ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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 single particle does not separate two spheres

Example

Let $d \geq 2$ and $0 < r_{a} < r_{b}$ . The one-particle initialization $μ_{0}$ that assigns unit mass to the origin has compact support ${0} \subset B (0, r_{b})$ , but its support does not separate $r_{a} S^{d - 1}$ from $r_{b} S^{d - 1}$ : a path from the inner to the outer sphere can avoid the origin.

Example references

Chizat and Bach, Theorem 3.3 and Definition of separation

Hatcher, Algebraic Topology, §2.2

Facts & Assumptions

Given: $d, r_{a}, r_{b}$ , and the support in the Example.

Verification

Given: Standard basis vectors $e_{1}, e_{2} \in R^{d}$ .

1.1givenconstruct

The path consisting of the radius- $r_{a}$ circular arc from $r_{a} e_{1}$ to $r_{a} e_{2}$ , the radial segment from $r_{a} e_{2}$ to $r_{b} e_{2}$ , and the radius- $r_{b}$ circular arc from $r_{b} e_{2}$ to $r_{b} e_{1}$ is continuous after linear reparameterization of its three pieces. Its endpoints lie on the required spheres and every point on it has norm at least $r_{a} > 0$ .

2.1givenstep 1.1∎

The path therefore avoids $spt μ_{0} = {0}$ while joining the two spheres. By the definition of separation, this support does not separate them.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, Chapter 2 (standard reference, not scraped)