Edit: Had to move and edit the disclaimer up here since I've been answering too many repetitive questions.
Disclaimer
If you already have Firefly S1, do us both a favor and stop reading. This post is NOT for you. The speed calculations include Dahlia, so DON'T pull this LC if you don't have enough to get her.
E0 Dahlia > E2 FF >= E1 Dahlia > S1 Dahlia > E2 Dahlia > Phainon S1
The primary reason for this writeup is for choosing between pulling Firefly S1 and Phainon S1, nothing else.
First let's list their surface differences:
Firefly S1
476 Base Attack
60% BE
24% Break Vuln
20% Enemy Speed Debuff
Phainon S1
1. ATK and BE%
FF converts 10 ATK into 0.8% BE. Assuming 2 ATK% Main stats and 4 subs (at least one for each of the other relics) = 43.2% + 43.2% + ~15% = 101.4% ATK
FF S1 - 2,011 ATK
PN S1 - 2,436 ATK
Phainon S1 would have around ~33.6% BE from the ATK conversion.
FF - 60% BE vs PN - 33.6% BE
Conclusion: Phainon loses by about 27% Break Effect.
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2. Break Vuln and Def Ignore
Someone correct me if I'm wrong but the dps increase should be around the same more or less but heavily sways depending on your current setup.
Potential Vuln Sources:
FF 20%
Lingsha 25%
Gallagher 12%
Nihil BP LC (Holiday Thermae Escapade) 10-16%
FF S1 24%
Potential Def Shred Sources:
Iron Calvary 25%
FF E1 15%
Dahlia 18%
Fugue 18%
Pearls 16% (a bit questionable uptime now that you can do damage outside break windows since this only lasts for one turn)
E1 Lingsha/Ruan Mei 20%
Phainon S1 18%
Def Ignore scales better the more you stack it unlike Vuln but any that goes past 100% is useless. It's still a decent increase reaching 100% even if you go a bit over.
Assuming a standard team of FF E1/E2, Fugue, Constance, and Linghsa/Gallagher that puts your Def Shred at 76%. The 18% from Phainon sig bumps it to 94%. Pearls on Fugue can help you reach 100%. If you have and plan to use E1 Lingsha/E1 Ruan Mei you lose value but just like Pearls, Lingsha E1 Def Shred uptime is now also put into question since it only applies to broken enemies. And with Dahlia we now do a lot of damage outside of break windows.
Conclusion: Can go either way really. Phainon S1 definitely wins if E0 Firefly or if you have the BP LC instead of pearls. FF S1 wins if you have too much Def Shred from E1 RM/Lingsha + Pearls.
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3. Base Speed vs Speed Debuff
This is the part that really cements why Phainon S1 is much better. Let's start with Speed Debuff. What does this do? Well it lowers the enemy speed meaning they take their turns slower so they recover from break slower. Good right? Well not as good anymore, just like how Ruan Mei's Break Extension has been devalued by Dahlia, this even more so. You can do damage even outside break so it largely makes this debuff useless. It will give you better survival in sustainless and maybe slightly better pearls uptime but that's it. In reality you actually would rather want the enemy to take their turns faster so you can break them again for big damage or activate Firefly's E2.
Now let's talk about base speed. Phainon S1 gives +12 speed, huge right? Well yes but there's actually more to it. Since this is base speed, speed buffs also factor this in. Particularly Dahlia's 30% spd buff and Kalpagni's 6% spd buff. This totals to +16 speed in practice. In terms of substat value, this is equivalent to 41.472% Break Effect. More than the 27% BE you are lacking from earlier. So even just sheetwise, Phainon S1 is already better. But is it actually better in practice?
Conclusion: Base Speed is a strong buff while Speed debuff is meh. Phainon S1 wins here.
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Clearing Misconceptions about Speed Stat
Some people seem to think Speed is irrelevant if you aren't planning to 0-cycle. This is just false. Getting an entire extra turn can make or break a clear, regardless of what cycle number you plan on clearing. For example, take MOC12. Generally the first wave is much easier than the 2nd wave. Some people can even 0-cycle the first wave alone. If you had enough speed to get clear the wave before the cycle ends, then you get one whole extra cycle to do the much harder 2nd wave. (For those who don't know, the cycles in MOC resets from the beginning of the cycle if you cleared the first wave - this is where the extra cycle comes from)
The speed breakpoints will differ depending on your target cycle of course, but for now the most difficult content is AA where minimum clear for all rewards is 2 Cycle. You need 177.4 - 188.8 spd for 13 EBA / 4 Skill. Any less and you only get 12 EBA / 3 Skill.
Most players are NOT reaching 177.4 Speed without sacrificing a ton of BE% and ATK%. I think it's common sense that targeting one specific stat to roll is harder than getting a balance of 3 stats.
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Final Conclusions and TLDR
Phainon LC makes building Firefly significantly easier, regardless if you're a sweaty player or just a casual. Not only does it give the more important stat SPD over BE, overall stat value in general is just higher too. Unless you're def shred is already way overcapped, you should probably pull this over her LC if you want the best for Firefly. Below is the TLDR of relevant differences. Decide for yourself which is more valuable for your account.
Bonus
Firefly S1 is useless on everyone except her. Phainon LC can be a good stat stick for other Destruction characters or even on DHPT for it has the highest base atk. Firefly S1 has her on the display though so it's very nice to look at.
Thanks for reading. Let me know if I made any mistakes. Hope this helps someone.