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.

Two-series null centering with a final block prefix

Example

Apply moving-mode Null-constrained block bootstrap for independent series in $H = R$ to the series $(0, 2, 4)$ and $(10, 12, 14, 16)$ , with $b_{1} = b_{2} = 2$ . Their observed means are $2$ and $13$ , but both bootstrap series have conditional mean ${\overset{ˉ}{Y}}_{P} = 58 / 7$ .

Verification

Given: The two fixed series and block lengths in the Example.

[F1] Null-constrained block bootstrap for independent series subtracts the within-series mean, then the candidate average at each within-block position, and adds the pooled mean; it retains exactly $n_{i}$ observations.

[F2] Exact null centering of the K-sample block bootstrap gives the common conditional mean for every retained observation and hence every series mean.

1.1F1givenalgebra

Here $M = 7$ , ${\overset{ˉ}{Y}}_{P} = (6 + 52) / 7 = 58 / 7$ , $N_{1} = 2$ , $N_{2} = 3$ , and $k_{1} = k_{2} = 2$ . The first residual series is $(- 2, 0, 2)$ ; its candidate blocks are $(- 2, 0)$ and $(0, 2)$ , with position averages $(- 1, 1)$ . After subtracting those averages the two candidates are $(- 1, - 1)$ and $(1, 1)$ . The second residual series is $(- 3, - 1, 1, 3)$ ; its candidates are $(- 3, - 1), (- 1, 1), (1, 3)$ , also with position averages $(- 1, 1)$ , and become $(- 2, - 2), (0, 0), (2, 2)$ .

2.1F1F2step 1.1algebra∎

Let $V_{1}, V_{2}$ be independent uniforms on ${- 1, 1}$ and $W_{1}, W_{2}$ independent uniforms on ${- 2, 0, 2}$ , all four independent. The outputs are $(58 / 7 + V_{1}, 58 / 7 + V_{1}, 58 / 7 + V_{2})$ and $(58 / 7 + W_{1}, 58 / 7 + W_{1}, 58 / 7 + W_{2}, 58 / 7 + W_{2})$ . The first output retains only the first entry of its final block. Since all four offsets have mean zero, $E^{*} {\overset{ˉ}{Y}}_{1}^{*} = E^{*} {\overset{ˉ}{Y}}_{2}^{*} = 58 / 7$ , explicitly realizing [F2] despite the unequal observed means and unequal sample sizes.

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