ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Uniform endpoint: ordinary bootstrap failure and the m-out-of-n remedy

Example

Take IID $U (0, 1)$ observations and estimate the unknown upper endpoint $1$ by the sample maximum $M_{n}$ . Compare the ordinary bootstrap root $n (M_{n}^{*} - M_{n})$ with the size- $m$ roots $m (M_{m, n}^{*} - M_{n})$ and $m (M_{I} - M_{n})$ when $m \to \infty$ and $m / n \to 0$ .

Facts & Assumptions

Given: The IID uniform sample and the three roots in the statement.

[F1]

Ordinary bootstrap failure for a uniform support endpoint, Proof step 1.2: the unique sample maximum is selected by at least one ordinary bootstrap draw with conditional probability $1 - (1 - 1 / n)^{n} \to 1 - e^{- 1}$ , giving the bootstrap root a conditional atom at zero.

[F2]

m-out-of-n bootstrap and subsampling recover the uniform endpoint root, Proof step 1.1: for every sample size $k$ , the actual root $k (M_{k} - 1)$ has CDF $0$ for $x < - k$ , $(1 + x / k)^{k}$ for $- k \leq x < 0$ , and $1$ for $x \geq 0$ ; these CDFs converge to the continuous limit $F (x) = e^{x}$ for $x < 0$ and $1$ for $x \geq 0$ .

[F3]

m-out-of-n bootstrap and subsampling: With-replacement and without-replacement resampling are distinct laws, and the resampled root uses the size- $m$ normalization.

[F4]

m-out-of-n bootstrap and subsampling recover the uniform endpoint root, Statement: for either resampling method, the conditional CDF converges uniformly in probability to $F$ when $m \to \infty$ and $m / n \to 0$ .

Verification

1.1F1F2F3given

By [F3], the ordinary root is $n (M_{n} - 1)$ and its bootstrap root is centered at $M_{n}$ with the same $n$ -scale. By [F1], the conditional bootstrap root has an atom at zero with limiting mass $1 - e^{- 1}$ ; by [F2], the sampling-root law converges to the continuous CDF $F$ .

1.2F3given

For the smaller-resample methods the corresponding endpoint root is $m (M_{m} - 1)$ , and the conditional roots are centered at the full-sample maximum but scaled by $m$ .

2.1F2F4step 1.2∎

By [F4], for either replacement design the conditional CDF converges uniformly in probability to the CDF of the actual size- $m$ sampling-root limit, so the smaller-resample approximation is first-order valid in this model. This conclusion is specific to the displayed uniform endpoint model and does not give uniform coverage over other laws.

Depends on

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