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.

Same state dynamics with distinct observation kernels

Example

Take $T = 1$ with scalar state and observation, $X_{0} \sim N_{1} (0, 1)$ , $F_{1} = 1$ , $Q_{1} = 0$ , and $R_{1} = 1$ . Let $q_{1} \sim N_{1} (0, 0)$ and $r_{1} \sim N_{1} (0, 1)$ be independent of each other and of $X_{0}$ . For $c \in {1, 2}$ , set $H_{1} = c$ and use the equations $X_{1} = X_{0} + q_{1}, Y_{1} = c X_{1} + r_{1} .$ The two models have the same initial and transition laws, but their observation kernels are $G_{1}^{(c)} (x, \cdot) = N_{1} (c x, 1)$ .

Facts & Assumptions

Given: The two one-step model specifications in the Example.

[A1]

AC is required for the stated Gaussian laws and the finite path-law supplier. (The Axiom of Choice)

[F1]

The general model law is the finite iterated product of its initial, transition and observation kernels. (General finite-horizon state-space models by kernels)

[F2]

The linear-Gaussian definition specifies Gaussian marginal laws, positive-semidefinite covariances, mutually independent noises and the displayed equations. (Linear-Gaussian state-space models with independent noises)

[F3]

The finite path has the exact kernel-product law and a multivariate normal law with its stated mean and covariance. (The finite linear-Gaussian state-space path is jointly Gaussian)

[F4]

A multivariate normal vector has its specified mean and covariance; singular covariance is allowed. (Multivariate normal law, including singular covariance)

[F5]

Independent random elements have independent generated sigma-algebras. (Independent random elements)

[F6]

Expectations of products of integrable functions of independent random variables factor. (Expectations factor over finite products of independent random variables)

[F7]

Covariance is bilinear on finite linear combinations and equals $E [X Y] - E [X] E [Y]$ for square-integrable variables. (Variance and covariance identities for random variables)

[F8]

Covariance and variance are defined for square-integrable real random variables by centered second moments. (Moments, variance, and covariance on a probability space)

[F9]

The finite-horizon kernel theorem gives a unique law with the ordered iterated-integral probabilities. (Finite-horizon state-space kernel law exists and is unique)

[F10]

A scalar normal law with zero variance is the point mass at its mean. (Standard normal and normal laws)

Verification

Given: The parameters and independent Gaussian noises in the Example.

1.1A1F1F2F3F4F9F10given

By [F10], the degenerate noise satisfies $q_{1} = 0$ almost surely, so both models have $X_{1} = X_{0}$ and the same transition kernel $K_{1} (x, A) = 1_{A} (x)$ . Each set of kernels defines the state-space model. The finite-horizon theorem and the path-law theorem supply its joint law. At $x = 1$ , the two observation-kernel means are $1$ and $2$ , so the kernels themselves are distinct.

1.2F2F3F4F5F6F7F8algebra

In the model indexed by $c$ , $Y_{1} = c X_{0} + r_{1}$ , with both variables centered, $Var (X_{0}) = Var (r_{1}) = 1$ , and $Cov (X_{0}, r_{1}) = 0$ by independence and factorization of expectations. Covariance bilinearity gives $Cov (X_{0}, Y_{1}) = c Var (X_{0}) + Cov (X_{0}, r_{1}) = c$ . Thus the two joint laws have cross-covariances $1$ and $2$ .

2.1F3F4F8step 1.2contradiction∎

Both joint laws have finite second moments. If the laws were equal, the covariance of their first and third coordinates would be equal. Step 1.2 gives different values, so the joint laws are different.

Depends on

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Sources

Nicolai Meinshausen, Time Series Analysis: Notes for the course HS 2022 (standard reference, not scraped)
Dewei Wang, STAT 720 Time Series Analysis, Spring 2015 lecture notes (standard reference, not scraped)