I think GP is saying that GGP assumes "tone" is one direction, in the sense there exists a vector V representing "tone direction", and you can scale "tone" independently by multiplying that vector with a scalar - hence, 1 dimension.
I'd say this assumption is both right and wrong. Wrong, because it's unlikely there's a direction in embedding space corresponding to a platonic ideal of "tone". Right, because I suspect that, for sufficiently large embedding space (on the order of what goes into current LLMs), any continuous concept we can articulate will have a corresponding direction in the embedding space, that's roughly as sharp as our ability to precisely define the concept.
I'd say this assumption is both right and wrong. Wrong, because it's unlikely there's a direction in embedding space corresponding to a platonic ideal of "tone". Right, because I suspect that, for sufficiently large embedding space (on the order of what goes into current LLMs), any continuous concept we can articulate will have a corresponding direction in the embedding space, that's roughly as sharp as our ability to precisely define the concept.