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.

Positive observation noise with a singular rank-one prediction

Example

Assume AC. Let $T = {a, b}$ with distinct points, $ϕ (a) = 1$ , $ϕ (b) = 2$ , $m = 0$ , and $k (s, t) = ϕ (s) ϕ (t)$ . Use the canonical cylinder-space Gaussian-process prior with this mean and kernel. For $X = X_{*} = (a, b)$ , set $v = (1, 2)^{T}$ , $K_{X X} = K_{* *} = v v^{T}$ , and $R = I_{2}$ , so $A_{X} = K_{X X} + R = I_{2} + v v^{T} ≻ 0$ . With data $y = (1, 0)^{T}$ , the latent posterior mean is $v / 6$ and the latent posterior covariance is $K_{X X} / 6$ , which is singular.

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, feature values $ϕ (a) = 1, ϕ (b) = 2$ , kernel $k (s, t) = ϕ (s) ϕ (t)$ , zero mean, ordered lists $X = X_{*} = (a, b)$ , noise covariance $R = I_{2}$ , and $y = (1, 0)^{T}$ .

[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 finite observation model defines Gram matrices, the observation vector, and noise covariance. Finite Gaussian-process regression observation model

[F2]

For positive-definite observed covariance, the posterior latent mean and covariance are $m_{*} + K_{* X} A_{X}^{- 1} (y - m_{X})$ and $K_{* *} - K_{* X} A_{X}^{- 1} K_{X *}$ . Finite Gaussian-process regression posterior and predictive law

[F3]

The observed matrix of PSD Gram and noise matrices is invertible exactly on the positive-definite branch. Positive definiteness is exactly the inverse condition for the observed GP covariance

Verification

Given: The finite model in the Example.

1.1A1F0F1F3givenconstructalgebra

For any finite list $t_{1}, \dots, t_{r} \in T$ and real $c_{i}$ , the Gram form for $k (s, t) = ϕ (s) ϕ (t)$ is $(\sum_{i} c_{i} ϕ (t_{i}))^{2} \geq 0$ . Thus [A1, F0] supplies the canonical prior. At $X = (a, b)$ this gives $K = v v^{T} = (\begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array})$ . Therefore $A_{X} = I_{2} + v v^{T}$ satisfies $c^{T} A_{X} c = ∥ c ∥^{2} + (v^{T} c)^{2} > 0$ for every nonzero $c$ ; [F3] confirms the inverse branch applies.

2.1F2step 1.1algebra

The identity $(I_{2} + v v^{T}) v = (1 + v^{T} v) v = 6 v$ gives $A_{X}^{- 1} v = v / 6$ , and hence $A_{X}^{- 1} y = A_{X}^{- 1} (1, 0)^{T} = (5 / 6, - 1 / 3)^{T}$ . Since $K = v v^{T}$ , $K A_{X}^{- 1} y = v (v^{T} A_{X}^{- 1} y) = v / 6$ . Formula [F2] therefore gives the stated posterior mean.

3.1F2step 1.1algebra∎

Also $K A_{X}^{- 1} K = v (v^{T} A_{X}^{- 1} v) v^{T} = (5 / 6) v v^{T}$ , because $v^{T} A_{X}^{- 1} v = 5 / 6$ . Thus the posterior covariance is $K - (5 / 6) K = K / 6$ . The nonzero vector $(2, - 1)^{T}$ is annihilated by $K / 6$ , while $K / 6 \neq 0$ , so this covariance is singular.

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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)