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.

Grid Fourier stability alone does not instantiate Lax equivalence

Example

For periodic upwind updates at fixed $0 < ν < 1$ , every individual grid family is power stable in its grid norm. The family on the varying spaces $C^{N}$ is not, as stated, an instance of the common-space Lax theorem.

Facts & Assumptions

Given: The periodic upwind updates on $G_{N} = C^{N}$ for varying $N$ , with fixed $0 < ν < 1$ .

[F1]

A common-space time-scheme family consists of bounded maps $S_{ν} : X \to X$ on one Banach space $X$ (Common Banach-space interface for a linear IVP and its time schemes).

[F2]

Consistency, stability, and convergence in that interface use the same norm of $X$ (Common Banach-space interface for a linear IVP and its time schemes).

[F3]

Under Dependent Choice, a consistent scheme in that common-space interface is convergent on finite time intervals if and only if it is stable there (Lax equivalence for a common Banach-space scheme).

[F4]

For each N-point periodic grid, the upwind update is power stable when $0 < ν < 1$ (Scheme-specific periodic power stability for scalar advection).

[A1]

Dependent Choice is the axiom assumed by [F3] (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Verification

1.1givenF4

By [F4], each N-point upwind update has $∥ T_{N}^{m} ∥_{2, h \to 2, h} \leq 1$ for all $m \geq 0$ when $0 < ν < 1$ .

1.2givenF1F2

The declared operators have domains and codomains $G_{N} = C^{N}$ , which vary with N; no fixed Banach space or projection/reconstruction lift is supplied in this example. Thus they do not meet [F1]'s fixed-space interface, and no same-space norm for comparing their errors is supplied under [F2].

2.1F1F2F3A1step 1.2∎

The theorem [F3] has the common-space interface and consistency among its hypotheses. Since step 1.2 shows that this example supplies no such interface, the grid power bounds alone do not imply convergence by [F3]. A separate common-space lift and its norm/consistency checks would be needed; this example makes no claim that such a lift cannot be constructed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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.

Sources

Lloyd N. Trefethen, Finite Difference and Spectral Methods, Chapter 4, §4.2 (standard reference, not scraped)