TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-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.

The residual and orthogonality certifier encloses the full ordered spectrum

Statement

If Certified Hermitian eigenvalue intervals from residual and Gram bounds returns certified with outward radius bound $\overset{ˉ}{η}$ and intervals $I_{k}$ , then for every admitted Hermitian $A$ its $k$ -th increasing eigenvalue lies in $I_{k}$ . Each $I_{k}$ contains the symmetric interval $[θ_{(k)} - \overset{ˉ}{η}, θ_{(k)} + \overset{ˉ}{η}]$ . Inconclusive output makes no spectral claim.

Facts & Assumptions

Given: A nonempty Hermitian family and a certified algorithm return with validated bounds $δ, ρ, M_{A}, M_{Θ}, τ$ , outward radius bound $\overset{ˉ}{η}$ , and returned intervals $I_{k}$ .

[F1]

Outward interval calculations enclose every exact family residual, Gram defect and norm upper bound (Outward-rounded operations are inclusion-isotone and enclose DAG evaluations).

[F2]

The algorithm checks the hypotheses of the near-orthonormal residual enclosure and returns $\overset{ˉ}{η} \geq ρ + (M_{A} + M_{Θ}) τ$ with each $I_{k}$ containing $[θ_{(k)} - \overset{ˉ}{η}, θ_{(k)} + \overset{ˉ}{η}]$ (Certified Hermitian eigenvalue intervals from residual and Gram bounds, Residual and near-orthogonality data for an approximate Hermitian eigenbasis).

[F3]

Such residual and orthogonality data enclose every ordered Hermitian eigenvalue within $ρ + (M_{A} + M_{Θ}) τ$ (Near-orthonormal residual data enclose all ordered Hermitian eigenvalues).

[F4]

The operator 2-norm is at most the Frobenius norm (Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison).

Proof

1.1F1F2F4algebra

By [F1], the reported interval bounds dominate the corresponding Frobenius norms for every admitted $A$ , and [F4] makes them upper bounds for the operator norms in the residual and orthogonality certificate. The algorithm accepts only if these bounds satisfy each inequality in that definition, including $δ < 1$ and $1 - \sqrt{1 - δ} \leq τ$ . Thus [F2] supplies valid certificate data uniformly for the whole family.

2.1F2F3step 1.1algebra∎

Apply [F3] to each $A$ . The theorem gives $∣ λ_{k} (A) - θ_{(k)} ∣ \leq ρ + (M_{A} + M_{Θ}) τ \leq \overset{ˉ}{η}$ for every increasing index $k$ , with the same validated radius bound for all members. Hence each eigenvalue lies in the centered interval contained in $I_{k}$ by [F2], and therefore lies in $I_{k}$ .

Depends on

Used by

Cited to discharge well-definedness by Certified Hermitian eigenvalue intervals from residual and Gram bounds.

Dependency tree · two levels

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Sources

Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §§5.1–5.6 (standard reference, not scraped)
Siegfried M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010), §§5.1–5.5 (standard reference, not scraped)