MAR1D: First-Person Mario

Correct. That's why this game is 3D and not 1D.

how is this 1d when there is a second dimension albeit small?

Yes, it wouldn't be 3D, but what you are alluding to is projection (X onto Y, or more generally instead of X which would be orthogonal only, a 1D plane of any orientation - AKA line segment) ultimately you are still looking at a 1D image, but it has a 2D shape projected onto it that changes depending on both your orientation and position, so that you can perceive part of the 2D shape's surface as you move.

And this would be more physically correct than either Mario or Mar1D.

The generalisation, starting in 3D, is that you can project the surface of a 3D object onto a 2D plane (which is how we see)... we get a little bit more by gauging depth through stereo separation but ultimately we only get to see one 3D projection onto a 2D plane at any point in time, i.e the entire 3D surface is not accessible, we can't see behind it, or inside it, and the perception of 3D is constructed in our mind from a combination of general learned/evolved intuition of 3D space and shape and temporal samples for a particular scenario e.g looking around the object from different angles.

Projection can extend to higher and lower dimensions, e.g in 4D space the surface of a 4D object can be projected onto a 3D "plane"; and as you are suggesting, for 2D space you can project a 2D object onto a 1D plane. The projection changing depending on the orientation and position of either the shape or eye/camera.

Normal 2D game rendering doesn't really make any physical sense as a projection unless you consider them to be a narrow 3D world (consider the fact that you can see the entire surface of a square and inside of the square, but Mario the character cannot possibly see this, only a small part of the surface).

But this is all based on a "projection" with the assumption of a lot of opacity... if each particle received from the projection also contained accurate enough depth information and was a vector of all depths traversed for some limit, then full 3D could be perceived in a single 2D projection within a 3D world i.e if you could see "through" objects while still being able to sample each depth.

I think it would require one eye above the other to have depth perception in his 2d world.

Eyes can’t even be side-by-side as that axis doesn’t exist.

Having said that, the human brain is capable of reconstructing a 3d perception with only one eye through learnt understand and interpreting the picture change over time as you move. I image it’s the same for Mario, but in 2d.

Ha, that's more or less what I was hoping the link would be.