In a comment on that post, Juliette of Honeysuckle Faire wrote the following:
My husband calls this "pushing vs. pulling". I know that sounds completely opposite of what you are talking about [I had been talking about pulling against a pull by the horse as creating a brace], but really I think it is right. He always uses setting the table as an example: some people push the plates down, slamming them loudly. Or you can "pull" the plates up as you are putting them down. The same with shutting a door - you can pull it shut loudly or push it back as you pull it toward you. It is this balance that I use when riding or leading. I lead Pie in and he is hot. I try to equally push and pull in a perfect balance so it seems like I am not even holding the lead - it is like he is walking in beside me without being restrained. If he jumps at something, I try to go with it in a push/pull movement so there is no tension. I try to do the same when I ride.
I wanted to particularly highlight this comment because it said some things about bracing in a way I didn't. Thank you Juliette - this comment really got me thinking more about braces and how to undo them, or not create them, so things can be accomplished with softness. I think what Juliette's husband is talking about when he says pushing is very much the same thing I'm saying when I use the word brace - a brace can be a push as well as pulling against pressure. I think what he's saying is that when a brace is a push - like his examples of slamming a plate down or pulling a door shut in a loud way - that you need to avoid the push by softly using a pull. One example that I've heard Mark Rashid use, which is almost the same as the one Juliette's husband used, is walking through a door and letting it slam loudly behind you - that's not soft - or instead going through the door and using your hand to allow it to ease shut quietly - that's soft and the pull that Juliette's husband was talking about.
I also like Juliette's example of keeping the push and pull in balance so the lead line, or reins, are alive - no tension but mentally not limp either - live connections between you and the horse. That's what softness is, to me - it isn't an absence of bracing (as in a nothing), it's a live and attentive engagement without bracing.
Thank you again Juliette for your comment - I hope my thoughts are consistent with yours!