# Flow Duality and Source Geometry for Categorical Generation

> Source: <https://arxiv.org/abs/2609.10863>
> Published: 2026-09-11 04:00:00+00:00

arXiv:2609.10863v1 Announce Type: new 
Abstract: Continuous and discrete flow matching are usually treated as separate constructions. This paper identifies a duality between them: projecting continuous convex-interpolant paths with one-hot targets through a position-wise argmax yields discrete convex-interpolant paths. The result requires source laws with appropriate coordinate symmetry and boundary regularity, and it makes the continuous source distribution an explicit design choice for categorical generation. We derive the induced discrete interpolation behavior for Gaussian, bounded-uniform, and centered negative-exponential sources, showing that different source geometries lead to qualitatively different transition timing and vocabulary-size dependence. Small visual diagnostics and a short language-modeling pilot suggest that these source-design effects can also appear in learned transports and early generative quality.