ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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 symmetric random walk on the circle

Example

Assume countable choice and let $T = R / Z$ carry its normalized Haar probability $m_{T}$ . Fix $0 < h < 1 / 2$ . The symmetric random walk that adds $+ h$ or $- h$ with equal probability has kernel $P (x, A) = \frac{1}{2} 1_{A} (x + h) + \frac{1}{2} 1_{A} (x - h), A \in B (T),$ where addition is modulo one. This is a probability kernel reversible with respect to $m_{T}$ , hence $m_{T}$ is invariant. The transition law is atomic; no transition density is used.

Facts & Assumptions

Given: The circle $T = R / Z$ , its normalized Haar probability $m_{T}$ , and $0 < h < 1 / 2$ .

[A1]

Countable choice ${A C}_{ω}$ is assumed only through the existing normalized-circle-measure interface; no full Axiom of Choice is assumed. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The one-dimensional torus carries the normalized Borel probability $m_{T}$ ; the supplied construction explicitly assumes ${A C}_{ω}$ . (The one-dimensional torus and its normalized Haar integral)

[F2]

Each translation $τ_{y} ([t]) = [t + y]$ is continuous. (The one-dimensional torus and its normalized Haar integral)

[F3]

Each translation $τ_{y}$ preserves $m_{T}$ . (The one-dimensional torus and its normalized Haar integral)

[F4]

A probability kernel has probability-measure rows and measurable event evaluations. (Measure kernel and probability kernel)

[F5]

The product sigma-algebra is generated by measurable rectangles. (The product sigma-algebra and its finite iterates)

[F6]

The joint flow measure satisfies the kernel iterated-integral identity for every nonnegative product-measurable function. (A prior and model kernel define a joint law)

[F7]

For a measure-preserving map $T$ and nonnegative measurable $g$ , $\int g \circ T d m = \int g d m$ . (Measure-preserving transformations and systems, Integral invariance under measure-preserving maps)

[F8]

Reversibility means equality of the joint flow measure with its coordinate transpose. (General-state reversibility by symmetric flow measure)

[F9]

Reversibility implies invariance. (General-state reversibility implies invariance)

Verification

Given: $T = R / Z$ , $m_{T}$ , $0 < h < 1 / 2$ , and $P (x, A) = \frac{1}{2} 1_{A} (τ_{h} x) + \frac{1}{2} 1_{A} (τ_{- h} x)$ .

Proof technique: Verify the kernel directly, write its joint flow on measurable sets, and change variables by the measure-preserving translations.

1.1A1F1F2F4given

For fixed $A \in B (T)$ , $x \mapsto 1_{A} (τ_{\pm h} x)$ is measurable by [F2]; the two translated points are distinct since $0 < 2 h < 1$ , each row has total mass one, and disjoint-event indicators add pointwise, so [F4] makes $P$ a probability kernel.

2.1F5F6step 1.1given

For product-measurable $C$ , the graph maps $Γ_{\pm h} (x) = (x, τ_{\pm h} x)$ are measurable because rectangle preimages are measurable by [F5]; with $g_{\pm} (x) = 1_{C} (x, τ_{\pm h} x)$ , the joint-law formula [F6] gives $Λ_{m_{T}, P} (C) = \frac{1}{2} \int g_{+} d m_{T} + \frac{1}{2} \int g_{-} d m_{T}$ .

3.1F3F5F6F7F8step 2.1given

The swap map is measurable by [F5]. The identities $τ \circ Γ_{h} = Γ_{- h} \circ τ_{h}$ and $τ \circ Γ_{- h} = Γ_{h} \circ τ_{- h}$ and [F6] give $Λ_{m_{T}, P}^{⊤} (C) = \frac{1}{2} \int g_{-} (τ_{h} x) d m_{T} + \frac{1}{2} \int g_{+} (τ_{- h} x) d m_{T}$ . By [F3] and [F7] this equals the expression for $Λ_{m_{T}, P} (C)$ in step 2.1, for every product-measurable $C$ .

4.1A1F8F9step 3.1∎

By [F8], symmetry makes $P$ reversible; [F9] then makes $m_{T}$ invariant. Countable choice is inherited only through the supplied circle measure; the kernel calculation uses no further choice.

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