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 $\Delta^{d-1}$, realized isometricallyas the positive orthant $\mathcal{S}^{d-1}_+$ of the round sphere via $x_i\mapsto\sqrt{x_i}$. Aposterior over completions $p_t(x_1\mid x_t)$ 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 $O(1/\kappa^2)$ remainder, replacing an earlier probabilistic concentrationargument and recovering its $\kappa\to\infty$ limit as the special case $K=1$. Second,for a $K\ge2$-component mixture-vMF posterior, we prove that its marginal velocity fieldcannot equal any single-target field $\log_{(\cdot)}(\mu^\star)$ on $\mathcal{S}^{d-1}_+$: exactly,with no error terms, in the limit $\kappa_k\to\infty$. Multimodality is therefore astructural obstruction that no single-target field can remove. We do not address herethe Fisher-Rao boundary-explosion problem near $\partial\mathcal{S}^{d-1}_+$ or the relationship tothe $\alpha$-Flow family.