You’re confusing information with sequence.
You can always break up any sequence of symbols into smaller fragments and rearrange them, producing a new sequence. But then someone like you can come along and declare that the sequence isn’t new, it’s just rearranged fragments of already existing sequence
I could rearrange
CACACAGAGAGA
into
GAGAGACACACA
And you’d say there’s no new information, because it’s just the first sequence broken in half and the latter half put before the former.
But we could do that again, beak it up into smaller bits like CA and GA, and rearrange them.
GACAGACAGACA
And you could still say no new information, because it’s still just rearranged already existing seqeunce. All the CAs and GAs were all there to begin with.
And we could do it again, break it up into individual letters A, G, and C.
CCCAAAGGGAAA
And you could still say no new information, because it’s still just rearranged already existing sequence. All those As, Gs, and Cs were there to begin wtih.
Which reveals the absurdity of what you’re saying. So no, rearrangement really is new information. That is the only sensible position to take.
The problem is that you haven’t defined what constitutes “new” information to begin with, that means you can just make up the rules as you go along and keep denying that any example we come up with counts as new information. And the one you seem to be working from here is absurd.