The 4D sphere makes sense here because its surface is 3-dimensional. That means I can project the model from 4D sphere back to 3D in a bijective (one-to-one) way.

You could project from 5D down to 3D, but the dimensional mismatch breaks the bijection - you'd lose information or overlap points. However, a 4D → 5D → 4D projection would preserve structure, though it gets harder to visualize.

I chose 3D ↔ 4D specifically because curved 3D space is much more intuitive and has direct physical meaning - it corresponds to positively curved space (see e.g. https://en.wikipedia.org/wiki/Shape_of_the_universe#Universe... )