ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 mixed-sign vorticity is not a probability for W1

Example

On $R^{2}$ , let $ω = δ_{0} - δ_{(1, 0)}$ . It is a nonzero finite signed measure of total mass zero and total variation two. The probability $W_{1}$ in The coupling definition of the 1-Wasserstein distance cannot be applied to $ω$ .

Facts & Assumptions

Given: The signed measure $ω$ of the Example.

[F1]

$W_{1}$ is defined only for probability measures with finite first moment (The coupling definition of the 1-Wasserstein distance).

[F2]

$d_{B L}$ is the dual supremum over functions bounded by one with Lipschitz constant at most one (Bounded-Lipschitz dual discrepancy for finite signed measures).

[F3]

Total variation is the supremum over measurable partition sums (The total variation |nu|(E) from countable measurable partitions).

[F4]

The reverse triangle inequality gives $∣ ∣ x ∣ - ∣ y ∣ ∣ \leq ∣ x - y ∣$ for the Euclidean norm (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

Verification

Given: $ω = δ_{0} - δ_{(1, 0)}$ .

1.1givenF1F3algebra

Among any countable partition, at most two cells have nonzero signed mass, each of magnitude one; the partition into the two singleton atoms and their complement attains the sum two. Thus $∣ ω ∣ (R^{2}) = 2$ by [F3], while its signed mass is $1 - 1 = 0$ . Therefore $ω$ is neither a nonnegative probability measure nor normalizable by its net mass, so the domain condition in [F1] fails.

2.1F2F4step 1.1algebra∎

The function $ϕ (x) = \min (1, ∣ x ∣)$ is bounded by one and 1-Lipschitz by [F4], with $ϕ (0) = 0$ and $ϕ (1, 0) = 1$ . Hence $∣ \int ϕ d ω ∣ = 1$ , so $d_{B L} (ω, 0) \geq 1$ . This finite signed dual quantity remains meaningful by [F2], whereas probability $W_{1} (ω, 0)$ is undefined.

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