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.

Forward Euler at and above the parabolic CFL threshold

Example

Take the one-dimensional periodic heat equation with $κ > 0$ on an even $N$ -point grid, and let $μ = κ Δ t / h^{2}$ . The alternating initial data $U_{j}^{0} = (- 1)^{j}$ keep their magnitude when $μ = 1 / 2$ and grow geometrically when $μ > 1 / 2$ under forward Euler.

Verification

Given: An even integer $N \geq 2$ , periodic spacing $h = L / N$ , $Δ t > 0$ , and the forward-Euler scheme with diffusivity $κ > 0$ .

[F1] Fully discrete θ heat finite-difference scheme: the periodic update is $U^{n + 1} - U^{n} + Δ t A_{h}^{p e r} U^{n} = 0$ when $θ = 0$ and the source is zero.

[F2] Periodic heat-stencil eigenvalues and θ amplification factor: mode $ℓ = N / 2$ has eigenvalue $4 κ / h^{2}$ and multiplier $1 - Δ t λ_{ℓ}$ .

[F3] Forward Euler heat CFL stability and sharp Nyquist instability: on even periodic grids $μ \leq 1 / 2$ gives nonexpansion, while $μ > 1 / 2$ makes the Nyquist multiplier less than $- 1$ .

1.1F1F2given

The vector $ϕ_{j} = (- 1)^{j}$ is the mode $ℓ = N / 2$ , so [F2] gives its multiplier $g = 1 - 4 μ$ .

2.1F2F3step 1.1algebra∎

At $μ = 1 / 2$ , $g = - 1$ , and hence $U_{j}^{n} = (- 1)^{n} ϕ_{j}$ has unchanged norm; [F3] shows all other Fourier modes are also nonexpansive. If $μ > 1 / 2$ , then $g < - 1$ , so $∥ U^{n} ∥_{h} = ∣ g ∣^{n} ∥ ϕ ∥_{h}$ grows geometrically.

Remarks

The endpoint is stable but not damping for the Nyquist mode. This example is periodic and does not assert a finite-boundary result.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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.