ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

The Koopman matrix for a two-point swap

Example

For the two-point probability space with masses $1 / 2, 1 / 2$ and swap T, the Koopman operator sends $(a, b)$ to $(b, a)$ , has matrix $(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , and preserves every real or complex Lp norm for $1 \leq p \leq \infty$ .

Facts & Assumptions

[F1]

Koopman acts by composing each function with T The Koopman operator.

[F2]

Every probability-preserving pullback preserves Lp norms, including infinity Koopman operators are linear isometries.

Verification

Given: The objects and hypotheses in the statement.

1.1

The cardinality measure $μ (E) = ∣ E ∣ / 2$ is countably additive, since a disjoint family has at most two nonempty members. The inverse images of $\emptyset, {0}, {1}, X$ are $\emptyset, {1}, {0}, X$ , with masses $0, 1 / 2, 1 / 2, 1$ . Hence the swap is measurable and preserves this probability measure. For $f (0) = a, f (1) = b$ , pullback gives $U_{T} f (0) = b, U_{T} f (1) = a$ , the displayed matrix formula.

F1given

2.1

For finite p, $∥ U_{T} (a, b) ∥_{p} = ((∣ b ∣^{p} + ∣ a ∣^{p}) / 2)^{1 / p} = ∥ (a, b) ∥_{p}$ . For infinity, both norms equal $\max {∣ a ∣, ∣ b ∣}$ . Thus the direct calculation, including zero coordinates, agrees with the general Koopman isometry theorem.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Einsiedler–Ward §2.4, pp.28–29; explicit finite specialization (standard reference, not scraped)