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.

Absolute-value boundary and squared-mean degeneracy

Example

For IID standard-normal data at mean zero, compare the two estimators $∣ {\overset{ˉ}{X}}_{n} ∣$ and ${\overset{ˉ}{X}}_{n}^{2}$ . The first has a kink at the boundary value zero. The second has zero first derivative and a nondegenerate second-order limit. Both ordinary bootstrap roots retain a random conditional distribution.

Facts & Assumptions

Given: AC, the IID normal model at mean zero, and the two statistic sequences.

[A1]

The Axiom of Choice: AC is the axiom that every family of nonempty sets has a choice function.

[F1]

Ordinary bootstrap failure for nonsmooth and first-order-degenerate mean functionals: Under AC and IID standard-normal observations, both correctly centered bootstrap root CDFs at zero stay separated in probability from the corresponding true root CDF value zero.

[F2]

m-out-of-n bootstrap and subsampling: The ordinary bootstrap is the size- $n$ with-replacement method, centered at the observed estimator and scaled by the actual root normalization.

Verification

1.1A1F1given

Under AC and the IID standard-normal model in [F1], the boundary mean is zero. The target for $∣ μ ∣$ is zero and $∣ \cdot ∣$ is not differentiable there; the target for $μ^{2}$ is also zero and its first derivative vanishes. The root normalizations are respectively $\sqrt{n}$ and $n$ .

2.1A1F1F2given∎

The ordinary bootstrap roots therefore are $\sqrt{n} (∣ {\overset{ˉ}{X}}_{n}^{*} ∣ - ∣ {\overset{ˉ}{X}}_{n} ∣)$ and $n (({\overset{ˉ}{X}}_{n}^{*})^{2} - {\overset{ˉ}{X}}_{n}^{2})$ , as follows from the centering and size- $n$ scale in [F2]. Theorem [F1] proves that each conditional CDF fails at the continuity point zero, despite the nondegenerate true limits $∣ Z ∣$ and $Z^{2}$ .

Depends on

Used by

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