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.

A two-state independence sampler

Example

Assume AC. On the finite discrete space $X = {0, 1}$ , let $π (0) = 3 / 4$ , $π (1) = 1 / 4$ , and use the independence proposal $Q (x, {0}) = Q (x, {1}) = 1 / 2$ for both current states $x$ . Relative to counting measure, the common-density rule gives $α (0, 1) = 1 / 3$ and $α (1, 0) = 1$ . The transition matrix is $P = (\begin{array}{cc} 5 / 6 & 1 / 6 \\ 1 / 2 & 1 / 2 \end{array}),$ and the two off-diagonal target flows are both $1 / 8$ .

Verification

Given: AC, the two-state target and the state-independent proposal in the Example.

[A1] The Axiom of Choice: AC states that every family of nonempty sets has a choice function.

[F1] Standard Borel spaces and Polish spaces are separable completely metrizable spaces: the discrete metric on a finite set is complete and separable, every subset is open, and the Borel sigma-algebra is the full power set; the definition therefore makes it standard Borel.

[F2] Counting measure on an arbitrary set and Counting measure is a measure: counting measure is a positive measure and assigns mass one to each singleton. Since this state space is finite, its total counting measure is finite and the one-set cover makes it sigma-finite by Finite, sigma-finite, and semifinite measures. Thus the target and proposal masses are densities with respect to counting measure.

[F3] Transition matrices and n-step probabilities: the entry $Q (x, y)$ is $Q (x, {y})$ .

[F4] Probability measures and probability spaces: a probability measure has total mass one.

[F5] Measure kernel and probability kernel: a probability kernel is a measurable family of probability measures with mass one at every current state. Each row's nonnegative singleton masses sum to one, so it defines a probability measure by finite addition; all event evaluations are measurable because the finite discrete state space has the full power-set sigma-algebra.

[F6] The common-density Metropolis–Hastings ratio: a common-density version $α_{c d}$ is $\min {1, π (y) Q (y, x) / [π (x) Q (x, y)]}$ when the forward mass is positive and is chosen as $1$ when that mass is zero.

[F7] Metropolis–Hastings kernel: accepted proposal mass is retained at the proposed state and rejected mass is added to the current-state self-transition.

[F8] Metropolis–Hastings flow is reversible and invariant: the constructed kernel is reversible and its target is invariant.

1.1F1F4given

The target masses sum to one, and the discrete metric makes the finite state space standard Borel.

1.2F2F3F5given

For each current state, the two proposal masses sum to one, so the row is a probability measure. Every event-evaluation function is measurable on the finite discrete space, hence [F3] and [F5] give a probability kernel. By [F2], target and proposal masses are their counting-measure densities.

2.1A1F6step 1.2algebra

The proposal masses cancel in [F6]. Hence $α (0, 1) = \min {1, (1 / 4) / (3 / 4)} = 1 / 3$ and $α (1, 0) = \min {1, (3 / 4) / (1 / 4)} = 1$ .

3.1F7step 2.1algebra

From state $0$ , the accepted mass to $1$ is $(1 / 2) (1 / 3) = 1 / 6$ and the remaining $5 / 6$ stays at $0$ . From state $1$ , the proposal to $0$ is accepted with mass $1 / 2$ , and the proposal to $1$ has mass $1 / 2$ . Applying [F7] gives the displayed matrix.

4.1F8step 3.1algebra∎

The off-diagonal flows are $π (0) P (0, 1) = (3 / 4) (1 / 6) = 1 / 8$ and $π (1) P (1, 0) = (1 / 4) (1 / 2) = 1 / 8$ . Diagonal flows agree with themselves, so [F8] gives reversibility and invariance.

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