Stochastic interpolants unify flows and diffusions, popular generative modeling frameworks. A primary hyperparameter in these methods is the interpolation schedule that determines how to bridge a standard Gaussian base measure to an arbitrary target measure. We prove how to convert a sample path of a stochastic differential equation (SDE) with arbitrary diffusion coefficient under any schedule into the unique sample path under another arbitrary schedule and diffusion coefficient. We then extend the stochastic interpolant framework to admit a larger class of point mass schedules in which the Gaussian base measure collapses to a point mass measure. Under the assumption of Gaussian data, we identify lazy schedule families that make the drift identically zero and show that with deterministic sampling one gets a variance-preserving schedule commonly used in diffusion models, whereas with statistically optimal SDE sampling one gets our point mass schedule. Finally, to demonstrate the usefulness of our theoretical results on realistic highly non-Gaussian data, we apply our lazy schedule conversion to a state-of-the-art pretrained flow model and show that this allows for generating images in fewer steps without retraining the model.

A wiring diagram is a labeled directed graph that represents an abstract concept such as a temporal process. In this article, we introduce the notion of a quasi-skeleton wiring diagram graph, and prove that quasi-skeleton wiring diagram graphs correspond to Hasse diagrams. Using this result, we designed algorithms that extract wiring diagrams from sequential data. We used our algorithms in analyzing the behavior of an autonomous agent playing a computer game, and the algorithms correctly identified the winning strategies. We compared the performance of our main algorithm with two other algorithms based on standard clustering techniques (DBSCAN and agglomerative hierarchical), including when some of the data was perturbed. Overall, this article brings together techniques in category theory, graph theory, clustering, reinforcement learning, and data engineering.

We present a concise, self-contained derivation of diffusion-based generative models. Starting from basic properties of Gaussian distributions (densities, quadratic expectations, re-parameterisation, products, and KL divergences), we construct denoising diffusion probabilistic models from first principles. This includes the forward noising process, its closed-form marginals, the exact discrete reverse posterior, and the related variational bound. This bound simplifies to the standard noise-prediction goal used in practice. We then discuss likelihood estimation and accelerated sampling, covering DDIM, adversarially learned reverse dynamics (DDGAN), and multi-scale variants such as nested and latent diffusion, with Stable Diffusion as a canonical example. A continuous-time formulation follows, in which we derive the probability-flow ODE from the diffusion SDE via the continuity and Fokker-Planck equations, introduce flow matching, and show how rectified flows recover DDIM up to a time re-parameterisation. Finally, we treat guided diffusion, interpreting classifier guidance as a posterior score correction and classifier-free guidance as a principled interpolation between conditional and unconditional scores. Throughout, the focus is on transparent algebra, explicit intermediate steps, and consistent notation, so that readers can both follow the theory and implement the corresponding algorithms in practice.

The $λ$-superposition calculus is a successful approach to proving higher-order formulas. However, some parts of the calculus are extremely explosive, notably due to the higher-order unifier enumeration and the functional extensionality axiom. In the present work, we introduce an "optimistic" version of $λ$-superposition that addresses these two issues. Specifically, our new calculus delays explosive unification problems using constraints stored along with the clauses, and it applies functional extensionality in a more targeted way. The calculus is sound and refutationally complete with respect to a Henkin semantics. We have yet to implement it in a prover, but examples suggest that it will outperform, or at least usefully complement, the original $λ$-superposition calculus.

This paper introduces Truth-Aware Decoding (TAD), a verification-oriented decoding scheme that aligns neural language generation with knowledge bases. Situated in the tradition of probabilistic program semantics for sequence models, TAD augments modern instruction-tuned systems with a lattice of semantic guards that operate at decode time. Our contributions are fourfold: (i) a constraint-based semantics that renders oracle filtering as a program-logic judgment, (ii) a proof that greedy selection enjoys local likelihood dominance under sound and complete guards (Theorem 2.7), (iii) an entropy-style invariant that quantifies factual risk via knowledge-aware safe mass, and (iv) a multi-agent operational calculus with verified Lean artefacts to certify implementation behaviour. Numerical and algorithmic case studies confirm that the resulting guardrails reduce hallucinations without sacrificing throughput, yielding a pragmatic bridge between large-scale empirical models and formal verification.

We investigate mathematically the notion of incoherence: a structural issue with reinforcement learning policies derived by naive goal-conditioning of autoregressive models. We focus on the process of re-training models on their own actions, that is, fine-tuning offline-learned policies with online RL. We prove that it decreases incoherence and leads to an improvement in return, and we aim to characterize the resulting trajectory of policies. By re-framing standard notions of control-as-inference and soft Q learning, we establish a three-way correspondence with two other ways of understanding the iterative re-training process: as folding the posterior into the reward and, in the deterministic case, as decreasing the temperature parameter; the correspondence has computational content via the training-inference trade-off. Through soft-conditioning generative models, we discuss the link between incoherence and the effective horizon.