Flow Matching with von Mises-Fisher Posteriors: An Irreducibility Theorem

Please don't cite it. The author found that the proof of some important theorems have some ambiguity, so the correctness is questionable. Disclosure: The proof verification involves AI-assistance. We study flow matching on the Fisher-Rao simplex Δᵈ⁻¹, realized isometricallyas the positive orthant Sᵈ⁻¹₊ of the round sphere via xᵢ↦√xᵢ. Aposterior over completions pₜ(x₁ xₜ) modeled by a single von Mises-Fisher(vMF) law is unimodal by construction, so it cannot represent a genuinely multimodalposterior: under the flow-matching loss it is driven toward a single mean direction thatneed not track any true mode. We give two exact, non-asymptotic results. First, for asingle vMF component, we compute the posterior-averaged log map in closed form up to anexplicit O(1/κ²) remainder, replacing an earlier probabilistic concentrationargument and recovering its κ→∞ limit as the special case $K=1$. Second,for a K≥2-component mixture-vMF posterior, we prove that its marginal velocity fieldcannot equal any single-target field log(·)(μ⁾ on Sᵈ⁻¹₊: exactly,with no error terms, in the limit κₖ→∞. Multimodality is therefore astructural obstruction that no single-target field can remove. We do not address herethe Fisher-Rao boundary-explosion problem near ᵈ⁻¹₊ or the relationship tothe α-Flow family.