ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

The length-five all-ones vector through Bluestein

Example

Take $N = 5$ and $x = (1, 1, 1, 1, 1)$ . Choose $M = 16$ , the least power of two at least $2 N - 1 = 9$ . The direct unnormalised DFT is $X = (5, 0, 0, 0, 0)$ , and the exact Bluestein chirp convolution with this padding represents the same vector. This example checks the prime-length indexing and the padding threshold; it makes no floating-point accuracy claim.

Facts & Assumptions

Given: $N = 5$ , $M = 16$ , and $x_{j} = 1$ for $0 \leq j < 5$ .

[F1]

The Bluestein identity represents the DFT through the zero-padded cyclic chirp convolution with padding $M \geq 2 N - 1$ (Bluestein’s chirp identity embeds an arbitrary DFT in a padded cyclic convolution).

[F2]

$e^{z + w} = e^{z} e^{w}$ and $e^{2 π i n} = 1$ for integers $n$ ; $e^{z} = 1$ exactly when $z \in 2 π i Z$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

Verification

1.1givenF1algebra

Since $2 N - 1 = 9$ , the greatest power of two below $9$ is $8$ and the next is $16$ , so $M = 16$ is the least qualifying power. It satisfies the no-aliasing condition [F1].

1.2givenF2algebra

The direct unnormalised transform is $X_{k} = \sum_{j = 0}^{4} e^{- 2 π i j k / 5}$ . For $k = 0$ , all five terms equal $1$ , so $X_{0} = 5$ . For $1 \leq k \leq 4$ , let $r = e^{- 2 π i k / 5}$ . The addition law in [F2] gives $r^{5} = e^{- 2 π i k} = 1$ . If $r = 1$ , the kernel statement in [F2] gives $- 2 π i k / 5 \in 2 π i Z$ , so $k / 5$ is an integer, contradicting $1 \leq k \leq 4$ . Therefore $r \neq 1$ , and $(1 - r) \sum_{j = 0}^{4} r^{j} = 1 - r^{5} = 0$ gives $X_{k} = 0$ after division by $1 - r$ .

2.1step 1.1step 1.2F1∎

By [F1], the exact chirp convolution with $M = 16$ equals this same DFT; hence its output is $(5, 0, 0, 0, 0)$ .

Depends on

Used by

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Sources

C. Sidney Burrus, Fast Fourier Transforms, Chapter 5 §5.2 (standard reference, not scraped)