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 two-member Gaussian ensemble has a fluctuating covariance

Example

Let $X_{1}, X_{2}$ be independent $N (0, 1)$ forecast members. The divisor- $N$ sample covariance used by Perturbed-observation ensemble Kalman filter at $N = 2$ is random and differs from the true forecast variance with positive probability.

Facts & Assumptions

Given: Independent $X_{1}, X_{2} \sim N (0, 1)$ and the divisor- $N$ covariance convention of Perturbed-observation ensemble Kalman filter.

[A1]

AC is assumed in the standard-normal construction. The Axiom of Choice

[F1]

The standard normal has density $(2 π)^{- 1 / 2} e^{- x^{2} / 2}$ , which is strictly positive on every real interval. Standard normal and normal laws

[F2]

$N (0, 1)$ has mean zero and variance one under the specified multivariate-normal convention. Multivariate normal law, including singular covariance

[F3]

The EnKF covariance is $N^{- 1} \sum_{i} (X_{i} - \overset{ˉ}{X}) (X_{i} - \overset{ˉ}{X})^{T}$ . Perturbed-observation ensemble Kalman filter

Verification

Given: The hypotheses in the Example.

1.1F3givenalgebra

For $N = 2$ , $\overset{ˉ}{X} = (X_{1} + X_{2}) / 2$ , so $X_{1} - \overset{ˉ}{X} = (X_{1} - X_{2}) / 2$ and $X_{2} - \overset{ˉ}{X} = - (X_{1} - X_{2}) / 2$ . By [F3], $C_{2} = \frac{1}{2} ((X_{1} - \overset{ˉ}{X})^{2} + (X_{2} - \overset{ˉ}{X})^{2}) = \frac{(X_{1} - X_{2})^{2}}{4} .$

2.1A1F1F2givenstep 1.1algebra∎

The event $A = {- 0.1 < X_{1} < 0.1, 3 < X_{2} < 3.1}$ has positive probability: each interval has positive probability by [F1], and the variables are independent. On $A$ , $X_{2} - X_{1} > 2.9$ , so step 1.1 gives $C_{2} > {2.9}^{2} / 4 = 2.1025 > 1$ . The true variance of $N (0, 1)$ is $1$ by [F2]; hence the finite-ensemble covariance differs from it with positive probability.

Depends on

Used by

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Sources

K. J. H. Law, A. M. Stuart, and K. C. Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 4 §4.2.3 (standard reference, not scraped)
Geir Evensen, Femke C. Vossepoel, and Peter Jan van Leeuwen, Data Assimilation Fundamentals, Chapter 8 (standard reference, not scraped)