> So I guess the 126-dimensional shape actually also is in 127-dimensional space then
Sometimes you need more dimensions to embed the manifold. For a 2-dimencional object, the most famous example is the Klein bottle https://en.wikipedia.org/wiki/Klein_bottle You can construct one of them in 3-dimmension only if you cheat. Yhey look nice and you can buy a few cheating-versions. But you can embed the Klein bottle in 4-dimensions (without cheating).
For the manifold in the article, I'm not sure how many additional dimensions you need. Perhaps 127 (n+1) is enough or perhaps you need 252 (2n) or perhaps something in between. You can always embed an n-dimensional manifold in the 2n space, but that is the worst case. https://en.wikipedia.org/wiki/Whitney_embedding_theorem
Sometimes you need more dimensions to embed the manifold. For a 2-dimencional object, the most famous example is the Klein bottle https://en.wikipedia.org/wiki/Klein_bottle You can construct one of them in 3-dimmension only if you cheat. Yhey look nice and you can buy a few cheating-versions. But you can embed the Klein bottle in 4-dimensions (without cheating).
For the manifold in the article, I'm not sure how many additional dimensions you need. Perhaps 127 (n+1) is enough or perhaps you need 252 (2n) or perhaps something in between. You can always embed an n-dimensional manifold in the 2n space, but that is the worst case. https://en.wikipedia.org/wiki/Whitney_embedding_theorem