ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 hand calculation of Arnoldi shows the Hessenberg factorization entry by entry

Example

Take

$A = (\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array}), v_{1} = e_{1} .$

The first two Arnoldi steps give

$v_{2} = e_{2}, v_{3} = e_{3}, {\overset{ˉ}{H}}_{2} = (\begin{array}{cc} 2 & 1 \\ 1 & 2 \\ 0 & 1 \end{array}),$

so with $Q_{2} = [e_{1} e_{2}]$ and $Q_{3} = [e_{1} e_{2} e_{3}]$ one has

$A Q_{2} = Q_{3} {\overset{ˉ}{H}}_{2} .$

Facts & Assumptions

Given: The matrix $A$ and start vector $v_{1} = e_{1}$ .

[L1]

Before breakdown, Arnoldi produces an orthonormal Krylov basis and the factorization $A Q_{m} = Q_{m + 1} {\overset{ˉ}{H}}_{m}$ (Before breakdown, Arnoldi produces an orthonormal Krylov basis and a rectangular upper-Hessenberg factorization).

Verification

technique · direct

1.1

Since $A v_{1} = 2 e_{1} + e_{2}$ , the first Arnoldi coefficients are $h_{11} = 2$ , $h_{21} = 1$ , and $v_{2} = e_{2}$ . Next, $A v_{2} = e_{1} + 2 e_{2} + e_{3}$ , so $h_{12} = 1$ , $h_{22} = 2$ , $h_{32} = 1$ , and $v_{3} = e_{3}$ .

L1algebra

2.1

Therefore $Q_{2} = [e_{1} e_{2}]$ and $Q_{3} {\overset{ˉ}{H}}_{2} = [A e_{1} A e_{2}] = A Q_{2}$ . The vectors are orthonormal exactly as [L1] predicts, and the Hessenberg entries are visible column by column.

L1step 1.1∎

Depends on

Before breakdown, Arnoldi produces an orthonormal Krylov basis and a rectangular upper-Hessenberg factorization

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra (standard reference, not scraped)