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 rank-deficient universal-kriging design

Example

On the finite product space $Ω = {- 1, 1}^{3}$ with equal probability on each of its eight outcomes, let $ϵ_{s}$ be the coordinate indexed by $s \in {0, 1, 2}$ . For unrestricted $β \in R^{3}$ , define $X (s) = β_{0} + β_{1} s + β_{2} s^{2} + ϵ_{s}$ . Observe at $s_{1} = 0$ and $s_{2} = 1$ , and predict at $s_{0} = 2$ , using $p (s) = (1, s, s^{2})^{⊤}$ . The observation design matrix $P = (\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 1 \end{array})$ has rank two, while $p (s_{0}) = (1, 2, 4)^{⊤}$ . The universal-unbiasedness constraint has no solution, although the observation covariance is $Σ = I_{2}$ .

Facts & Assumptions

Given: The equal-weight product space, the three coordinate variables, the quadratic trend, and the observation and target sites.

[F1]

Simple, ordinary, and universal kriging: for trend design matrix $P$ and target trend vector $p_{0}$ , universal-unbiased weights must satisfy $P^{⊤} λ = p_{0}$ .

[F2]

Finite probability spaces, outcome weights, events, and event probabilities and Real random variables on finite probability spaces and their finite distributions: the eight outcomes have weights $1 / 8$ , and each coordinate variable is a function on that finite space.

[F3]

Expectation of a real random variable on a finite probability space: expectations are finite weighted sums.

[F4]

Variance, standard deviation, and covariance on a finite probability space: covariance is the expectation of a centered product.

Verification

Given: The eight equally weighted sign triples and the stated trend.

1.1givenF2F3F4algebra

For each coordinate $ϵ_{s}$ , four outcomes have value $1$ and four have value $- 1$ , so [F2] and [F3] give $E [ϵ_{s}] = 0$ and $E [ϵ_{s}^{2}] = 1$ . For distinct coordinates, each of the four sign pairs occurs twice, so [F4] gives $Cov (ϵ_{s}, ϵ_{t}) = 0$ . Hence the observation-error covariance is $I_{2}$ , which is positive definite.

2.1F1step 1.1algebra∎

The two rows of $P$ are linearly independent, but its last two columns are equal, so its rank is two rather than three. For every $λ = (λ_{1}, λ_{2})^{⊤}$ , $P^{⊤} λ = (λ_{1} + λ_{2}, λ_{2}, λ_{2})^{⊤}$ ; its last two coordinates cannot equal the target's $2$ and $4$ . Thus no weight vector satisfies $P^{⊤} λ = p (s_{0})$ , so no linear predictor is unbiased for every $β$ .

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