I don't think it was ever stated to be multiplicative, and it doesn't have to be multiplicative to make it true that the stronger the mage gets the stronger the Spirit gets. IMO this doesn't make much sense when you take into account the Spirits have their own reserves of magical power that exists even without any owners. We've already seen what happens when Lucy's powers doubled with second origin, and it wasn't like Loke's or Aquarius' powers doubled along with Lucy. Gemini + Lucy's Urano Metria wasn't 4x the power of the time she used against Angel. CSK didn't boost Lucy's magical power with his own either. He gave Lucy Aquarius' star dress, which gave Lucy access to Aquarius' MP. CSK has a much higher MP reserve than Lucy herself, so of course summoning him to fight for her would increase Lucy's tier overall.Stronger Celestial Wizard canonically equals stronger Celestial Spirit.So yeah, its exactly the same as scaling Roaring Thunder or Zeroth Longbow
True, however unlike Roaring Thunder and Zeroth Longbow, CSK was shown to boost Lucy several tiers beyond her norm, allowing her to simultaneously fight Jackal and Mard Geer 2v1. Defeating the former and getting the better of the latter before her MP ran out. And we saw how weak she was against Jackal alone even with Wendy on her side in their previous fight, the increase in power is colossal.
As for how the MP increase works, i view it as multiplicative. If Lucy's strength doubles or triples, then so too does her spirit's, pretty much like any other spell.
Also, if CSK is ever summoned again(and there is a good chance he wont be), he will have to be top tier, and i mean top top, not Natsu top. Enough to at least hold off Zeref/Acno. Cause right now there is no one left to fight. If Lucy is sacrificing a key sometime this arc(prob Loke), it will definitely mean something. A colossal amount of speculation on my part i know, but if CSK appears again imo its a safe bet that he'll be one of the strongest characters in the manga.
Speculation works both ways. There's an equal likelihood that the Spirit King can simply get oneshotted to simply further cement Acnologia and Zeref's hype, and that something else will be used to actually defeat them. Anna Heartfilia was the strongest Celestial Mage to have ever lived, and even she admitted that there was nothing that could be done when she had all 12 golden keys that could defeat Acnologia. Zeref should also know about the Spirit King and know what Celestial Spirit Magic is, and he still opted to go for Fairy Heart against Acnologia. There's a fair possibility that Zeref and Acnologia will tear each other apart when it comes to the actual fight, and that the good guys can just finish them off with a plot device like pushing them into the time rift.
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