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.

翻译：连续状态生成采样器（包括扩散模型和流匹配模型）通过连续逆向动力学演化，但其样本常经历突发的定性变化：轨迹锁定至模态，语义替代方案坍缩，且狭窄时间窗口内的微小扰动可产生显著的下游效应。本文从几何角度阐释此类类相变行为。我们将去噪过程视为自由能景观上的梯度下降，并证明尖锐相变出现在投影焦点附近——此时数据支撑集上的最近点投影不再唯一。受此视角启发，我们提出临界边界检测器（CBD），作为分数方向不稳定性的实用诊断工具。在玩具模型、标准扩散模型及潜在文本到图像扩散模型中，CBD 能够定位模态锁定位置、预测干预敏感窗口，并在几何敏感区域支持定向控制。我们的结果建立了数据几何与扩散生成动力学之间的关联。