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.

Periodic and symmetric padding give different Haar coefficients

Example

Start with the odd-length data $(0, 0, 1)$ . Define two explicit padding rules to length four. Periodic endpoint padding appends the first entry, giving $x_{P} = (0, 0, 1, 0)$ . Symmetric endpoint padding repeats the final entry, giving $x_{S} = (0, 0, 1, 1)$ . Applying the first Haar stage to the last pair gives $(a, d) = (1 / \sqrt{2}, 1 / \sqrt{2})$ for $x_{P}$ and $(a, d) = (\sqrt{2}, 0)$ for $x_{S}$ . The coefficients change because the boundary rule changes the pair presented to the filter.

Verification

Given: The two length-four padded vectors above and the Haar pair map of Finite periodic Haar transform.

[F1] Finite periodic Haar transform: a pair $(u, v)$ maps to $((u + v) / \sqrt{2}, (u - v) / \sqrt{2})$ .

1.1givenF1algebra

For the periodic padding, the last pair is $(1, 0)$ ; substitution gives average $(1 + 0) / \sqrt{2} = 1 / \sqrt{2}$ and detail $(1 - 0) / \sqrt{2} = 1 / \sqrt{2}$ .

2.1givenF1step 1.1algebra∎

For the symmetric padding, the last pair is $(1, 1)$ ; substitution gives average $(1 + 1) / \sqrt{2} = \sqrt{2}$ and detail $(1 - 1) / \sqrt{2} = 0$ . The two coefficient pairs differ.

Depends on

Finite periodic Haar transform

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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