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arXiv · 2606.13191

The Geometry of Phase Transitions in Generative Dynamics via Projection Caustics

Ryosuke Sakamoto·Kotaro Sakamoto
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Abstract

Continuous-state generative samplers, including diffusion and flow-matching models, evolve through continuous reverse-time dynamics, yet their samples often undergo abrupt qualitative changes: trajectories commit to modes, semantic alternatives collapse, and small perturbations in narrow time windows can produce large downstream effects. This paper develops a geometric account of such phase-transition-like behaviour. We view denoising as gradient descent on a free energy landscape and show that sharp transitions arise near projection caustics, where the nearest-point projection onto the data support ceases to be unique. Motivated by this perspective, we introduce the Critical Boundary Detector (CBD), as practical diagnostics for score-direction instability. Across toy models, standard diffusion models, and latent text-to-image diffusion models, CBD localises mode commitment, predicts intervention-sensitive windows, and supports targeted control in geometrically sensitive regions. Our results connect geometry of data and dynamics of diffusion generation.

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Ryosuke Sakamoto, Kotaro Sakamoto. 2026-06-11. The Geometry of Phase Transitions in Generative Dynamics via Projection Caustics. https://arxiv.org/abs/2606.13191

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