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.

A polynomial scale tail is outside the stated theorem

Example

Let $G$ be the mixture with weight $1 / 2$ on $Unif [1, 2]$ and weight $1 / 2$ on the Pareto law with density $2 s^{- 2} 1_{[2, \infty)} (s)$ . This scale law has a polynomial upper tail.

Facts & Assumptions

Given: The scale law $G$ just defined and the upper-tail premise in Dirichlet normal-mixture posterior contraction.

[F1]

Dirichlet normal-mixture posterior contraction: the theorem requires $1 - G (s) ≲ e^{- β s^{γ}}$ for some $β > 0$ , $γ > 1$ as $s \to \infty$ .

Verification

1.1givenalgebra

The Pareto density integrates to one because $\int_{2}^{\infty} 2 s^{- 2} d s = 1$ , so $G$ is a probability measure. For $s < 1$ , $G (s) = 0$ , which satisfies every required lower-tail upper bound near zero.

1.2givenalgebra

For $s \geq 2$ , the uniform component has no mass above $s$ and the Pareto survival function is $2 / s$ ; the mixture weight $1 / 2$ therefore gives $1 - G (s) = 1 / s$ .

2.1F1step 1.2givenalgebra∎

For any $β > 0$ and $γ > 1$ , $e^{β s^{γ}} / s \to \infty$ , so $1 / s$ is not bounded above by a constant multiple of $e^{- β s^{γ}}$ . Thus this prior is outside the theorem's upper-tail hypotheses. This example says nothing about whether its posterior contracts or is inconsistent.

Depends on

Dirichlet normal-mixture posterior contraction

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Subhashis Ghosal and Aad W. van der Vaart, Posterior convergence rates of Dirichlet mixtures at smooth densities, discussion following Theorem 1 (standard reference, not scraped)