}$ in the Example and the boundary values $u_{0} = u_{3} = 0$ , $D_{h} u_{1} = (2 u_{1} - u_{2}) / h^{2}$ and $D_{h} u_{2} = (2 u_{2} - u_{1}) / h^{2}$ , which gives the displayed matrix $A_{h}$ . For any vectors $v, w \in R^{2}$ and scalars $a, b$ , matrix multiplication gives $A_{h} (a v + b w) = a A_{h} v + b A_{h} w$ , so $A_{h}$ is linear.

2.1givenF1algebra∎

The set $C$ is nonempty because $(0, 0) \in C$ . The candidate has $z_{1} = - 1 < 0$ , so $z \notin C$ and the validator rejects it before the step that solves $A_{h} z = f_{h}$ . This verifies only the stated discrete admissibility check and makes no continuum PDE or solver-convergence claim.

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.