Reasoning traces of an LDT look similar to a discrete diffusion process, but we train the model in-distribution of its own search states using a state-conditional supervision target. We borrow from the theory of abstract interpretation (https://en.wikipedia.org/wiki/Abstract_interpretation): At each step, the supervision target is given by a lattice approximation of a set of compatible solutions. This enables a novel regime of on-policy supervised training on datasets of the form (x, {y1, .., yk}), where each input x is paired with a set of valid solutions y1, …, yk.