Typically when people use the phrase core to describe league builds, it's usually what items are required for your champion to "function" at an efficient level.

Your core items are typically your mythic + boot selection. I believe if you reread that paragraph, it doesn't specify the quantity. The mythic + keystone is supposed to reflect the way you intend to play that game.

For example, my goal in this hypothetical game is to kill the enemy carry which is a squishy champion. To fulfill this purpose, I've opted to build Ludens Tempest + sorc boots purely for damage output.

If my goal isn't to kill a specific carry, then my itemization would open up to include Everfrost, Crown, or Liandrys as possible alternatives. Think about what your champion needs to do to succeed this game and what your team needs to do to succeed. Is the only damage threat a Vayne? She has to die! Do they have lots of survivability but lack damage? Liandrys does great at consistent dps. Is most of their damage situational / all in? Crown/Everfrost does a great job at disabling that.

Likewise, the difference between Predator, Comet, Electrocute all enable different playstyles for Syndra. The goal is to IDENTIFY what you want to do from champion select and how to achieve that goal. Do I need more flat damage, do I need more consistent dps, do I need utility? Am I roaming this game or focusing on waveclear?

If you want to include a third item as a "core", then Dark Seal is best item in the game no cap.

Typically never, cosmic drive and demonic embrace are not as efficient as Void Staff / Seraphs / Deathcap purchases. Electrocute is great for mid, if you're playing bot lane you might want to try comet though.

A standard build into MR would be - Ludens / Sorcs / Voidstaff
A standard scaling would be - Ludens / Sorcs / Seraphs
A standard tank buster would be - Liandrys / Sorcs / Void (Seraphs)

Haste/Utility.

Horizon focus/Rab have the most value early due to linear haste making your spike harder, but once you've gone through situational items like Banshees/Void/Zhonyas, all the other AP items are pretty trash. Most players run Phase Rush for utility because she has 0 mobility, Cosmic Drive replaces that while offering 30 Haste and pushing her up to 35% CDR.

Arcane Comet has a lower skill requirement making it easier for lane phase but has less value than Electrocute or Phase Rush. Part of becoming experienced with a champion is sacrificing ease for higher skill expression. Phase Rush and Electrocute cover up weaknesses or add onto her strengths. (Arcane Comet in this situation finds itself outvalued post-lost chapter. 6 or 7 minutes into the game) I've seen comet a few times in worlds, so I understand the idea but I feel that you sacrifice too much to make it optimal.

Thanks for your comment!

I'll be honest, Syndra jungle isn't the worst. Her clear used to be pretty decent with Old Q, with the nerfs you'll be very mana hungry until jungle item but her DPS is still the same. Predator helps with gank.

Thank you!

He is challenger... more or less. His challenger is on BakaEric2.