ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Soft thresholding shrinks a retained coefficient while hard best-m retention does not

Example

In the standard orthonormal basis, let $c = (3, 1)$ , choose $m = 1$ , and use soft threshold $λ = 1$ . Hard best-1 retention gives $(3, 0)$ and error norm $1$ . Soft thresholding gives $(2, 0)$ and error norm $\sqrt{2}$ : it both shrinks the retained coefficient and discards the smaller one.

Verification

Given: $c = (3, 1)$ , $m = 1$ , $λ = 1$ , and the standard orthonormal basis $U = I_{2}$ .

[F1] Hard m-term retention and soft thresholding of finite coefficients: $H_{m}$ keeps the $m$ largest absolute coordinates, while $S_{λ} (c)_{i} = sign (c_{i}) (∣ c_{i} ∣ - λ)_{+}$ .

[F2] Hard m-term retention is optimal for an orthonormal coefficient basis: hard retention has squared error equal to the discarded squared-coefficient sum and is best among one-term approximations.

1.1givenF1F2algebra

The coordinate magnitudes are $3 > 1$ , so [F1] gives $H_{1} (c) = (3, 0)$ . The discarded coordinate is $1$ , hence [F2] gives hard squared error $1$ .

2.1givenF1step 1.1algebra∎

The soft formula gives $S_{1} (c) = (2, 0)$ . Its residual is $(1, 1)$ , whose squared norm is $1^{2} + 1^{2} = 2$ , so the soft error norm is $\sqrt{2} > ⁣ 1$ .

Depends on

Used by

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Sources

Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd ed., Chapter 9 (standard reference, not scraped)
Gilbert Strang and Kevin Amaratunga, Wavelets and Filter Banks, MIT OpenCourseWare 18.327, Handout 15 (standard reference, not scraped)