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.

An affine conditional path has the stated velocity field

Statement

Let $d \geq 1$ , fix $x_{1} \in R^{d}$ , and let $t \mapsto μ_{t} (x_{1}) \in R^{d}$ and $t \mapsto σ_{t} (x_{1}) \in (0, \infty)$ be continuously differentiable on $[0, 1]$ , with one-sided derivatives at the endpoints. For $z \in R^{d}$ , define the affine map $ψ_{t} (z ∣ x_{1}) : = σ_{t} (x_{1}) z + μ_{t} (x_{1}) .$ For each $t$ , this map is a diffeomorphism with inverse $ψ_{t}^{- 1} (x ∣ x_{1}) = (x - μ_{t} (x_{1})) / σ_{t} (x_{1})$ . The unique vector field satisfying $\frac{d}{d t} ψ_{t} (z ∣ x_{1}) = u_{t} (ψ_{t} (z ∣ x_{1}) ∣ x_{1})$ for every $z \in R^{d}$ is $u_{t} (x ∣ x_{1}) = \frac{{\dot{σ}}_{t} (x_{1})}{σ_{t} (x_{1})} (x - μ_{t} (x_{1})) + {\dot{μ}}_{t} (x_{1}) .$

Statement references

Lipman et al., §4 Theorem 3 and Appendix A

Holderrieth and Erives, §3.2 Example 10

Fjelde, Mathieu, and Dutordoir, Gaussian conditional path derivation

Facts & Assumptions

Given: $d$ , $x_{1}$ , and the continuously differentiable functions $μ_{t} (x_{1})$ and $σ_{t} (x_{1}) > 0$ in the Statement.

[F1]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : vectors in $R^{d}$ have coordinatewise addition and scalar multiplication.

Proof

Given: The hypotheses in the Statement.

1.1givenF1algebra

Fix $t \in [0, 1]$ . Since $σ_{t} (x_{1}) > 0$ , solving $x = σ_{t} (x_{1}) z + μ_{t} (x_{1})$ gives the unique inverse $z = (x - μ_{t} (x_{1})) / σ_{t} (x_{1})$ . Both the map and inverse are continuously differentiable in $z$ , so $ψ_{t} (\cdot ∣ x_{1})$ is a diffeomorphism.

2.1givenstep 1.1algebra

For fixed $z$ , differentiation gives $\frac{d}{d t} ψ_{t} (z ∣ x_{1}) = {\dot{σ}}_{t} (x_{1}) z + {\dot{μ}}_{t} (x_{1})$ . Substitute the inverse expression from step 1.1 to obtain $\frac{d}{d t} ψ_{t} (z ∣ x_{1}) = \frac{{\dot{σ}}_{t} (x_{1})}{σ_{t} (x_{1})} (ψ_{t} (z ∣ x_{1}) - μ_{t} (x_{1})) + {\dot{μ}}_{t} (x_{1}) .$ Thus the displayed $u_{t}$ satisfies the flow equation.

3.1step 1.1step 2.1algebra∎

If another field ${\tilde{u}}_{t}$ satisfies the same equation for every $z$ , then for any $x$ choose the explicit inverse value $z = ψ_{t}^{- 1} (x ∣ x_{1})$ . The flow equation forces ${\tilde{u}}_{t} (x ∣ x_{1}) = \frac{{\dot{σ}}_{t} (x_{1})}{σ_{t} (x_{1})} (x - μ_{t} (x_{1})) + {\dot{μ}}_{t} (x_{1})$ , proving uniqueness.

Source notes

The proof is the explicit affine inversion and differentiation argument. The cited paper and two independent treatments were inspected for the same formula and schedule convention.

Depends on

The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

Used by

The Gaussian conditional affine map is quadratic-cost optimal Theorem

Dependency tree · two levels

14 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 Theorem 3 and Appendix A (standard reference, not scraped)
Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.2 Example 10 (standard reference, not scraped)
Tor Fjelde, Emile Mathieu, and Vincent Dutordoir, An Introduction to Flow Matching, Gaussian conditional path derivation (standard reference, not scraped)