CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

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A preconditioner can worsen the condition number that actually controls CG

Statement refuted

Every invertible, or even every symmetric positive-definite, preconditioner improves the condition number that governs CG.

Facts & Assumptions

Given: The residual-and-error maps for preconditioning and the symmetric positive-definite CG transform.

[F1]

Symmetric preconditioning replaces $A$ by the transformed operator $C^{- 1} A C^{- *}$ (Invertible preconditioners give equivalent linear systems, with the transformed residuals and errors written explicitly).

[L1]

The CG bound is governed by the spectral condition number of the transformed operator (Symmetric positive-definite preconditioning preserves a Hermitian positive-definite CG problem, and the CG bound uses the transformed condition number).

Counterexample

technique · direct calculation

1.1F1algebra

Take $A = I_{2}, M = diag (1, 1 / 100) .$ Then $A$ is already Hermitian positive definite with $κ_{2} (A) = 1.$ Since $M = C C^{*}$ with $C = diag (1, 1 / 10)$ , [F1] gives the transformed operator $C^{- 1} A C^{- *} = diag (1, 10) I_{2} diag (1, 10) = diag (1, 100) .$

2.1L1step 1.1algebra∎

The transformed spectral condition number is therefore $κ_{2} (C^{- 1} A C^{- *}) = 100 > 1 = κ_{2} (A) .$ So this symmetric positive-definite preconditioner worsens the condition number that actually appears in the CG bound from [L1]. The refuted statement is false.

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.

Sources

Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (standard reference, not scraped)