CounterexampleConstruction: 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.

A repeated unit root causes a linearly growing parasitic mode

Statement refuted

Every consistent two-step linear multistep method is zero-stable.

Facts & Assumptions

Given: The method and consistency conventions of Linear multistep methods, consistency, order, and the root condition.

[F1]

Taylor moment conditions characterize the order of a linear multistep method says consistency is equivalent to $ρ (1) = 0$ and $ρ^{'} (1) = σ (1)$ .

Counterexample

Take $ρ (ζ) = (ζ - 1)^{2}$ and $σ (ζ) = 0$ , so the recurrence is $Y_{n + 2} - 2 Y_{n + 1} + Y_{n} = 0$ . This is a consistent first-order two-step method, but its double unit root is not zero-stable.

1.1F1algebra

Here $ρ (1) = 0$ and $ρ^{'} (1) = 0 = σ (1)$ , so [F1] proves consistency.

1.2givenalgebra

The characteristic polynomial has a root $1$ of multiplicity two, so it violates the root condition in Linear multistep methods, consistency, order, and the root condition.

1.3givenconstruct

Apply the recurrence to $y^{'} = 0$ , whose exact solution is $y = 0$ , with $Y_{0} = 0$ and $Y_{1} = h$ . The sequence $Y_{n} = n h$ satisfies the recurrence because its second difference is zero.

2.1step 1.3algebra∎

On the grid $h = T / N$ , the starting error is $h \to 0$ , but $∣ Y_{N} - y (T) ∣ = N h = T$ . Thus convergence fails even for starting errors of order $h$ , and the proposed claim is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.