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.

An unobserved unit prevents universal unbiased estimation

Example

Let $U = {1, 2}$ and let the design always sample $s = {1}$ . Compare the fixed populations $y = (0, 0)$ and $y^{'} = (0, 1)$ . The second unit has $π_{2} = 0$ ; both populations produce the same observed record $({1}, 0)$ , while their totals are $0$ and $1$ . This is the two-unit instance of A zero inclusion probability prevents total identification and universal unbiased estimation.

Facts & Assumptions

Given: The fixed two-unit population set $U = {1, 2}$ , the degenerate design supported only on ${1}$ , and the two value functions shown above.

[F1]

Finite populations and sampling designs: A design law specifies the sample support, inclusion probabilities, observed record, and total while holding population values fixed.

[F2]

A zero inclusion probability prevents total identification and universal unbiased estimation: A zero first-order inclusion probability gives equal observed laws for some populations with different totals and rules out an estimator unbiased for all finite populations.

[F3]

Statistics and estimators: An estimator is a statistic, hence a function of the observed record with the specified real target.

[F4]

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

Verification

1.1givenF1algebra

The only supported sample is ${1}$ , so $π_{2} = 0$ . Under both $y$ and $y^{'}$ , the record is $({1}, 0)$ , but $T (y) = 0$ and $T (y^{'}) = 1$ .

2.1F1F2F3F4step 1.1algebra∎

Any observed-record estimator has one value $d$ at $({1}, 0)$ , so its design expectation is $d$ under both populations. It cannot equal both totals $0$ and $1$ . This explicit calculation verifies the instance asserted by [F2], using the unbiasedness definition [F4].

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