According to everybody on the internet, you have to hit each weekly Level 7 boss, while it is unshielded, with the Worldline Zero sword.
According to some people, the attack you use NEEDS to be Tesseract. Some say it NEEDS to be a "ground" Tesseract, and not an "air" Tesseract, others say nothing about them or even what the specific differences between the two are (if it starts on the ground, but ends in the air, which of the two is it!?!?).
According to some people, the attack you use doesn't have to be Tesseract, because "their friend got it by doing light attacks".
According to some people, all that matters is that the boss is unshielded when you hit it. Others seem to think that you need to hit the boss in specific places for it to count ("you have to hit that ogre on the back").
Some people say you only have to hit the boss once for it to work, others say that you have to hit the boss over and over and over again until it pops, with no indications whatso-blam!-ingever on how many -blam!-ing hits that's supposed to take.
Some say it doesn't count if other players are shooting the boss when you hit it, some say the boss can't be tethered, melting pointed, or booped when you hit it either, yet if this is true, barely anybody seems to mention this at -blam!-ing all!
So, unless someone can give me an accurate description of what the -blam!- needs to happen for this Catalyst to progress, then I insist that Bungie change it to something simpler, like "Escalation Protocol kills", because having to go through this bullshit each week for 5 weeks straight is just stupid as -blam!-, and the perk is so lackluster that it isn't even -blam!-ing worth it!
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Im at 40% with mine.. All I did was land hits on the boss, lights, heavies, didnt bother with the perk of the sword. I learned that being the only one doing damage to the boss helped in getting the percentage... the last 20 I got I was with a group of randoms, all in party, and I just said "hey.. i need to hit the boss with the sword, can you guys hold off doing damage?" they agreed.. i got my solo hits in, and the percentage dropped for me, and we went about farming. Iv not had trouble getting it done so far. I wish you luck in getting yours working.