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.

Exact two-coefficient Haar compression of a four-entry step vector

Example

For the four-entry step vector $x = (1, 1, 0, 0)$ , the finite periodic Haar transform has coefficient vector $(1, 1, 0, 0)$ in the order “coarse average, coarsest detail, finest details.” Retaining the first two coefficients reconstructs $x$ exactly. Among one-coefficient approximations the squared error is $1$ .

Verification

Given: $x = (1, 1, 0, 0)$ and the transform convention of Finite periodic Haar transform.

[F1] Finite periodic Haar transform: each pair $(u, v)$ maps to average/detail $((u + v) / \sqrt{2}, (u - v) / \sqrt{2})$ , and output order is coarse coefficient followed by detail blocks from coarse to fine.

[F2] Orthogonality, inverse, and Parseval identity for the finite periodic Haar transform: the transform is orthogonal with inverse transpose and preserves squared Euclidean norm.

[F3] Hard m-term retention is optimal for an orthonormal coefficient basis: keeping $m$ largest coefficients is best among all $m$ -coefficient reconstructions and squared error is the discarded squared sum.

1.1givenF1algebra

The first pair $(1, 1)$ yields approximation $\sqrt{2}$ and detail $0$ ; the second pair $(0, 0)$ yields approximation and detail $0$ . Transforming the remaining approximation pair $(\sqrt{2}, 0)$ gives coarse coefficient $1$ and coarsest detail $1$ . In the declared ordering the full vector is therefore $(1, 1, 0, 0)$ .

2.1givenF2F3step 1.1algebra∎

Keeping the two nonzero coefficients gives the original vector by the inverse in [F2]. Keeping only one leaves one coefficient of magnitude $1$ discarded; [F3] gives squared error $1$ .

Depends on

Used by

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Sources

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