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.

Distant alternatives can prevent uniform testability

Example

On $X = {0, 1}$ with counting measure, let $p_{0} (0) = p_{0} (1) = 1 / 2$ . Use labels $0, 1, 2, \dots$ , with $p_{0}$ at label $0$ and, for $j \geq 1$ , $p_{j} (1) = 1 / 2 + δ_{j}, p_{j} (0) = 1 / 2 - δ_{j}, δ_{j} : = 1 / (j + 4) .$ Give label $0$ prior probability $1 / 2$ and label $j$ prior probability $2^{- (j + 1)}$ . Use the discrete metric $ρ (j, k) = 0$ if $j = k$ and $1$ otherwise, so every alternative is distance one from label $0$ . Take observations IID with the law at the selected parameter label.

Facts & Assumptions

Given: The model, prior, parameter metric, and IID law in the Example.

[F1]

KL projection sets and relative KL neighborhoods: the relative KL neighborhoods are defined using $K (θ) - K^{*}$ .

[A1]

Assume AC through the imported KL convention The Axiom of Choice. The displayed finite density family is explicit.

[F2]

Total variation distance for probability laws: $∥ P - Q ∥_{T V} = \sup_{A} ∣ P (A) - Q (A) ∣$ .

[F3]

Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric: a metric separates distinct points and obeys the triangle inequality.

Verification

1.1A1F1F3algebra

For $δ_{j} = 1 / (j + 4)$ , direct substitution gives $K L (P_{0} ∥ P_{j}) = \frac{1}{2} \log \frac{1 / 2}{1 / 2 + δ_{j}} + \frac{1}{2} \log \frac{1 / 2}{1 / 2 - δ_{j}} = - \frac{1}{2} \log (1 - 4 δ_{j}^{2}) > 0.$ Thus $Θ^{*} = {0}$ . Every relative KL neighborhood contains $0$ and therefore has prior mass at least $1 / 2$ , while every $j \geq 1$ lies in the distance-one alternative set.

1.2F2givenalgebra

Fix $n$ and choose $j_{n} \geq n^{2}$ . For one coordinate define a joint law of $(X, Y)$ by masses $1 / 2 - δ_{j_{n}}$ at $(0, 0)$ , $1 / 2$ at $(1, 1)$ , and $δ_{j_{n}}$ at $(0, 1)$ , with zero mass at $(1, 0)$ . Its marginals are $P_{0}$ and $P_{j_{n}}$ , and its disagreement probability is $δ_{j_{n}}$ . Take the $n$ -fold product of this finite joint law. By the union bound, the coupled vectors differ with probability at most $n δ_{j_{n}} \leq 1 / n$ . For every event $A$ , the indicators of ${X_{1 : n} \in A}$ and ${Y_{1 : n} \in A}$ can differ only when the vectors differ, so by [F2], $∥ P_{0}^{n} - P_{j_{n}}^{n} ∥_{T V} \leq 1 / n .$

2.1F2step 1.2algebra∎

For probability laws $P, Q$ on the finite set ${0, 1}^{n}$ , $∥ P - Q ∥_{T V} = \sum_{z : P (z) > Q (z)} (P (z) - Q (z))$ : the set of points with positive difference attains the supremum over events, and no event can sum more than all positive differences. Hence for every test $ϕ_{n}$ with $0 \leq ϕ_{n} \leq 1$ , $P ϕ_{n} - Q ϕ_{n} = \sum_{z} ϕ_{n} (z) (P (z) - Q (z)) \leq \sum_{z : P (z) > Q (z)} (P (z) - Q (z)) = ∥ P - Q ∥_{T V} .$ Taking $P = P_{0}^{n}$ , $Q = P_{j_{n}}^{n}$ and rearranging gives $P_{0}^{n} ϕ_{n} + P_{j_{n}}^{n} (1 - ϕ_{n}) \geq 1 - ∥ P_{0}^{n} - P_{j_{n}}^{n} ∥_{T V} \geq 1 - \frac{1}{n} .$ If $P_{0}^{n} ϕ_{n} \to 0$ , then $\sup_{j \geq 1} P_{j}^{n} (1 - ϕ_{n}) \geq P_{j_{n}}^{n} (1 - ϕ_{n}) \to 1$ . Since the selected minimizer is the true law, $Q_{j} = P_{j}$ here. Thus no test sequence has both error limits required by the KL-projection theorem.

Depends on

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Sources

R. V. Ramamoorthi, K. Sriram, and R. Martin, On Posterior Concentration in Misspecified Models, §1, pp. 760–761 (standard reference, not scraped)
B. J. K. Kleijn and A. W. van der Vaart, The Bernstein-Von-Mises Theorem under Misspecification, §4.1, p. 378 (standard reference, not scraped)