Title: Information Lattice Learning. Abstract: Information Lattice Learning (ILL) is a general framework to learn decomposed representations, called rules, of a signal such as an image or a probability distribution. Each rule is a coarsened signal used to gain some human-interpretable insight into what might govern the nature of the original signal. To summarize the signal, we need several disentangled rules arranged in a hierarchy, formalized by a lattice structure. ILL focuses on explainability and generalizability from "small data", and aims for rules akin to those humans distill from experience (rather than a representation optimized for a specific task like classification). This paper focuses on a mathematical and algorithmic presentation of ILL, then demonstrates how ILL addresses the core question "what makes X an X" or "what makes X different from Y" to create effective, rule-based explanations designed to help human learners understand. The key part here is \emph{what} rather than tasks like generating X or predicting labels X,Y. Typical applications of ILL are presented for artistic and scientific knowledge discovery. These use ILL to learn music theory from scores and chemical laws from molecule data, revealing relationships between domains. We include initial benchmarks and assessments for ILL to demonstrate efficacy.