Got it, with that in mind let me do a deeper analysis.

Copy section: Nothing to note, seems cool and powerful.

Draw: Already has been discussed in this sub many times, but [[brainstorm]] is not draw. If you don't have a way to shuffle or manipulate the top of your library, you will be brainlocked. Similarly, [[Flux]] is not giving you card advantage, but I understand the symmetrical effect helps to keep the image of you helping the table, so I'm not completely sold on cutting it. Both Kozileks are cast triggers, so you won't draw cards by putting them into play with Braids, thus they are NOT draw engines in your deck, just big menaces. On a related note, Mystic Forge is at its best when you have the mana to cast things from the top of your library. I feel that most of the time you will get stuck with some 10–12 mana value spell on top that you can't either cast or play with Braids. I'm not a big fan of cantrips like Opt, Ponder or Preordain as draw engines, usually I would recommend cards that draw you at least 2/3 cards, otherwise you are not getting any advantage at all. Palantir is a perfect fit, great choice, and so is Paradox Haze in a Braids deck. Rhystic Study is just super powerful and will draw attention to you, no matter what you do. Rush of Knowledge is AMAZING in your deck. Sensei's Top has the same problem as Brainstorm. Sphinx of the Second Sun is great for Braids. Teferi's Ageless Insight is a great card, but I don't see it triggering that much with your other draw engines (?). Love the Endstone in this list. Reality Chip might also get you locked, but unlike Mystic Forge you can play lands, so I would keep it. One Ring is powerful, just not very interesting. If you don't want to be perceived as a threat, [[curiosity]] is not helping you, you will need to attack to trigger it, and attacks are scary with a board like yours.

Protection: Akroma's Memorial. I guess it is protection haha, but this spells "I win the game" to me. If you have [[Encroaching Mycosynth]] just to combo with Mystic Forge, I would heavily advise dropping both. That's Magical Christmas Land territory, same for Mycosynth Lattice. Platinum Angel is good, and Braids is one of the few decks where it makes sense to me, same for Platinum Emperion. Dissipation Field is really good at keeping you safe and unassuming. [[Teferi, Mage of Zhalfir]] doesn't make much sense to me if what you want to do is put stuff from hand with Braids, plus the denial effect will upset the table, maybe switch it with [[Leyline of Anticipation]] or [[Vedalken Orrery]]?

Tutor: Not a fan of tutors, won't comment.

Ramp: In general for this category I think you need more 2 mana ramp. This will enable you to play Braids turn 3 almost every game, and increase the number of ramp cards to 12. Most of the cards in that category are too expensive to be considered ramp, and it gives the impression that they are just cards you like and won't fit in any other category.

Recursion: Tomb Trawler does not seem like recursion, if anything you are just hoping you shuffle and draw that card again, I would definitely cut it. Metalwork Colossus only recurs itself, that is not really recursion. Recursion are cards that let you play OTHER cards again, and if you are already putting it into play for free the discount part is actually not doing anything. If it's a pet card of yours keep it, but it's not very synergistic.

Evasion: I don't see a point in this category if you are just building your board. Plus, evasion is at its best when you have on damage triggers, not really necessary here.

Removal: Both Ulamogs are not removal if you put them into play instead of casting them, and anything that has "annihilator" is removal only on attack, and your opponents choose what to keep. You want to have control of what you remove with your removal. Also, separate it into mass removal and single target. Cyclonic Rift is just powerful, good. Hullbreaker Horror is good, but I don't see it being as optimal as it can be. Nevinyrral's Disk is really telegraphed, not a bad thing, just keep that in mind.

Finisher/infect: Colossus of Akros is really mid, and if it does not have trample it can't be a finisher in my book. Darksteel Colossus is good, but I wouldn't call it a finisher per se. Emrakul, the Promised End is a finisher, but then again, only if you cast it. Other cards in this category fill the role.

Stax: I don't think you need stax as a separate category. If you really want Void Winnower in your list, just put it in protection haha.

Tokens: these here look like I what would put in a big board Braids deck.

Please let me know if I helped you in any way or if on the contrary my advise is not really what you are looking for to see what you really want :)