http://tromp.github.io/cl/cl.html
which links to an explanation of the graphical notation, to my corresponding IOCCC entry from 2012, and to a Wikipedia page on Binary Lambda Calculus that has since been deleted.
Is Lambda calculus a good match for algorithmic information theory. The versions of AIT I've read involved either Turing machines, hand-waving of computation or recursive function. Lambda calculus seems more complex and thus harder to integrate with an AIT that mostly needs to know that strings are programs, that not all programs terminate, that some programs can give a substring of their string as outputs and so-forth.
For AIT, the binary encodings used in this paper have the nice property that they're complete and prefix-free, meaning that every binary string either decodes to a term (perhaps using only an initial portion of the string), or can be extended (arbitrarily) until it gives rise to a term.
At first glance, it feels like it would be hard for lambda terms to "give a substring of their string as outputs", since that would require re-encoding a subterm. However, I don't think that's as bad as it sounds. For example, consider that a Turing machine can only move its head along the tape one cell at a time (we could use any finite step size, but we usually stick to one), which makes it difficult/impossible to "preserve" a section of tape. Hence Turing machines which "give a substring of their string as outputs" might sound simple, but would probably be somewhat complicated to ensure that any "corruption" caused when traversing the tape gets "undone" afterwards.
(Unless they're multitape machines, like monotone turing machines!)
See this paper on how to cluster using compression: https://arxiv.org/pdf/cs/0312044.pdf