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.

Rejection creates a self-transition

Example

Assume AC. On the finite discrete space $X = {0, 1}$ , let $π (0) = 2 / 3$ , $π (1) = 1 / 3$ , and let the proposal deterministically switch to the other state. Then $α (0, 1) = 1 / 2$ and $α (1, 0) = 1$ , so $P = (\begin{array}{cc} 1 / 2 & 1 / 2 \\ 1 & 0 \end{array}) .$ The rejected proposal from state $0$ is recorded as a self-transition of probability $1 / 2$ .

Verification

Given: AC, the two-state target and the deterministic switch 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: a finite discrete space is standard Borel because its discrete metric is complete and the space is separable.

[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: the rule adds rejected proposal mass to the current state's 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

The proposal switches state with probability one in each row, so each row defines a probability measure. Every event-evaluation function is measurable on the finite discrete space, hence [F3] and [F5] give a probability kernel. By [F2], the displayed target and proposal masses are densities with respect to counting measure.

2.1A1F6step 1.2algebra

The reverse proposal masses are both one. Formula [F6] gives $α (0, 1) = \min {1, (1 / 3) / (2 / 3)} = 1 / 2$ and $α (1, 0) = \min {1, (2 / 3) / (1 / 3)} = 1$ . Thus the accepted mass from state $0$ to $1$ is $1 / 2$ , leaving rejected mass $1 / 2$ .

3.1F7F8step 2.1algebra∎

Applying the rejection rule [F7] gives $P (0, 0) = 1 / 2$ , $P (0, 1) = 1 / 2$ , $P (1, 0) = 1$ , and $P (1, 1) = 0$ . The off-diagonal target flows are both $1 / 3$ , so detailed balance holds and [F8] gives invariance.

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