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.

Inexact local solves enter through spectral-equivalence constants

Example

Take $A = I_{2}$ and the two coordinate subspaces. Exact local matrices are $A_{1} = A_{2} = 1$ , so the exact preconditioner is $B = I_{2}$ and its condition number is $1$ . Choose inexact local matrices ${\tilde{A}}_{1} = 4$ and ${\tilde{A}}_{2} = 1 / 2$ . They satisfy $\frac{1}{4} A_{i} ⪯ {\tilde{A}}_{i} ⪯ 4 A_{i}$ , while $\tilde{B} = diag (1 / 4, 2)$ and $κ_{A} (\tilde{B} A) = 8$ . The theorem's bound is $(β / α) C_{0} C_{1} = 16$ .

Facts & Assumptions

Given: $A = I_{2}$ , $V_{i} = span {e_{i}}$ for $i = 1, 2$ , and the inexact local matrices above.

[F1]

The additive Schwarz theorem gives $κ_{A} (\tilde{B} A) \leq (β / α) C_{0} C_{1}$ under $α A_{i} ⪯ {\tilde{A}}_{i} ⪯ β A_{i}$ (Additive Schwarz condition bound from stable decomposition and synthesis).

[F2]

The Schwarz definition gives $B = \sum_{i} R_{i} A_{i}^{- 1} R_{i}^{T}$ and $\tilde{B} = \sum_{i} R_{i} {\tilde{A}}_{i}^{- 1} R_{i}^{T}$ (Finite-dimensional Schwarz subspace-correction data).

Verification

1.1F2givenalgebra

The coordinate decomposition is stable with $C_{0} = 1$ , and its synthesis bound is $C_{1} = 1$ because $∥ z_{1} e_{1} + z_{2} e_{2} ∥_{2}^{2} = z_{1}^{2} + z_{2}^{2}$ . Exact local matrices equal $1$ , so [F2] gives $B = I_{2}$ and condition number $1$ .

1.2F2givenalgebra

The inexact local inverses are $1 / 4$ and $2$ , so [F2] gives $\tilde{B} = diag (1 / 4, 2)$ . Its eigenvalues are $1 / 4$ and $2$ , hence $κ_{A} (\tilde{B} A) = 8$ . The choices $α = 1 / 4$ and $β = 4$ satisfy the local order bounds.

2.1F1step 1.1step 1.2algebra∎

Substitution of $C_{0} = C_{1} = 1$ , $α = 1 / 4$ , and $β = 4$ into [F1] gives the rigorous upper bound $16$ , which contains the directly computed value $8$ .

Depends on

Used by

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Sources

Jinchao Xu, Iterative Methods by Space Decomposition and Subspace Correction, SIAM Review 34(4) (1992) (standard reference, not scraped)
Peter Bastian, Lecture Notes on Parallel Solution of Large Sparse Linear Systems, University of Heidelberg (2015) (standard reference, not scraped)