LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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 centered law and its affine dilation have an optimal affine coupling

Statement

Let $μ$ be a Borel probability measure on $R^{d}$ with finite second moment, mean $\int x μ (d x) = 0$ , and $m_{2} : = \int ∥ x ∥_{2}^{2} μ (d x) < \infty$ . Fix $a \in R^{d}$ and $s > 0$ , define $T (x) : = a + s x$ , and let $ν$ be the law of $T (X)$ when $X$ has law $μ$ . For every Borel probability measure $γ$ on $R^{d} \times R^{d}$ whose coordinate marginals are $μ$ and $ν$ , $\int_{R^{d} \times R^{d}} ∥ y - x ∥_{2}^{2} γ (d x, d y) \geq ∥ a ∥_{2}^{2} + (1 - s)^{2} m_{2} .$ The law of $(X, T (X))$ under $μ$ attains equality. Thus $T$ is an optimal transport map for the squared Euclidean cost between $μ$ and its translated dilation $ν$ .

Statement references

Lipman et al., §4.1

Peyré and Cuturi, §2.6 Remark 2.31

Facts & Assumptions

Given: The Borel probability measure $μ$ , its zero mean and finite second moment $m_{2}$ , the affine map $T (x) = a + s x$ , and a coupling $γ$ with the stated marginals.

[F1]

Probability measures and probability spaces: $μ$ and $γ$ are countably additive probability measures on their stated Borel spaces.

[F2]

Law or distribution of a random element: the law of a Borel map $T$ under $μ$ assigns to each Borel set $B$ the value $μ (T^{- 1} (B))$ .

[F3]

Change of variables for expectation: integrating a nonnegative or integrable function against the law of a random element equals integrating its composition against the original probability measure.

[F4]

The law of a random element is a probability measure: the law of a measurable random element is a Borel probability measure.

[F5]

A continuous map has Borel preimages of Borel sets: a continuous map between Euclidean spaces has Borel preimages of Borel sets.

[F6]

The Lebesgue integral is linear on $L^{1} (μ)$ : the integral is linear for integrable scalar components.

[F7]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $∥ x ∥_{2}^{2} = ⟨ x, x ⟩$ on $R^{d}$ , with the coordinatewise Euclidean inner product.

[F8]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ ⟨ x, z ⟩ ∣ \leq ∥ x ∥_{2} ∥ z ∥_{2}$ for Euclidean vectors.

[F9]

Cauchy-Schwarz for random variables: for square-integrable real random variables $U, V$ , $E ∣ U V ∣ \leq (E U^{2})^{1 / 2} (E V^{2})^{1 / 2}$ .

[F10]

Expectation of a nonnegative or integrable random variable: expectation on a probability space is integration against its probability measure.

Proof

Given: The hypotheses in the Statement.

1.1givenF1F2F3F4F5F6F7F8F9F10

The affine maps $T (x) = a + s x$ and $x \mapsto (x, T (x))$ are continuous, so their Borel preimages are Borel by [F5]; their laws are probability measures by [F2, F4]. Let $(X, Y)$ have joint law $γ$ . The first marginal gives $E_{γ} X = 0$ and $E_{γ} ∥ X ∥_{2}^{2} = m_{2}$ . The second marginal is the law of $a + s X$ under $μ$ , so change of variables [F3] and linearity [F6] give $E_{γ} (Y - a) = 0$ and $E_{γ} ∥ Y - a ∥_{2}^{2} = s^{2} m_{2}$ . These second moments make all displayed costs finite; Cauchy–Schwarz [F8, F9] makes the cross term integrable.

2.1step 1.1F8F9algebra

Expand $Y - X = a + (Y - a) - X$ . The expectation of the cross term with $a$ vanishes because $E_{γ} (Y - a) = E_{γ} X = 0$ , so $E_{γ} ∥ Y - X ∥_{2}^{2} = ∥ a ∥_{2}^{2} + (1 + s^{2}) m_{2} - 2 E_{γ} ⟨ X, Y - a ⟩ .$ By [F8], pointwise $∣ ⟨ X, Y - a ⟩ ∣ \leq ∥ X ∥_{2} ∥ Y - a ∥_{2}$ . Applying [F9] to the two scalar norms gives $E_{γ} ⟨ X, Y - a ⟩ \leq \sqrt{m_{2} s^{2} m_{2}} = s m_{2}$ . Substitution proves the lower bound.

3.1step 2.1F2F3F4F5F6F10algebra∎

The law of $(X, T (X))$ for $X \sim μ$ is a coupling with marginals $μ$ and $ν$ by [F2, F4, F5]. Under this coupling $Y - a = s X$ , so $E ⟨ X, Y - a ⟩ = s m_{2}$ . The expansion in step 2.1 therefore gives cost $∥ a ∥_{2}^{2} + (1 - s)^{2} m_{2}$ , attaining the lower bound.

Source notes

The cited flow-matching paper states the Gaussian special case. The local proof above establishes the more general centered finite-second-moment dilation case directly by Cauchy–Schwarz and does not import Brenier's theorem.

Depends on

Used by

The Gaussian conditional affine map is quadratic-cost optimal Theorem

Dependency tree · two levels

46 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yaron Lipman et al., Flow Matching for Generative Modeling, §4.1 (standard reference, not scraped)
Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §2.6 Remark 2.31 (standard reference, not scraped)