My understanding is that they are picking a genome (in whole or in part synthetic / designed by them), a set of starting enzymes or transcription factors or the DNA sequence of those (not sure) (also in whole or in part synthetic), then running a computation by letting DNA transcription happen for some number of cycles, and finally read the output by some signaling protein like fluorescence that will either be present or absent depending on how you designed the initial conditions.

To me, Trumpet sounds more like a compiler or circuit/program synthesizer than an operating system, since my understanding is you have some circuit in mind and the goal is to find a good genome (and/or the enzymes, not super sure yet) that can implement it. This means finding enzymes that will accurately promote or block transcription at a target location. And also checking against potential problems as the computation/reaction progresses since in a deep circuit depth you would presumably have many more kinds of things floating around that could interfere. I didn't see any mention on the circuit efficiency here, like how many base pairs per gate as # of gates increases, which I would guess would get larger faster than linear.

They mention the signal amplification of many genomes being transcribed which is really neat (like error correction). Though makes me wonder if the output signal has a defined or expected duration of stability. If it is stable forever, it is like a fixed point or attractor, but I suspect unless under much greater design or control, things would change. And what is the latency of these gates?

With cell-free, it seems plausible to design from-scratch genomes using a database of known proteins and that could be really cool. And funny to think about applying evolutionary search to determine the fittest genome for the job. From there we go to unicellular and then up to multicellular!