Ergodic Generative Flows

Both the paper and the candidate solution aim to prove the same universality statement: if an EGF family contains a summably L2‑mixing star forward policy π*→ and its outflow class is L2‑dense, then the family is universal. The paper states this as Theorem 3.4 and sketches a rigorous version in Appendix B (Theorem B.1) where a fixed‑point construction is used for ν_{T} = ∑_{t=0}^{T} (Finit − Fterm)(π*→)^t, followed by a positivity shift. However, the proof takes the step “∑t ∥ε_t∥^2_{L2} < ∞ ⇒ ∑t ε_t converges in L2” without justification; square‑summability of L2 norms alone does not imply L2 convergence of the series in a Hilbert space. This step appears in the B.1 proof around equations (40)–(45), where convergence of ν_T and a bound on ∥ν_∞∥_2 are asserted from ∑t ∫ (dε_t/dλ)^2 < ∞ without orthogonality or an absolute convergence argument . In addition, the proof intermixes the invariant measure λ̂ (used to define the L2‑mixing coefficients) with a background measure λ, but does not supply the bounded Radon–Nikodym assumptions needed to transfer L2 bounds between L2(λ̂) and L2(λ) . The candidate solution gives an elegant Poisson/Stein equation approach, attempting to solve (I−P*)f = h via a Neumann series R = ∑n≥0 (P*)^n on L2_0(λ̂). Yet it also assumes convergence using ∑n √γ_n < ∞ (implicitly), while the paper’s “summably L2‑mixing” assumption is ∑n γ_n < ∞; the candidate’s series control relies on ∑n √γ_n < ∞, which is not implied by ∑n γ_n < ∞. The candidate additionally transfers L2‑density from L2(λ) to L2(λ̂) using only equivalence λ̂ ≃ λ, which is insufficient without essential boundedness of dλ̂/dλ and dλ/dλ̂. Therefore, under the paper’s stated assumptions, both proofs have gaps. Strengthening assumptions (e.g., a spectral gap with ∥P∥_{L2_0→L2_0} < 1, or summability of √γ_n, and bounded change of measure between λ and λ̂) would repair both proofs. The statements about applications to tori and spheres rest on standard spectral‑gap results and look directionally right, but the universality reduction depends on the gap above .