This build is on-hit Lucian to take the most advantage of his new passive amp and more viability in bot lane. The more you can get stacks the better your early trades and all ins will be which is why we want people like Taric, Yuumi, Lulu, and Sona. You can be a lot more aggressive and with PTA plus your OP synergy in bot lane you'll win a lot of trades that aren't caitlyn. First we'll go into the pros and cons of this build and Lucian as a whole after the changes, then we can talk about the items and the game phase.

-High Damage Output: With these new changes you do incredible auto damage in all phases of the game. You have a surprising amount of early damage gain on ally buff autos because of it scaling with 10% of your total AD that you overwhelm duos that don't have immediate sustain. As the game continues that damage only gets higher.

-Great Scaling Power: With these new changes to Lucian he's now a lot better at scaling through a match and wrecking teamfights. The On-Hit build only makes it better(which we'll go over in the next couple chapters). You have so much compensation with your team synergy that all it takes is one good teamfight to get back in the match.

-Good Passive Abuse: Even with our focus being the new Vengeance Passive, we're getting so much attack speed and guinsoos, so not only do you do lots of damage with autos, you're attacking so fast that E+auto fires 3 shots instead of 2 which can lead to some crazy kills. This build is easily the best for abusing new lucian.

-Little To No Sustain: Outside of Dorans Blade you don't have any sustain, making healing supports even more valuable. Unfortunately, this also means without your support you have no way to recover mid fight or in a side lane and have to be a lot more careful about your actions. The only form of sustain with this build outside of dorans blade is if you replaced Wit's End with Blade Of The Ruined King in Low AP matchups.

-R neglect: New lucian R shoots more shots based on his crit [22(+25% Crit)] and with full crit will fire up to around 47 shots. But since we're buying Guinsoos Rageblade we lose out on all crit value for Ultimate. This leaves your ult at a permanent 22 shots. You still do around 2800 damage but that's only about 1/2 of the damage youd get otherwise.

-squishy: This one is obvious. You're a very squishy champion because you're an adc. We take advantage of this with Lord Dominiks but you're still susceptible to assassins rolling their face on the keyboard and one shotting you. Luckily, the best supports for new lucian have pretty good healing and sustain so you can get away with being squishy more often.

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  This one is pretty obvious. It's easily his best Keystone and should be run on him every single game. His double hit passive and new auto passive makes for some incredible early trades and opens up one shot opportunities late game.
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  Presence Of Mind is good for early mana retention until you get manamune and will allow you to go in on trades more often. It's a solid tool to make sure you aren't running out of mana too often.
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  Since attack speed and on-hit are the name of the game with this build, it only makes sense to run Alacrity as it gives % attack speed over the duration of the match to help you scale.
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  Coup De Grace is just some added damage so you can secure your kills with ease and not have to worry as much about the enemy adc surviving with 1 hp.
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  Sudden Impact is another no brainer for lucian as his dash has a cooldown reduction built in and each rank the mana cost lowers, allowing for more dash spam and more sudden impact activation.
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  Eyeball collection will gives us permanent bonus AD and add on to the stupid high damage we get from Q and autos.
  

  Alternative Rune Page

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  Hail Of Blades is a fun substitute when you're feeling bored of PTA, and would actually work fairly well with on-hit Lucian since it's 3 shots in rapid succession added onto our already insane attack speed.
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  Ultimate Hunter is a nice touch for Lucian as his Ultimate does a lot of damage even when we're not focusing on it. It's also a nice wave clear tool and can help interrupt and contest dragons and barons. In general it's great to have up as soon as possible.
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  Biscuits will be a fine touch of survivability for early game.
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  It's a free stopwatch. That's about it.

Kraken Slayer is his best mythic period. It pairs with PTA and deals true damage every 3 hits which is extremely easy to do with 2.2 attack speed and Guinsoos Rageblade. Kraken also has good AD and is a great item for early AD. Once you get Kraken you should be able to win any trade as long as it's not something stupid like a 1v3. In general, the item is just amazing for on-hit and attack speed as it gives tons of both.

Berserker Greaves are another attack speed item that gives a good early game boost and movement speed of course. It's also the only pair of boots that makes sense for ADC Lucian and the attack speed feeds into the on-hit build super well.

Manamune is incredible for the on-hit properties, the total mana, and the total AD gain. You stack tear quickly so you can get Muramana around 25 minutes and optimize your mana to AD ratio so you get the most value as soon as possible. It's another staple item on Lucian as his mana problems are pretty apparent up until mid game.

Guinsoos Rageblade gives incredible attack speed(45% to be exact) and since we're going crit heavy the conversion gives you a bonus 120 physical damage per hit as well as the 2nd passive(seething strike) interacting with Wit's End for more magic damage on hit. It's also a very good item for its low cost of 2600 gold.
Note: as far as I can tell guinsoos does not effect his new passive auto damage.

Wit's End also gives incredible attack speed and is naturally a great on-hit item for its passive damage. it's good for ap comps and can stop ap supports from completely killing you late game if you play smart. It also pairs with Guinsoos for a nice on-hit interaction and only adds on to the ****ery of on-hit lucian. Even with the nerf in 11.17 it's still a good late game item which is why we're buying it later on in the build instead of rushing it.

Lord Dominiks is a great item for late game as it will make your overwhelming power known to tanks and bruisers alike. The crit goes into guinsoos and the bonus % physical damage is amazing since the only people with lower max health than you would be another ADC or a few supports. This opens up bonus damage for people like mages and assassins late game even if it's not the highest percentage you could get. If there is ANYONE that has a chance of beating you in your match, the moment you get LDR you have a chance to beat them as long as it's not an overwhelming ****fest like a 40/0 Katarina that jumps on a dagger and literally gets a Penta.

In the event that the enemy team has little to no AP, you can also buy Blade Of The ruined King as the 6% current health physical damage will help kill beefier opponents, you get lifesteal, attack speed, and Blade Of The Ruined King's 3rd hit passive corresponds with Guinsoos 3rd hit passive to make the damage more effective. It's also only 100 gold more expensive so it wouldn't put much of a dent in your gold comparatively.

In the event that you need to get Grevious Wounds, Mortal Reminder is the best item as it's extremely cheap, gives crit which applies to Guinsoos Rageblade, and it gives attack speed to somewhat make up for the attack speed loss from replacing an item. Grevious Wounds is a bit iffy right now but if you feel like it's necessary then absolutely go for it. This is also a good alternative item to Wit's End as Lucian already buys it from time to time. Though If I were to have a preferred alternative item to Wit's End I'd go with Essence Reaver.

Early Game should mostly be spent farming and pulling off quick trades while working with your support to get the most value from your new passive stacks. If you can outright win lane then do it as soon as possible so later on you can show no mercy to enemy tanks and bruisers. Eventually you just become Rambo running in to kill the whole squad.

Mid Game is going to be for farming to split bot or for grouping. Either way do your best to keep up on gold as the mid game items are fairly cheap, allowing for a faster full build. Once you get Guinsoos you'll be good for most of mid game considering the interations with Guinsoos.

Late Game is going to be for taking advantage of your new passive to crush teamfights and take objectives at great speeds. By now you should be full build meaning you 2 tap anyone that's not a tank. Your only issue is that you have no sustain and R damage is a bit low, but we make up for that with crazy damage and excellent kiting skills. If it actually takes to late game just focus on proper positioning and targeting the right people

Below will be an Excel Sheet of around 30 test games I played to test the build and find new synergies. I believe these are some of the most bat**** stats I've gotten in a series of matches in a long time and make me believe Lucian is going to be nuts. For Clarification, I am not a mathematician so If there's any incorrect math or a number where it shouldn't be just tell me and I'll get around to fixing it as soon as possible. Menace to LoL Lucian Excel Sheet

In Conclusion, On-Hit Lucian is the go to build with his new changes even if R gets no value. You get so much bonus damage from items and PTA that every 3 autos you're absolutely chunking people in every phase of the game. Your already good auto weave combos are easier to hit and even just E+auto can give some amazing results. Get a good support who can give you sustain and you're set to go absoultely ape****. I'm expecting the viability of this build to last a single patch since I think they'll tweak lucian a bit to make him less OP, but for now enjoy the build and glhf!