Research directions in condensation: varieties of objectivity

Summary

This is the first part of a survey of various ways that I’d like to see work on the theory of condensation develop. Condensation is a mathematical theory dealing with the organization of descriptions of the world into conceptual parts; some of the existing work on it is presented in the paper (Eisenstat 2025). This sequence will draw from that paper the definitions of random variable models and latent variable models, the notation for indexing subfamilies of variables in such models, and the objectivity theorem, Theorem 6.8. In the condensation paper, Section 3, Ideas, gives an overview of all of this. For more context on condensation, readers can refer to LessWrong posts including (Demski 2025, 2026; Gillen and Chiang 2026; Kirchner 2026). The first two posts of this sequence will introduce some central directions of current work—mainly, the concepts of almost perfect condensation and Kolmogorov (or algorithmic-information) condensation—which will be used in other sections, but the different parts are mostly independent beyond that. I’d encourage those making a serious effort on any of these problem to contact me for further thoughts and coordination. Thanks to Kaarel Hänni and James Cook for some of the ideas behind this research program, and to James Cook and Jeremy Gillen for comments on this document. 1. Varieties of objectivity For the theory of condensation to be useful, we’d like hypotheses that we can assume about latent variable models that are general enough so that such latent variable models exist in cases of interesting random variable models, but narrow enough to imply objectivity-like conclusions—statements of approximate uniqueness along the lines of Theorem 6.8 of (Eisenstat 2025). We’ll refer to hypotheses of this kind as condensation properties. To know that such properties apply to interesting data, we’d like to come up with interesting random variable models satisfying them. (We can also look for interesting string models, which will be the analogue of random variable models when we introduce Kolmogorov condensation in the next post.) Work in these directions will also affect the problems discussed in later sections, which almost all either need to assume the existence of a latent variable model (or a latent string model in the Kolmogorov case) with good condensation properties, or else ask for such latent variable models in some particular context. 1.1. Almost perfect condensation One strong condensation property is almost perfect condensation. I’ve defined this somewhat differently in different places, and there isn’t yet a good reference for this. We’ll look at a definition here, and we’ll discuss informally the consequences that follow from and motivate the definition, without giving a theorem statement or proof sketch. Definition 1. Suppose that mjx-math { display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0; } mjx-container[jax="CHTML"][display="true"] { display: block; text-align: center; margin: 1em 0; } mjx-container[jax="CHTML"][display="true"][width="full"] { display: flex; } mjx-container[jax="CHTML"][display="true"] mjx-math { padding: 0; } mjx-container[jax="CHTML"][justify="left"] { text-align: left; } mjx-container[jax="CHTML"][justify="right"] { text-align: right; } mjx-msub { display: inline-block; text-align: left; } mjx-mrow { display: inline-block; text-align: left; } mjx-mo { display: inline-block; text-align: left; } mjx-stretchy-h { display: inline-table; width: 100%; } mjx-stretchy-h > * { display: table-cell; width: 0; } mjx-stretchy-h > * > mjx-c { display: inline-block; transform: scalex(1.0000001); } mjx-stretchy-h > * > mjx-c::before { display: inline-block; width: initial; } mjx-stretchy-h > mjx-ext { /* IE */ overflow: hidden; /* others */ overflow: clip visible; width: 100%; } mjx-stretchy-h > mjx-ext > mjx-c::before { transform: scalex(500); } mjx-stretchy-h > mjx-ext > mjx-c { width: 0; } mjx-stretchy-h > mjx-beg > mjx-c { margin-right: -.1em; } mjx-stretchy-h > mjx-end > mjx-c { margin-left: -.1em; } mjx-stretchy-v { display: inline-block; } mjx-stretchy-v > * { display: block; } mjx-stretchy-v > mjx-beg { height: 0; } mjx-stretchy-v > mjx-end > mjx-c { display: block; } mjx-stretchy-v > * > mjx-c { transform: scaley(1.0000001); transform-origin: left center; overflow: hidden; } mjx-stretchy-v > mjx-ext { display: block; height: 100%; box-sizing: border-box; border: 0px solid transparent; /* IE */ overflow: hidden; /* others */ overflow: visible clip; } mjx-stretchy-v > mjx-ext > mjx-c::before { width: initial; box-sizing: border-box; } mjx-stretchy-v > mjx-ext > mjx-c