ExampleConstruction: AI-generatedVerification: 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.

Truncated SVD denoising on a diagonal problem keeps the declared large singular directions and discards the noisy ones

Example

Let

$A = diag (4, 10^{- 2}), b = [\begin{array}{c} 8 \\ 1 \end{array}], τ = 10^{- 1} .$

Then the truncated pseudoinverse keeps the first singular direction and drops the second, so

$A_{τ}^{+} b = [\begin{array}{c} 2 \\ 0 \end{array}] .$

Facts & Assumptions

Given: The diagonal matrix $A$ , the data vector $b$ , and the threshold $τ = 10^{- 1}$ .

[L1]

The truncated singular-value pseudoinverse keeps the reciprocal $1 / σ_{i}$ exactly when $σ_{i} > τ$ (The truncated singular-value pseudoinverse obtained by discarding singular values below a declared numerical-rank threshold).

[L2]

Truncated SVD is the hard spectral filter among the regularisation methods (Truncated SVD and Tikhonov regularisation act as hard and smooth spectral filters on the singular components).

Verification

technique · computation

1.1

The singular values of $A$ are $4$ and $10^{- 2}$ . Since $4 > τ$ but $10^{- 2} \leq τ$ , [L1] gives $A_{τ}^{+} = diag ⁣ (\frac{1}{4}, 0) .$

L1algebra

2.1

Multiplying by $b$ yields $A_{τ}^{+} b = diag ⁣ (\frac{1}{4}, 0) [\begin{array}{c} 8 \\ 1 \end{array}] = [\begin{array}{c} 2 \\ 0 \end{array}] .$

step 1.1algebra

3.1

The second data component is discarded entirely because it lies in the singular direction below the declared threshold, exactly as [L2] predicts.

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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