e⟨Khx,Qhy′  ⟩dμ(y′  ) Vhy dμ(y) Theorem [Furuya, de Hoop, Peyré]: Let be -continuous on a compact . Γ⋆ : 𝒫 (Ω) × Ω → ℝd Wass2 × ℓ2 Ω ⊂ ℝd For any there exists and such that ε N (θ1 , …, θN ) Γθ [μ](x) := MLPθ (x) or ∀(μ, x) ∈ 𝒫 (Ω) × Ω, |Γ⋆[μ](x) − ΓθN ⋄ ⋯ ⋄ ΓθN [μ](x)| ≤ ε with and . token dimensions ≤ 4d H ≤ d fixed dimensions, arbitrary # tokens. Masked transformers: works in progress. Novelties: Previous works: [Yun, Bhojanapalli, Singh Rawat, Reddi, Kumar, 2019] , dimension n → H = 2 ∼ [Agrachev, Letrouit 2019] abstract genericity hypothesis (Lie algebra/control) → Discrete tokens: transformers are universal Turing machines: e.g. [Elhage et al 2021]