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.

A certainty unit and a census under Horvitz–Thompson estimation

Example

Let $U = {1, 2, 3}$ with fixed values $(a, b, c)$ . First use the design that chooses ${1, 2}$ or ${1, 3}$ with probability $1 / 2$ each. Unit $1$ is a certainty unit: $π_{1} = 1$ , while $π_{2} = π_{3} = 1 / 2$ . The Horvitz–Thompson total is $a + 2 b$ or $a + 2 c$ , according to the sample. In the census design, which always selects $U$ , every $π_{i} = 1$ and the Horvitz–Thompson total and mean equal the population total and mean exactly.

Facts & Assumptions

Given: A finite population $U = {1, 2, 3}$ with fixed real values $y_{1} = a$ , $y_{2} = b$ , $y_{3} = c$ ; either the two-sample design described above or the census design.

[F1]

Finite populations and sampling designs: Inclusion probabilities are design probabilities, and the total and mean hold the values fixed.

[F2]

The Horvitz–Thompson total and mean are design-unbiased under positive inclusion probabilities: With all first-order inclusion probabilities positive, the inverse-inclusion weighted total and its population-size-scaled mean are unbiased.

[F3]

Expectation of a real random variable on a finite probability space: Expectations are finite weighted sums over the design support.

[F4]

Unbiased estimators and UMVU estimators: An estimator is unbiased when its expectation equals its estimand at every parameter value.

Verification

1.1givenF1algebra

Under the two-sample design, unit $1$ is in both support samples, unit $2$ only in the first, and unit $3$ only in the second. Therefore $π_{1} = 1$ and $π_{2} = π_{3} = 1 / 2$ . The weighted total is $a + 2 b$ for sample ${1, 2}$ and $a + 2 c$ for sample ${1, 3}$ , so unit $1$ contributes its observed value $a$ without expansion.

2.1F1F2F3F4step 1.1algebra

The design expectation of the displayed total is $\frac{1}{2} (a + 2 b) + \frac{1}{2} (a + 2 c) = a + b + c = T (y)$ by [F3]. Since each $π_{i} > 0$ , [F2] also identifies its divided-by- $3$ form as the unbiased estimator of ${\overset{ˉ}{y}}_{U}$ .

3.1givenF1algebra∎

In the census design, the only supported sample is $U$ with probability $1$ , so every $π_{i} = 1$ . Thus the total estimator is $\sum_{i \in U} y_{i} = T (y)$ on the sole outcome, and its mean version is $T (y) / 3 = {\overset{ˉ}{y}}_{U}$ .

Depends on

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