ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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 small mode is lost by an unorthogonalized power pass

Example

Use the explicitly defined thresholded dyadic arithmetic that rounds any computed scalar of magnitude below $2^{- 70}$ to zero and computes all other operations exactly. For $A = diag (1, 2^{- 10})$ , $Ω = I_{2}$ , and power count $q = 4$ , the unorthogonalized product $(A A^{*})^{q} A Ω$ loses its second mode, while the deterministic QR iteration instructions from Reorthogonalized randomized power subspace iteration, executed with this supplied $Ω$ and the stated arithmetic, retain both coordinate modes. Here the Gaussian sampling step is omitted: this is a fixed-sketch arithmetic example, not a probabilistic claim or a general stability guarantee.

Facts & Assumptions

Given: The diagonal matrix, identity start block, power count, and the stated scalar rounding rule.

[F1]

After its sampling step, the power procedure inserts an orthonormal-basis computation after each application of $A^{*}$ and $A$ (Reorthogonalized randomized power subspace iteration).

Verification

1.1givenalgebra

Without intermediate QR, the second diagonal entry of $(A A^{*})^{q} A Ω$ is $(2^{- 10})^{2 q + 1} = 2^{- 90} < 2^{- 70}$ , whereas the leading entry is $1$ . The stated threshold therefore rounds the second column to zero, leaving rank $1$ .

2.1step 1.1F1givenalgebra∎

With QR after each multiplication, the first sample is $A Ω = diag (1, 2^{- 10})$ . Both columns are nonzero and orthogonal; because $2^{- 10} > 2^{- 70}$ , exact column normalization yields $Q_{0} = I_{2}$ . Each subsequent $A^{*} Q$ and $A \tilde{Q}$ is again $diag (1, 2^{- 10})$ , so the same QR step returns $I_{2}$ and neither mode is rounded away.

Depends on

Reorthogonalized randomized power subspace iteration

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

Halko, Martinsson, and Tropp, Finding Structure with Randomness, Algorithm 4.4 and Remark 4.3 (standard reference, not scraped)
Martinsson and Tropp, Randomized Numerical Linear Algebra: Foundations & Algorithms, §11.6 (standard reference, not scraped)