Today, many of us discovered a new quest marker over Lakshmi-2 called "Not Forged In Light". Is there a coincidence it appeared only the second time Paradox is the daily heroic? I don't think so. Here's my step-by-step guide.
1. Collect the three memory fragments, beat the Blighted Descendent Minotaur, and collect the Cult Ghost in the Paradox daily heroic mission. This can only be done when it's the daily (like Black Spindle can only be obtained when Lost to Light is the daily). Here's a guide from Datto: https://www.youtube.com/watch?v=JfWI3l1xTlI
2. Go see Lakshmi-2 (FWC vendor) in the Tower. The quest will go cold after this. If you are doing it today, Oct. 20, it will go cold until Paradox is the daily again, or the next weekly reset. This is just a theory, but makes sense because of how it became available to us who did it when Paradox was the daily about 2 weeks ago. Just keep your eyes out for progress if you complete it today.
3. When prompted, see Lakshmi-2 in the Tower. She will give you a quest called "Not Forged in Light". This will require you to pledge allegiance to FWC and donate materials to get reputation with FWC. Turn in Armor Materials and Motes to get reputation.
4. Once this is done, you will need to kill a powerful Taken Minotaur, collect it's Simulation Core, and shatter it. You can do the Praedyth's Memories event in the Paradox daily mission, or you can Patrol Venus and kill the Blighted Descendent that will spawn in the Ember Caves. When you kill it, you'll receive the Simulation Core. All you have to do is dismantle it.
5. Go talk to Lakshmi-2 again. She'll give you the next step, which is to get a Fireteam together and go kill Atheon. You'll need to get his Eye and shatter it. You can do either Normal or Hard difficult, it doesn't matter.
6. You'll see a mission pop up called Shadow of Twilight. You'll spawn into the Twilight Gap Crucible map, with pools of Light scattered across the map. You're looking for a normal chest that will contain items for you to progress. If you step out of the pools of Light, you get a debuff that will turn your screen black. If you screen goes completely black, you die. The chest spawns randomly but the pools are constant. Hop from pool to pool until you find the chest. Some people found it by the special ammo close to C spawn, others where C flag is and also where C heavy spawns, so I'd check there first. Once you find it, it's time for the last step
7. Now you must complete "Blood of the Garden", a 290 light level mission in the Black Garden. This one's hard. Really hard. You're gonna want your buddies in there to complete this. You have to summon a Taken Ultra by keeping his Anger level at 100%. Taken Goblins will increase it 1-2%, but when the Minotaurs spawn, you'll be able to increase it pretty fast. Thing is, Minotaurs won't stop spawning, no matter what. Only advice I can give is to focus on the boss and get him down as fast as possible. It gets hectic, but you're done after this. Go see Lakshmi and get your shiny new No Time To Explain.
No Time To Explain is a mid rate of fire, mid impact pulse rifle (much like Smite of Merain or Nirwen's Mercy) with perks that are full auto and precision hits directly refund ammo back in the magazine (doesn't pull from your reserves, it just shows up much like Black Hammer's White Nail perk pre-2.0). Hope this helps and clears up any confusion.
EDIT: Yes, you only need to wait a week for the quest to pop up after Paradox. I also solo'd Blood of the Garden to receive mine, took a bit of patience but I did it. Using Dark Drinker was what saved my butt. Thank you guys for the great feedback and using this guide, I'm glad I could help!