ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 zero Ritz residual can belong to an excited eigenstate

Example

Under Countable Choice, take $A = diag (- 1, 1)$ and the unit Ritz vector $u = e_{2}$ . Its Rayleigh quotient is $μ = 1$ and its residual norm is $ρ = ∥ (A - μ I) u ∥ = 0$ , yet the ground eigenvalue is $- 1$ . The residual certifies spectral proximity but does not identify the ground-state branch.

Facts & Assumptions

Given: Countable Choice, the finite Hermitian residual theorem, $A = diag (- 1, 1)$ , and $u = e_{2}$ .

[A1]

Countable Choice as stated in The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

The residual bound places the Rayleigh value near some eigenvalue; the residual alone does not identify which eigenvalue or the lowest one (A Hermitian Ritz residual certifies spectral proximity and a gap-conditioned direction).

[F2]

A Hermitian matrix has real eigenvalues and an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

Verification

1.1F2givenalgebra

By [F2], the eigenvalues of $A$ are $- 1$ and $1$ , so its ground eigenvalue is $- 1$ . The vector $u = e_{2}$ is a unit eigenvector for $1$ , whence $μ = u^{*} A u = 1$ and $(A - μ I) u = 0$ . Thus $ρ = 0$ .

2.1A1F1step 1.1given∎

The general residual theorem [F1] therefore returns exact proximity to the spectrum, but this example's nearby eigenvalue is the excited value $1$ , not the ground value $- 1$ . A zero residual still cannot supply spectral ordering information.

Depends on

Used by

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Dependency tree · two levels

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