I got a 5x5 cube a while ago and was reading about how to solve it, and came across a nice, concise description of how one might invent moves which can be combined into algorithms which let you make incremental progress on a cube. I can't find it again, now!

Some of it was along the lines of (1) Figure out how to do something A to a single layer (while messing up the rest of the cube). (2) Observe that if you do A, then rotate the layer R, then undo A (A'), you have an operation B = A R A' which does something to only one layer of the cube. I assume that most moves in most common algorithms can be expressed in terms of a couple of fundamental techniques like this (probably using the words "commutator" and "conjugate"). Does someone have a link or reference that gives you the general meta-technique (even if it involves incompletely-specified things like "figure out an interesting sequence of moves in terms of their effect on a single layer")?

I'm interested in this because I'm not really interested in learning (again) and forgetting (again) any particular existing method for solving the cube, but it would be fun to be able to fiddle around with it in a less-than-random way and eventually arrive at a method, based on some higher-level principles.

Choosing a speedcubing method is an interesting algorithm design challenge. It's basically a tradeoff between speed and how many subalgorithms sequences you need to learn by heart. On the simplicity extremum there's the corner-3-cycle method, where you can solve it using zero hard coded sequence (pure reasoning). On the speed extremum, there's Fridrich or ZZ, which most top level speedcubers use but they require memorizing hundreds of sequences.

I would just like to point out that Sexy Move is an actually used piece of cuber terminology.

https://www.speedsolving.com/wiki/index.php/Sexy_Move

This YT-video explains the 8355 method really well https://youtu.be/zB8cKBYNTps . It's how I learned it. They call the sexy-move the fishing-move.

Wow this looks cool, I learned CFOP as a teenager and from showing it to people the number of algos (150-200+) is definitely the barrier for entry for most.

Solving a cube with 3 algos means most people can probably learn in an hour or two.

Is there hypothetically a single algorithm I can just repeat indefinitely and eventually get a solved cube?

I’m guessing no: you need conditional branches otherwise your single algorithm would just end up back where you started?