# The Geometry of Phase Transitions in Generative Dynamics via Projection Caustics

arXiv: https://arxiv.org/abs/2606.13191 · HTML: https://mlsanity.com/paper/2606.13191/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 3 of its 45 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 3 of 3)

- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [How Out-of-Equilibrium Phase Transitions can Seed Pattern Formation in Trained Diffusion Models](https://mlsanity.com/paper/2603.20092/) (2026)
- [Concurrence of Symmetry Breaking and Nonlocality Phase Transitions in Diffusion Models](https://mlsanity.com/paper/2605.04830/) (2026)

## Built on by (most cited first, 1 of 1)

- [Cross-Time Directional Selection in Diffusion Sampling](https://mlsanity.com/paper/2610.05199/) (2026)

## Abstract (arXiv)

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.