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"You're on a path. At the end of that path, there is a forest. Hiding in that forest, there is a Wanderer. You are here to kill her. If you don't, it will be the end of the world."

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"You're on a path. At the end of that path, there is a forest. Hiding in that forest, there is a Wanderer. You are here to kill her. If you don't, it will be the end of the world."
The Sunken King remains nameless
So, I wanted to actually try and find the Japanese version of the name Gryth, which was supposedly used in a cut item description of the Rotten's Soul prior to the release of Scholar of the First Sin.
I looked far and wide, but I could not find a single mention of it anywhere, except on a Japanese site/blog discussing the Four Old Ones and their connections to the Four Lords of Dark Souls 1, while awaiting the release of Dark Souls 3. In the section dedicated to the Rotten, this person has a crossed-out paragraph detailing the connection between it and the King of Shulva Gryth, which they wrote as "グリス" (Gurisu). So, hurrah, problem solved, right? Not quite, because why was the paragraph crossed out? I found my answer immediately after, in parentheses.
The person apologized for getting this wrong and taking the info at face value, citing that it was just a piece of misinformation. They linked to the source that confirmed it as such, and here it is:
Sooooo, guess Gryth is a complete fabrication then... except it's apparently not that easy? Zullie added another tweet under this one, mentioning that the name didn't come out of nowhere but was instead taken from a customer support article by Bandai Namco itself, detailing how to get to the Sunken King DLC. In that article, the boss of the Black Gulch was given the name "Gryth the Rotten" for seemingly no reason at all.
I really have to wonder where the name came from. Did Bamco just hallucinate it or something? It was revealed to them in a dream? What's the deal with this? It's hard to truly verify it since these posts do not exist anymore and were not archived, but I'm inclined to believe Zullie's not lying on this. Why do that when she is already coming clean about the fake descriptions?
But yeah, I suppose this means there's no real tie between the Rotten and the Sunken King, and that the latter still doesn't have a name. Makes sense since that would make him an outlier between his kingly nameless peers... but the name Gryth could still potentially be used? I mean, it did appear SOMEWHERE "official". Or even just as a little reference to this whole fiasco. It's fine as long as one's aware that the name and the lore that ties it to the Rotten are pretty much fake lol.
Artist
The Rotten - Dark Souls 2
The most hear-me-out to ever hear-me-out
I feel like generally coexisting in the same location as The Rotten is very trying. He doesn't actively chase anyone who falls into the Gutter, of course, but it is endless psychological pressure nonetheless. His powerful soul beckons, and the closer someone is to him the stronger it gets.
After all, people (that aren't just corpses) are in the Gutter for a reason... They were unwanted and thrown away by humanity. You have to choose to struggle instead of seeking his "embrace" that can end the pain, all the time. And it feels like something always knows why you're here. It is like telepathy.