ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

Noisy interpolation on two independent coordinates

Example

Assume AC and take $T = {a, b}$ with distinct points, mean $m = 0$ , and equality kernel $k (s, t) = 1$ if $s = t$ and $0$ otherwise. Use the canonical cylinder-space Gaussian-process prior with this mean and kernel. For $X = (a, b)$ , set $K_{X X} = (k (x_{i}, x_{j})) = I_{2}$ and observe with centered Gaussian noise $ε \sim N_{2} (0, I_{2})$ independent of $f$ , so $A_{X} = K_{X X} + I_{2} = 2 I_{2}$ , with data $y = (2, - 1)^{T}$ . Predict at $X_{*} = X$ . The latent posterior mean is $(1, - \frac{1}{2})^{T}$ and its covariance is $\frac{1}{2} I_{2}$ . If a centered Gaussian test-noise vector $η_{*} \sim N_{2} (0, R_{*})$ with $R_{*} = I_{2}$ is independent of the pair $(f, ε)$ , then the noisy predictive mean is unchanged and its covariance is $\frac{3}{2} I_{2}$ .

Example references

Rasmussen and Williams, GPML, Chapter 2 §2.2, printed pp. 13–18

Bishop, PRML, §6.4.2, printed pp. 306–309

Facts & Assumptions

Given: AC, the two-point set, equality kernel, zero mean, ordered training and test lists, centered Gaussian training noise $ε \sim N_{2} (0, I_{2})$ independent of $f$ , optional test noise $η_{*} \sim N_{2} (0, I_{2})$ independent of $(f, ε)$ , and observation $y$ as stated.

[A1]

AC is the axiom used by the canonical GP construction and finite conditioning interface. The Axiom of Choice

[F0]

Under AC, a symmetric positive-semidefinite kernel and mean define a canonical Gaussian process with the specified finite Gaussian laws. A mean and positive-semidefinite kernel define a Gaussian process

[F1]

The observation model permits finite ordered lists and defines $K_{A B}$ , $m_{X}$ , $Y = f_{X} + ε$ , and test noise independent of the pair $(f, ε)$ . Finite Gaussian-process regression observation model

[F2]

For $A_{X} ≻ 0$ , the posterior latent mean and covariance are $m_{*} + K_{* X} A_{X}^{- 1} (y - m_{X})$ and $K_{* *} - K_{* X} A_{X}^{- 1} K_{X *}$ ; test noise independent of the pair $(f, ε)$ adds its covariance. Finite Gaussian-process regression posterior and predictive law

[F3]

For PSD Gram and noise matrices, the observed covariance is invertible exactly when it is positive definite. Positive definiteness is exactly the inverse condition for the observed GP covariance

Verification

Given: The finite model in the Example.

1.1A1F0F1F2F3givenconstructalgebra

For any finite list $t_{1}, \dots, t_{r} \in T$ and scalars $c_{i}$ , the equality-kernel Gram form is $\sum_{t \in T} (\sum_{i : t_{i} = t} c_{i})^{2} \geq 0$ . Thus the kernel is positive semidefinite, and [A1, F0] supplies the stated canonical prior. At $X = (a, b)$ , $K = I_{2}$ , so $A_{X} = K + R = 2 I_{2} ≻ 0$ and the inverse required in [F2] exists by [F3].

2.1A1F2step 1.1algebra

Since $m_{X} = 0$ and $K_{* X} = K_{X X} = I_{2}$ , [F2] gives posterior mean $(2 I_{2})^{- 1} y = \frac{1}{2} (2, - 1)^{T} = (1, - \frac{1}{2})^{T}$ . It gives posterior covariance $I_{2} - I_{2} (2 I_{2})^{- 1} I_{2} = \frac{1}{2} I_{2}$ .

3.1F1F2step 2.1algebra∎

By the Example's hypothesis, $η_{*}$ is independent of the pair $(f, ε)$ , so [F2]'s test-noise clause applies: the conditional mean is unchanged and $I_{2}$ is added to the covariance, yielding $\frac{1}{2} I_{2} + I_{2} = \frac{3}{2} I_{2}$ .

Depends on

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Sources

Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, Chapter 2 §2.2 (standard reference, not scraped)
Christopher M. Bishop, Pattern Recognition and Machine Learning, §6.4.2 (standard reference, not scraped)