Pipeline-generated

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.

Random fields and variograms — Examples

1 · Prerequisites

2 · Summary

The candidate covariance $C (h) = 1 + h^{2}$ fails the two-site positive-semidefinite test. The quadratic variogram $γ (h) = h^{2}$ is realized by a random line $X (t) = t Z$ with $E [Z] = 0$ and $E [Z^{2}] = 2$ ; its increment calculation also gives the real zero-sum CND inequality. The witness has stationary increments but nonconstant level variance, so it is not wide stationary. No Schoenberg converse is used.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)Open item page →

A failed stationary covariance and a valid quadratic variogram

Example

On $R$ , the lag function $C (h) = 1 + h^{2}$ is not a stationary covariance. By contrast, $γ (h) = h^{2}$ is a valid order-two intrinsic variogram: if $Z$ takes the values $\pm \sqrt{2}$ with equal probabilities and $X (t) = t Z$ , then $\frac{1}{2} E [(X (t + h) - X (t))^{2}] = h^{2}$ . This field has stationary second-order increments but is not wide stationary.

Facts & Assumptions

Given: The real-lag candidate $C (h) = 1 + h^{2}$ and the finite probability space $Ω = {- 1, + 1}$ with equal weights.

[F1]

Random fields, stationarity, covariance, and variograms: a wide- stationary field has a covariance kernel depending only on lag; an order-two intrinsic field has displacement-only finite increment second moments.

[F2]

Covariance kernels of square-integrable fields are positive semidefinite: every finite covariance matrix of real field values has nonnegative real quadratic forms.

[F3]

Order-two intrinsic variograms are conditionally negative definite: for real coefficients summing to zero, the variogram quadratic form is nonpositive.

[F4]

Finite probability spaces, outcome weights, events, and event probabilities: the two outcomes each receive weight $1 / 2$ .

[F5]

Real random variables on finite probability spaces and their finite distributions: a real random variable on this space is a real-valued function on its two outcomes.

[F6]

Expectation of a real random variable on a finite probability space: expectation is the sum of the two outcome values weighted by $1 / 2$ .

[F7]

Moments, variance, and covariance on a probability space: for a square-integrable real variable, variance is the expected centered square.

Verification

1.1givenF2algebra

If $C$ were a stationary covariance, its matrix at the two sites $0$ and $1$ would be $M = (\begin{array}{cc} C (0) & C (- 1) \\ C (1) & C (0) \end{array}) = (\begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array}) .$ The covariance-kernel lemma [F2] would make this matrix positive semidefinite, but for $v = (1, - 1)^{T}$ , $v^{T} M v = - 2 < 0$ . Hence $C (h) = 1 + h^{2}$ is not a stationary covariance.

1.2givenF3algebra

For any real $a_{1}, \dots, a_{n}$ with $\sum_{i} a_{i} = 0$ and sites $s_{1}, \dots, s_{n}$ , $\sum_{i, j} a_{i} a_{j} (s_{i} - s_{j})^{2} = - 2 {(\sum_{i} a_{i} s_{i})}^{2} \leq 0.$ This directly verifies the zero-sum CND inequality for $γ (h) = h^{2}$ and is consistent with [F3].

1.3F4F5F6construct

Define $Z (ω) : = \sqrt{2} ω$ on $Ω = {- 1, + 1}$ . By the finite space and expectation definitions [F4]–[F6], $E [Z] = \frac{1}{2} \sqrt{2} - \frac{1}{2} \sqrt{2} = 0, E [Z^{2}] = \frac{1}{2} (2) + \frac{1}{2} (2) = 2.$

2.1givenF1F6step 1.3algebra

Define $X (t) : = t Z$ , so every coordinate has finite range and is square-integrable. For any admissible $t, h$ , $X (t + h) - X (t) = h Z$ , and $\frac{1}{2} E [(X (t + h) - X (t))^{2}] = \frac{1}{2} h^{2} E [Z^{2}] = h^{2},$ independent of $t$ . Its increment mean is also zero, so the stated intrinsic stationarity and variogram claim hold by [F1].

3.1givenF1F6F7step 1.3step 2.1algebra∎

The field mean is zero and $Var (X (t)) = 2 t^{2}$ by [F6]–[F7], so its variance is not constant in $t$ . Thus this intrinsically stationary field is not wide stationary, as claimed.