Here are some of my thoughts on the Fox match-up. I'm going to break it up into sections on various aspects of the match-up, and discuss them somewhat individually. For a brief overview, I want to say that I feel like this is the defining match-up of PM; it basically asks whether one of the newly designed PM characters can beat the best Melee character, which I think is pretty cool. It's also one of the most complex match-ups due to the slight imbalances between the characters, which make the match-up hard to evaluate, especially since both characters are obviously extremely good. I think the eventual outcome will be determined by how far Lucas's metagame can be pushed, and how well Fox can counter Lucas's tactics as they develop. I'll move along to my sections, but bear in mind they aren't meant to be a comprehensive guide to the match-up; just some of my broader conceptual thoughts on various things that I think will be important. Some of it will be in depth, some might not, some might be incorrect opinions that I need to be corrected on, etc. This is just a starting point for the discussion.
The Neutral Game
This is probably the most complicated part of this match-up, and I think it is the aspect that will ultimately determine who has the general advantage. Both characters are extremely fast both in terms of movement speed and their ability to execute attacks quickly. Both have amazing shield pressure options and good out of shield options, making the neutral game very dangerous, since you can't mess up your pressure or oos options without getting punished. I think that overall Fox is slightly safer in the close-range neutral game because of his CC shine and higher priority aerials, but Lucas makes up for it by controlling space very well with pkf and punishing harder than Fox does. This is going to start with some background and stats on different things, and then discuss general stuff about the neutral game.
Ground Movement
A general note here is that Fox has slightly better ground movement, while Lucas has significantly better aerial movement. Fox's run is slightly faster than Lucas's, but it doesn't make all that much of a difference in the neutral game anyway, since Lucas spends so much time in the air and never seems too slow to continue his combos. Lucas also has a much longer wavedash than Fox, and a superior DACUS, both of which expand his movement capabilities. Both characters can wavedash out of their down b, but Fox can do it faster -- 3 frames in shine + 3 frames in jumpsquat compared to Lucas's 7 frames in magnet + 4 frames of jumpsquat. However, Lucas is actually slightly safer when doing this as long as he gets his magnet out, since Fox has 5 frames after his last active hitbox before he can wavedash while Lucas only has 4. This slight frame advantage is probably not a big deal, but Lucas is also safer because he can reverse his momentum by b-reversing his down b, letting him start his retreat more quickly.
Despite all of this, Fox's shine gives him a lot of control whenever he's on the ground and you're low enough to hit. He can crouch cancel any repeating magnets and easily shine out of them, so you lose that option in some situations, which sucks since repeating magnets are really useful for pressure and building up damage. Shine is a real problem for Lucas because he has very low traction, which gives him a long wavedash but also causes him to slide very far when hit by shine, which makes it a lot easier for Fox to knock him off the stage with a few shines and get a gimp (but we'll talk more about that in the recovery section). Fox is also pretty capable of getting in on Lucas because of his speed and the fact that shine comes out on frame 1, so he’s one of the few characters that can seriously threaten Lucas in the neutral game.
Aerial Movement
Lucas's aerial movement capabilities are far better than Fox's. Lucas can do running sh magnets to retreating DJC fair or waveland back and control space very safely. He can't even be grabbed if he does this on a shield, and it's just generally super safe. Even without using magnets to glide forward when you jump, DJCs alone give you tons of mobility and the option to totally change your momentum. With magnets, you can not only glide slightly, but also reverse your momentum with a b-reverse for lots of tricky aerial movement. DJC pkf also lets you stay safe by moving away as you fire the pkf. Lucas can also attack out of airdodges, granting him invincible movement in any direction with the option to attack, which is pretty excellent. And lastly Lucas has all of his magnet pull tricks for following up on combos. This is pretty useful, but Fox is probably fast enough that he could also edgeguard anyone who got knocked far enough away that you had time to do a magnet pull, so it might not provide Lucas with any unique advantage over Fox. While Lucas is one of the best aerial characters, Fox is still very good in the air due to his strong aerials and extremely fast shffls. Fox’s nair will beat Lucas’s magnet and nair, unless Fox seriously mis-spaces and runs into the magnet without hitting you. Fox’s bair is similarly great at controlling space and outprioritizing Lucas’s magnet and nair.
General Neutral Game Strategy
Lucas’s pkf is extremely useful for controlling all aspects of the neutral game. They control space, can force opponents into shield, and they set up for combos if they connect. Fox has to be a lot more concerned about avoiding your pkfs than you do about avoiding his lasers, since lasers don’t cause hitstun to set up for the type of punishments that you can get off your projectiles. This helps Lucas force approaches, which may or may not be useful. I’ve played this matchup a decent amount from the Fox side, and prefer to constantly remain attacking, so Lucas doesn’t have much opportunity to pkf. Fox has an advantage when he is pressuring Lucas’s shield, and Lucas has an advantage while he is pressuring Fox’s shield, so it makes sense that both characters want to get in on the other and force the opponent to choose between shielding and getting hit (or rolling, which should usually lead to getting hit in this matchup). If Fox is going to be approaching anyways, it’s not that big of a deal that Lucas can force approaches with pkf. Lucas will often want to shoot repeated DJC pkfs at Fox anytime that it is safe to do so, as they provide pretty good pressure and lead into tons of good punish options if they hit.
At closer range, both characters want to apply pressure while staying safe. This is one of the key areas where the strengths of the characters create a slight imbalance. Fox has somewhat safer aerials since they have persistent hitboxes with high priority, while many of Lucas’s moves have lower priority and none are sex kicks. However, Lucas has better shield pressure, so if he can get in on Fox, he’s more likely to get a shield poke/break or force a roll that you can then capitalize on. So Lucas trades a bit of safety in the true neutral game for stronger pressure. One note on shield pressure is that Fox has a slightly faster grab and shine grab, which give him another small advantage to compensate for Lucas’s stronger pressure. Fox’s shine out of shield is also better than any of Lucas’s oos options, so Fox is better at defending against shield pressure while Lucas is better at applying it.
In terms of actual shield pressure options, you can look at the Lucas TAS Shield Pressure video and its description (it’s in the general thread). Most of the stuff in that video is not all that difficult, and you can do mixups with holding magnets if you can’t do some of the tech skill. Fox has standard Fox stuff like nair/bair -> shine -> shffl nair/bair or grab out of shine etc. The bottom line here is that Lucas has better shield pressure; Fox can’t punish Lucas’s perfect pressure, and can only roll or spot dodge to escape, but both of those options are pretty heavily punishable. No one can do perfect pressure yet, but really good pressure is already very possible. I’ve broken lots of shields with my Lucas, and I have barely scratched the surface of his shield pressure options. Even obvious stuff like magnet -> low sh magnet -> DJC aerial or magnet hold is really good, and his pressure is super unpredictable compared to Fox. (The things I’m saying in this paragraph should make it apparent how amazing Lucas is).
Combo Game
Both characters have really strong and dynamic combos that can pretty much destroy the other character. This won’t be a full guide to Lucas combos, but some stuff that works well against Fox includes up throw to up smash at low percent, up tilts, full hop double up airs, and standard combos with nairs, magnets, fairs, up airs, etc. You pretty much just want to use combos that work on fastfallers, and make sure not to get shined out of them. This isn’t really too hard since Lucas’s combos have lots of hitstun and are very difficult to escape. I don’t think Lucas really ever needs to drop a combo on Fox, so you should really try to always get a kill off the combo. Lucas has lots of ways to lead into a smash attack or bair, so he has plenty of opportunities to get the kill. Fox probably wants to go mostly for shines to knock Lucas off the stage, or up throw -> up air or maybe up smash at low percent. Much like Lucas, Fox also wants to use his standard fastfaller combos like up throw to up smash/up air. The only difference between Lucas and other fastfallers is that Lucas slides really far when he gets shined, so he can get knocked off the stage a lot more easily. He’s also not super heavy, so Fox may want to use bair or fsmash to go for some kills off the side on stages with high ceilings, especially since this can set up for edgeguards if Lucas survives. Overall the combo game is not clear yet. Fox’s combos are easier and more guaranteed in certain situations, but Lucas can continue his combos longer and they seem better at moving opponents off the stage or leading into kill moves (like any smash attack, bair, up throw, or fair at slightly higher percentages).
Edgeguarding and Recovery
Lucas probably has the advantage here, since he can go further off stage to gimp Fox, and has better option coverage from on stage than Fox does. Off stage, Lucas generally wants to go for bair spikes, while on-stage he can use dsmash to cover Fox’s attempts to go to the ledge. Unless Fox sweetspots perfectly, he’ll get hit by the dsmash and probably die. The first hit of dsmash also ends soon enough that if you realize Fox is going at a higher angle with up b, you can usually end the dsmash after the first hit in time to punish with something else, like up smash or bair. I don’t really see other people using dsmash to edgeguard, but it’s very good since its hitboxes are out for a while and very strong. Fox still has the ability to recover by mixing up side b and different angles of up b, but Lucas has a pretty strong edgeguarding game and can cover a lot of options. I could definitely see Lucas’s edgeguards become super consistent against spacies since he has so many ways to get the kill from both on stage and off stage.
Fox also has pretty strong edgeguards against Lucas, but I don’t think they’re quite as a strong as Lucas’s edgeguarding options, and I think Lucas has better tools for getting around Fox’s attempts to edgeguard. The biggest thing Lucas needs to look out for is shine, particularly when he is tethering back to the ledge, since his recovery path can be pretty predictable and leaves him very vulnerable at times. Fox can hang on the ledge to prevent you from reeling in to it, and just shine you if you tether to the ledge. I think Lucas needs to use tether cancels here in order to bait Fox into shining at a time where Lucas is safe and can still air dodge, and then either airdodge to the stage or tether to the ledge, depending on where Fox goes after he tries to shine spike you. Fox also has all his normal edgegaurd options like bairing from the ledge if you try to go over him, or staying on stage and punishing when you try to get off the ledge. Fox could also just hang on the ledge, which makes it very difficult to tether since you can easily be shined when you tether and you can’t reel in to the ledge, so Fox can still punish you when you do you jump back onto the stage. One thing that might be useful is tethering and then cancelling directly into a magnet to knock Fox off the ledge, but I haven’t really used this much yet. Using your up b is pretty risky, since Fox can shine you during the start up or can knock you out of the pk thunder 2 when you’re coming back to the stage. It’s just much slower and riskier than tethering, and I think it should generally only be used when necessary.
Concluding Notes
This match-up is pretty interesting since both characters are so good and have so many options against each other. I feel like one of Fox’s strengths in the matchup is his ability to keep Lucas out better than most other characters can, so Fox might have more of a defensive than offensive advantage against Lucas, which is somewhat unusual for Fox since he is normally the aggressor in most matchups. Lucas generally wants to set up combos by landing a safe pkf or connecting with some other attack when Fox is vulnerable, whereas Fox wants to utilize his priority to stuff Lucas’s moves and then probably up throw up air or shine, but lots of other stuff will be the best option in certain situations. They both combo each other super hard, and are pretty good at moving the other off stage and getting gimps/edgeguards, so every exchange is more important than it would be in most other match-ups. You need to try to stay safe, but one of the safest things you can do is get close to Fox and pressure him with your aerial capabilities, which give Fox little opportunity to retaliate since you have a decent frame advantage in lots of situations and Lucas can stay out of Fox’s range while applying this pressure. Overall it’s not clear who comes out on top at this point in time, as there are a multitude of strategies that each character will have to explore before we’ll know the most effective strategies and be able to figure out who might have an advantage.