Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.

Lucky breakdown occurs when the current Krylov space is already invariant

Example

Let

A=diag⁡(2,3,4),b=e1.

Then K1(A,b)=span⁡{e1} and A(K1(A,b))⊆K1(A,b), because Ae1=2e1. Arnoldi therefore breaks down exactly after the first step: the next residual is zero.

Facts & Assumptions

Given: The diagonal matrix A and start vector b=e1.

[L1]

Exact Arnoldi breakdown at the next step is equivalent to invariance of the current Krylov space (Exact Arnoldi breakdown is equivalent to invariance of the current Krylov space).

Verification

technique · direct
1.1

Here K1(A,b)=span⁡{e1} and Ae1=2e1∈K1(A,b), so the current Krylov space is A-invariant.

algebra
2.1

By [L1], that invariance is equivalent to exact Arnoldi breakdown at the next step. Concretely, the Arnoldi residual is w1=Ae1−⟨Ae1,e1⟩e1=2e1−2e1=0.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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